A geometrical approach to conjugate point classification for linear differential equations

Alexandru Nemeth
(original), Nonlinear analysis, ODEs (theory), paper

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A.B. Nemeth
Institutul de Calcul

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A.B. Németh, A geometrical approach to conjugate point classification for linear differential equations. Rev. Anal. Numér. Théor. Approx. 4(1975), no. 2, 137–152 (1976).

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Revue d’Analyse Numerique et de Theorie de l’Approximation

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Romanian Academy

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[9] Karlin, Samuel, Studden, William J., Tchebycheff systems: With applications in analysis and statistics. Pure and Applied Mathematics, Vol. XV Interscience Publishers John Wiley & Sons, New York-London-Sydney 1966 xviii+586 pp., MR0204922.

[10] \cyr Kreĭn, M. G.; \cyr Nudel’man, A. A. \cyr Problema momentov Markova i èkstremal’nye zadachi. (Russian) [The Markov moment problems, and extremal problems] \cyr Idei i problemy P. L. Chebysheva i A. A. Markova i ikh dal’neĭshee razvitie. [The ideas and problems of P. L. Čebyšev and A. A. Markov, and their further development] Izdat. “Nauka”, Moscow, 1973. 551 pp., MR0445244.

[11] A. Yu, Levin, Neoseilliacia rešnia uravnenia x⁽ⁿ⁾+p₁(t)x⁽ⁿ⁻¹⁾+⋯p_{n}(t)x=0 Uspehi Mat. Nauk 24, 43-96, 1969.

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[16] Pólya, G., On the mean-value theorem corresponding to a given linear homogeneous differential equation. Trans. Amer. Math. Soc. 24 (1922), no. 4, 312-324, MR1501228, https://doi.org/10.1090/s0002-9947-1922-1501228-5

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1975-Nemeth-A GEOMETRiCAL APPROACH TO CONJUGATE POINTCLASSIFICATION FOR LINEAR DIFFERENTIAL EQUATION

A GEOMETRICAL APPROACH TO CONJUGATE POINT CLASSIFICATION FOR LINEAR DIFFERENTIAL EQUATIONS

byA. B. NEMETH(Cluj-Napoca)

0. Definitions and results

If V V VVV is a vector space and v i ∈ V , i = 1 , … , m v i ∈ V , i = 1 , … , m v_(i)in V,i=1,dots,mv_{i} \in V, i=1, \ldots, mvi∈V,i=1,…,m, we shall denote by sp ( v 1 , … , v m ) v 1 , … , v m (v_(1),dots,v_(m))\left(v_{1}, \ldots, v_{m}\right)(v1,…,vm) the subspace of V V VVV spanned by the elements v 1 , … , v m v 1 , … , v m v_(1),dots,v_(m)v_{1}, \ldots, v_{m}v1,…,vm.
Denote by C n ( J ) C n ( J ) C^(n)(J)C^{n}(J)Cn(J) the vector space over R R R\mathbf{R}R of the real valued functions with continuous n n nnn-th derivatives on the (open, half closed or closed) interval J J JJJ of the real axis.
Definition 1. The n n nnn-dimensional linear subspace L n L n L_(n)L_{n}Ln in C ′ ′ ( J ) C ′ ′ ( J ) C^('')(J)C^{\prime \prime}(J)C′′(J) will be said to be a Chebyshev space (CSp) if any nonzero element in L n L n L_(n)L_{n}Ln has at most n − 1 n − 1 n-1n-1n−1 distinct zeros in J . A J . A J.AJ . AJ.A basis of a CSp is called a Chebyshev system (CS).
Definition 2. The n-dimensional linear subspace L n L n L_(n)L_{n}Ln in C n ( J ) C n ( J ) C^(n)(J)C^{n}(J)Cn(J) will be said to form an unrestricted Chebyshev space (UCSp) if any nonzero element of its has at most n − 1 n − 1 n-1n-1n−1 distinct zeros in J J JJJ counting multiplicities and a basis of L n L n L_(n)L_{n}Ln is called an unresiricted Chebyshev system (UCS).
Consider the differential equation
(1) x ( n ) + p 1 ( t ) x ( n − 1 ) + … + p n − 1 ( t ) x ′ + p n ( t ) x = 0 , (1) x ( n ) + p 1 ( t ) x ( n − 1 ) + … + p n − 1 ( t ) x ′ + p n ( t ) x = 0 , {:(1)x^((n))+p_(1)(t)x^((n-1))+dots+p_(n-1)(t)x^(')+p_(n)(t)x=0",":}\begin{equation*} x^{(n)}+p_{1}(t) x^{(n-1)}+\ldots+p_{n-1}(t) x^{\prime}+p_{n}(t) x=0, \tag{1} \end{equation*}(1)x(n)+p1(t)x(n−1)+…+pn−1(t)x′+pn(t)x=0,
where p i p i p_(i)p_{i}pi are continuous real valued functions defined on R R R\mathbf{R}R.
Definition 3. It is said that the points a , b ∈ R , a < b a , b ∈ R , a < b a,b inR,a < ba, b \in \mathbf{R}, a<ba,b∈R,a<b are conjugate for the differential equation (1), if the space L n L n L_(n)L_{n}Ln of solutions of (1) forms an UCSP of dimension n n nnn on [ a , b ) [ a , b ) [a,b)[a, b)[a,b), but this property fails on [ a , b ] [ a , b ] [a,b][a, b][a,b].
Remarks. (i) For a given differential equation (1), each point a ∈ R a ∈ R a inRa \in \mathbf{R}a∈R has a neighbourhood in which there are no conjugate points (see [5] p. 81, Proposition 1.).
(ii) If a a aaa and b b bbb are conjugate points for the differential equation (1), then the space L n L n L_(n)L_{n}Ln of solutions of it forms an UCSp also on the interval ( a , b a , b a,ba, ba,b ] (see [5], p. 102, Theorem 8).
In all which follows we shall consider that a = 0 a = 0 a=0a=0a=0 and b = 1 b = 1 b=1b=1b=1 are conjugate points for the differential equation (1).
From the disconjugacy theory for the linear differential equations it follows (see for ex. [5] p. 89, Proposition 5 and pp. 98-99) that the points 0 and 1 are conjugate points for the differential equation (1) if and only if there exists a solution of (1) with a zero of multiplicity ≧ n − k ≧ n − k >= n-k\geqq n-k≧n−k at 0 and a zero of multiplicity ≧ k ≧ k >= k\geqq k≧k at 1 for some k , 1 ≦ k ≦ n − 1 k , 1 ≦ k ≦ n − 1 k,1 <= k <= n-1k, 1 \leqq k \leqq n-1k,1≦k≦n−1, and there are no solutions with a similar property for two points a , b a , b a,ba, ba,b in [ 0 , 1 ) [ 0 , 1 ) [0,1)[0,1)[0,1) (or, by Remark (ii), for two points a , b ∈ ( 0 , 1 ] a , b ∈ ( 0 , 1 ] a,b in(0,1]a, b \in(0,1]a,b∈(0,1] ).
Definition 4. The conjugate points 0 and 1 for (1) are said to be of type k ( 1 ≦ k ≦ n − 1 ) k ( 1 ≦ k ≦ n − 1 ) k(1 <= k <= n-1)k(1 \leqq k \leqq n-1)k(1≦k≦n−1), if (1) has a solution with a zero of multiplicity ≧ k ≧ k >= k\geqq k≧k at 1 and a zero of multiplicity ≧ n − k ≧ n − k >= n-k\geqq n-k≧n−k at 0 , and there are no solution with a similar property for l < k l < k l < kl<kl<k.
THEOREM 1. Suppose that n ≧ 3 n ≧ 3 n >= 3n \geqq 3n≧3 and that 0 and 1 are conjugate points of the type k k kkk for the differential equation (1). Suppose that there exists a single solution (up to a multiplicative constant) with zero of multiplicity n − k − 1 n − k − 1 n-k-1n-k-1n−k−1 of 0 did (1) and definition can be (1) form a Chebyshev space on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], whose work extended with at most n − 3 n − 3 n-3n-3n−3 distinct points with the preserving of the property of L n L n L_(n)L_{n}Ln to be a Chebyshev space.
A geometrical version of a strengthened form of this theorem will be stated in the paragraph 4 of our paper.
In a recent note [14], starting from a result in [13] we have estabilished some properties of the space L n L n L_(n)L_{n}Ln of solutions of (1) on the closed interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], in the case when 0 and 1 are conjugate points for the differential equation (1). In [14] we have given between others a Chebyshev space of domain of definition he dimension the preserving of the can be extended exactly whil a som ( property to be a Chebyshev space). Our above Theore 1 extension of the Proposition 4 in [14] and is a contribution to the study of conjugate points, a study initiated by PH. HARTMAN [7] and A. YU. LEVIN [11]. The results in the mentioned papers concerns investigations about the analytical properties of the conjugate points [11], and a classification of the solutions in the neighbourhood of the conjugate points [7, 11], while our conjugate point classification is a contribution to the theory of Cheby-
shev spaces, which in the last time obtains an advance by the results of V. I. volikov [17, 18], s. karlin and w. studden [9], r. ZIEIKE [19, 20, 21], P. HADELER [8], YU. G. ABAKUMOV [1, 2], M. G. KREIN, and A. A. NUDEL'MAN [10] and the author [12, 13]. By the Theorem 1 and 2 in our paper becomes possible the constructions of some Chebyshev spaces defined on closed intervals and eventually a finite set of points outside this intervals, whose domain of definition can be extended with no point. This property implies other structural properties of the respective spaces This property implies other struction (see [12]). (From these properties it follows that the recent constructions of r. zielme; [21] furnish examples of Chebyshev spaces defined on closed and halfclosed intervals, whose domain of definition cannot be extended with any point.)
It remains open the problem if the extension of the domain of definition of the CSp-s in our theorems is evermore actually possible or not.
The method applied by us requires some results from the disconjugacy theory due to G. PÓLYA [16], PH. HARTMAN [6], O. ARAMA [4], A. YU. LEVIN [11], for which we have already used the reference monography of W. A. COPPEL [5]. But the essential step is the cal machinery which essentially is the same as that of f. NEUMAN [15] and yu. A. ABAKUMOV [ 1,2 ] and more concretly is the extension of our method in [13] to the differentiable case.

1. Geometrical auxiliaries

The geometrical method which we use is in fact the classical theory of the differentiable curves. We shall particularize in this paragraph some aspects of this theory for our special purposes.
  1. Let x 1 , … , x n x 1 , … , x n x_(1),dots,x_(n)x_{1}, \ldots, x_{n}x1,…,xn be elements of C n [ 0 , 1 ] C n [ 0 , 1 ] C^(n)[0,1]C^{n}[0,1]Cn[0,1]. Consider the mapping Φ : [ 0 , 1 ] → R n Φ : [ 0 , 1 ] → R n Phi:[0,1]rarrR^(n)\Phi:[0,1] \rightarrow \mathbf{R}^{n}Φ:[0,1]→Rn given by
(2) Φ ( t ) = ( x 1 ( t ) , … , x 1 ( t ) ) . (2) Φ ( t ) = x 1 ( t ) , … , x 1 ( t ) . {:(2)Phi(t)=(x_(1)(t),dots,x_(1)(t)).:}\begin{equation*} \Phi(t)=\left(x_{1}(t), \ldots, x_{1}(t)\right) . \tag{2} \end{equation*}(2)Φ(t)=(x1(t),…,x1(t)).
In all which follows we shall consider only the case when the Wronskian W ( x 1 , … , x n ; t ) W x 1 , … , x n ; t W(x_(1),dots,x_(n);t)W\left(x_{1}, \ldots, x_{n} ; t\right)W(x1,…,xn;t) is different from zero for any t t ttt in [0,1]. In this case Φ ( [ 0 , 1 ] ) Φ ( [ 0 , 1 ] ) Phi([0,1])\Phi([0,1])Φ([0,1]) will be the curve in R n R n R^(n)\mathbf{R}^{n}Rn given in the parametric form
(3) x i = x i ( t ) , i = 1 , … , n (3) x i = x i ( t ) , i = 1 , … , n {:(3)x^(i)=x_(i)(t)","i=1","dots","n:}\begin{equation*} x^{i}=x_{i}(t), i=1, \ldots, n \tag{3} \end{equation*}(3)xi=xi(t),i=1,…,n
This curve will be said to be the characteristic curve of the space sp ( x 1 , … , x n ) sp x 1 , … , x n sp(x_(1),dots,x_(n))\mathrm{sp}\left(x_{1}, \ldots, x_{n}\right)sp(x1,…,xn) in C n [ 0 , 1 ] C n [ 0 , 1 ] C^(n)[0,1]C^{n}[0,1]Cn[0,1]. It is obviously uniquelly determined up to a linear, nonsingular transformation.
Denote by R k R k R^(k)\mathbb{R}^{k}Rk a subspace of dimension k k kkk in R n R n R^(n)\mathbb{R}^{n}Rn. We say that the curve (3) (or Φ ( [ 0 , 1 ] ) Φ ( [ 0 , 1 ] ) Phi([0,1])\Phi([0,1])Φ([0,1]) ) has an intersection point of multiplicity l l lll with R k R k R^(k)\mathbf{R}^{k}Rk at the point Φ ( t 0 ) Φ t 0 Phi(t_(0))\Phi\left(t_{0}\right)Φ(t0), if
Φ ( j ) ( t 0 ) = ( x 1 ( j ) ( t 0 ) , … , x n ( j ) ( t 0 ) ) ∈ R k , j = 0 , … , l − 1 Φ ( j ) t 0 = x 1 ( j ) t 0 , … , x n ( j ) t 0 ∈ R k , j = 0 , … , l − 1 Phi^((j))(t_(0))=(x_(1)^((j))(t_(0)),dots,x_(n)^((j))(t_(0)))inR^(k),j=0,dots,l-1\Phi^{(j)}\left(t_{0}\right)=\left(x_{1}^{(j)}\left(t_{0}\right), \ldots, x_{n}^{(j)}\left(t_{0}\right)\right) \in \mathbf{R}^{k}, j=0, \ldots, l-1Φ(j)(t0)=(x1(j)(t0),…,xn(j)(t0))∈Rk,j=0,…,l−1
and
Φ ( l ) ( t 0 ) = ( x 1 ( l ) ( t 0 ) , … , x n ( l ) ( t 0 ) ) ≠ R k . Φ ( l ) t 0 = x 1 ( l ) t 0 , … , x n ( l ) t 0 ≠ R k . Phi^((l))(t_(0))=(x_(1)^((l))(t_(0)),dots,x_(n)^((l))(t_(0)))!=R^(k).\Phi^{(l)}\left(t_{0}\right)=\left(x_{1}^{(l)}\left(t_{0}\right), \ldots, x_{n}^{(l)}\left(t_{0}\right)\right) \neq \mathbf{R}^{k} .Φ(l)(t0)=(x1(l)(t0),…,xn(l)(t0))≠Rk.
If we denote by a i = ( a i 1 , … , a i n ) , i = 1 , … , n − k a i = a i 1 , … , a i n , i = 1 , … , n − k a_(i)=(a_(i)^(1),dots,a_(i)^(n)),i=1,dots,n-ka_{i}=\left(a_{i}^{1}, \ldots, a_{i}^{n}\right), i=1, \ldots, n-kai=(ai1,…,ain),i=1,…,n−k a basis of R n − k R n − k R^(n-k)\mathbf{R}^{n-k}Rn−k, the orthogonal complement of R k R k R^(k)\mathbf{R}^{k}Rk in R ′ ′ R ′ ′ R^('')\mathbf{R}^{\prime \prime}R′′, then the condition to have (3) a 11 a 11 a_(11)a_{11}a11 intersection point of multiplicity l l lll with R k R k R^(k)\mathbf{R}^{k}Rk at Φ ( t 0 ) Φ t 0 Phi(t_(0))\Phi\left(t_{0}\right)Φ(t0) may be interpreted analytically as follows:
The subspace of the space L n = sp ( x 1 … , x n ) L n = sp x 1 … , x n L_(n)=sp(x_(1)dots,x_(n))L_{n}=\operatorname{sp}\left(x_{1} \ldots, x_{n}\right)Ln=sp(x1…,xn) in C n [ 0 , 1 ] C n [ 0 , 1 ] C^(n)[0,1]C^{n}[0,1]Cn[0,1] spanned by the elements
(4) a i 1 x 1 + … + a i n x n , i = 1 , … , n − k (4) a i 1 x 1 + … + a i n x n , i = 1 , … , n − k {:(4)a_(i)^(1)x_(1)+dots+a_(i)^(n)x_(n)","quad i=1","dots","n-k:}\begin{equation*} a_{i}^{1} x_{1}+\ldots+a_{i}^{n} x_{n}, \quad i=1, \ldots, n-k \tag{4} \end{equation*}(4)ai1x1+…+ainxn,i=1,…,n−k
has the property that any its element has at t 0 t 0 t_(0)t_{0}t0 a zero of multiplicity at least k k kkk and it contains an element with zero of multiplicity at most k k kkk at t 0 t 0 t_(0)t_{0}t0.
From this in particular it follows that the space L n L n L_(n)L_{n}Ln is an UCSp on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] if and only if no subspace R n − 1 R n − 1 R^(n-1)\mathbf{R}^{n-1}Rn−1 in R n R n R^(n)\mathbf{R}^{n}Rn has with the curve Φ ( [ 0 , 1 ] ) Φ ( [ 0 , 1 ] ) Phi([0,1])\Phi([0,1])Φ([0,1]) more than n − 1 n − 1 n-1n-1n−1 intersection points, counting their multiplicities (see also [15]).
It follows also that the space of all elements in L n L n L_(n)L_{n}Ln which have a zero of multiplicity at least k k kkk at t 0 t 0 t_(0)t_{0}t0 is the space spanned by the elements (4), where a 1 , … , a n − k a 1 , … , a n − k a_(1),dots,a_(n-k)a_{1}, \ldots, a_{n-k}a1,…,an−k spans the space R n − k R n − k R^(n-k)\mathbf{R}^{n-k}Rn−k, the orthogonal complement of the space
R k = sp ( Φ ( t 0 ) , … , Φ ( k − 1 ) ( t 0 ) ) . R k = sp Φ t 0 , … , Φ ( k − 1 ) t 0 . R^(k)=sp(Phi(t_(0)),dots,Phi^((k-1))(t_(0))).\mathbf{R}^{k}=\operatorname{sp}\left(\Phi\left(t_{0}\right), \ldots, \Phi^{(k-1)}\left(t_{0}\right)\right) .Rk=sp(Φ(t0),…,Φ(k−1)(t0)).
Then if we want to determine the space of all elements in L n L n L_(n)L_{n}Ln which have zeros of multiplicity k i k i k_(i)k_{i}ki at t i , i = 1 , … , m t i , i = 1 , … , m t_(i),i=1,dots,mt_{i}, i=1, \ldots, mti,i=1,…,m, we have to consider the vectors Φ ( j ) ( t i ) , i = 1 , … , m , j = 0 , … , k i − 1 Φ ( j ) t i , i = 1 , … , m , j = 0 , … , k i − 1 Phi^((j))(t_(i)),i=1,dots,m,j=0,dots,k_(i)-1\Phi^{(j)}\left(t_{i}\right), i=1, \ldots, m, j=0, \ldots, k_{i}-1Φ(j)(ti),i=1,…,m,j=0,…,ki−1, the space R v R v R^(v)\mathbf{R}^{v}Rv spanned by them, the orthogonal complement R n − v R n − v R^(n-v)\mathbf{R}^{n-v}Rn−v of this space in R n R n R^(n)\mathbf{R}^{n}Rn, a basis a i = ( a i 1 , … , a i n ) a i = a i 1 , … , a i n a_(i)=(a_(i)^(1),dots,a_(i)^(n))a_{i}=\left(a_{i}^{1}, \ldots, a_{i}^{n}\right)ai=(ai1,…,ain), i = 1 , … , n − v i = 1 , … , n − v i=1,dots,n-vi=1, \ldots, n-vi=1,…,n−v of this space, and to consider the space spanned by the elements
(5) a i 1 x 1 + … + a i n x n , i = 1 , … , n − v . (5) a i 1 x 1 + … + a i n x n , i = 1 , … , n − v . {:(5)a_(i)^(1)x_(1)+dots+a_(i)^(n)x_(n)","quad i=1","dots","n-v.:}\begin{equation*} a_{i}^{1} x_{1}+\ldots+a_{i}^{n} x_{n}, \quad i=1, \ldots, n-v . \tag{5} \end{equation*}(5)ai1x1+…+ainxn,i=1,…,n−v.
  1. Let us consider the representation of the vector Φ ( t ) Φ ( t ) Phi(t)\Phi(t)Φ(t) in the form Φ ( t ) = Φ 1 ( t ) + Φ 2 ( t ) Φ ( t ) = Φ 1 ( t ) + Φ 2 ( t ) Phi(t)=Phi_(1)(t)+Phi_(2)(t)\Phi(t)=\Phi_{1}(t)+\Phi_{2}(t)Φ(t)=Φ1(t)+Φ2(t), where Φ 1 ( t ) ∈ R v Φ 1 ( t ) ∈ R v Phi_(1)(t)inR^(v)\Phi_{1}(t) \in \mathbf{R}^{v}Φ1(t)∈Rv and Φ 2 ( t ) ∈ R n − v Φ 2 ( t ) ∈ R n − v Phi_(2)(t)inR^(n-v)\Phi_{2}(t) \in \mathbf{R}^{n-v}Φ2(t)∈Rn−v. Then, if a i i = 1 , … , n − v a i i = 1 , … , n − v a_(i)i=1,dots,n-va_{i} i=1, \ldots, n-vaii=1,…,n−v are the vectors determined above, we have
    (6) a i 1 x 1 ( t ) + … + a i n x n ( t ) = ( a i , Φ ( t ) ) = ( a i , Φ 2 ( t ) ) , i = 1 , … n − v a i 1 x 1 ( t ) + … + a i n x n ( t ) = a i , Φ ( t ) = a i , Φ 2 ( t ) , i = 1 , … n − v quada_(i)^(1)x_(1)(t)+dots+a_(i)^(n)x_(n)(t)=(a_(i),Phi(t))=(a_(i),Phi_(2)(t)),i=1,dots n-v\quad a_{i}^{1} x_{1}(t)+\ldots+a_{i}^{n} x_{n}(t)=\left(a_{i}, \Phi(t)\right)=\left(a_{i}, \Phi_{2}(t)\right), i=1, \ldots n-vai1x1(t)+…+ainxn(t)=(ai,Φ(t))=(ai,Φ2(t)),i=1,…n−v,
Suppose now that a i , i = 1 , … , n − v a i , i = 1 , … , n − v a_(i),i=1,dots,n-va_{i}, i=1, \ldots, n-vai,i=1,…,n−v form an orthonormal basis in R n − v R n − v R^(n-v)\mathbf{R}^{n-v}Rn−v. After a rotation (and a respective change of the basis in L n L n L_(n)L_{n}Ln ) we may suppose that
(7) a i = ( δ i 1 , … , δ i ′ ′ ) , i = 1 , … , n − v , (7) a i = δ i 1 , … , δ i ′ ′ , i = 1 , … , n − v , {:(7)a_(i)=(delta_(i)^(1),dots,delta_(i)^(''))","quad i=1","dots","n-v",":}\begin{equation*} a_{i}=\left(\delta_{i}^{1}, \ldots, \delta_{i}^{\prime \prime}\right), \quad i=1, \ldots, n-v, \tag{7} \end{equation*}(7)ai=(δi1,…,δi′′),i=1,…,n−v,
where δ i j δ i j delta_(i)^(j)\delta_{i}^{j}δij is the Kronecker-symbol,
Let us denote
(8) x ~ i ( t ) = ( a i , Φ 2 ( t ) ) . (8) x ~ i ( t ) = a i , Φ 2 ( t ) . {:(8) tilde(x)_(i)(t)=(a_(i),Phi_(2)(t)).:}\begin{equation*} \tilde{x}_{i}(t)=\left(a_{i}, \Phi_{2}(t)\right) . \tag{8} \end{equation*}(8)x~i(t)=(ai,Φ2(t)).
In what follows we shall need the characteristic curve spanned by the functions (5). From (6) and our above assumptions (7) on the vectors a i a i a_(i)a_{i}ai, it follows that the characteristic curve Ψ ( [ 0 , 1 ] ) Ψ ( [ 0 , 1 ] ) Psi([0,1])\Psi([0,1])Ψ([0,1]) of these functions is the projection of the curve Φ ( [ 0 , 1 ] ) Φ ( [ 0 , 1 ] ) Phi([0,1])\Phi([0,1])Φ([0,1]) into R n − v R n − v R^(n-v)\mathbf{R}^{n-v}Rn−v.
3. We need also the following simple fact:
If the curve Φ ( [ 0 , 1 ] ) Φ ( [ 0 , 1 ] ) Phi([0,1])\Phi([0,1])Φ([0,1]) has with R k R k R^(k)\mathbf{R}^{k}Rk an intersection point of multiplicity l l lll at Φ ( t 0 ) Φ t 0 Phi(t_(0))\Phi\left(t_{0}\right)Φ(t0) and R s R s R^(s)\mathbf{R}^{s}Rs is a subspace of R k R k R^(k)\mathbf{R}^{k}Rk spanned by the vectors Φ ( t 0 ) Φ t 0 Phi(t_(0))\Phi\left(t_{0}\right)Φ(t0),
… , Φ ( s − 1 ) ( t 0 ) , s < l < k , R n − s … , Φ ( s − 1 ) t 0 , s < l < k , R n − s dots,Phi^((s-1))(t_(0)),s < l < k,R^(n-s)\ldots, \Phi^{(s-1)}\left(t_{0}\right), s<l<k, \mathbf{R}^{n-s}…,Φ(s−1)(t0),s<l<k,Rn−s is the orthogonal complement of R s R s R^(s)\mathbf{R}^{s}Rs in R n R n R^(n)\mathbf{R}^{n}Rn, p p ppp denotes the projection of R n R n R^(n)\mathbf{R}^{n}Rn onto R n − s , R k − s = p ( R ― k ) R n − s , R k − s = p R ¯ k R^(n-s),R^(k-s)=p( bar(R)^(k))\mathbf{R}^{n-s}, \mathbf{R}^{k-s}=p\left(\overline{\mathbf{R}}^{k}\right)Rn−s,Rk−s=p(R―k), then the tangent vector to the arc p ( Φ ( [ 0 , 1 ] ) ) p ( Φ ( [ 0 , 1 ] ) ) p(Phi([0,1]))p(\Phi([0,1]))p(Φ([0,1])) in the point 0 = p ( Φ ( t 0 ) ) 0 = p Φ t 0 0=p(Phi(t_(0)))0=p\left(\Phi\left(t_{0}\right)\right)0=p(Φ(t0)) is contained in R k − s R k − s R^(k-s)\mathbf{R}^{k-s}Rk−s.
To verify this we consider the Taylor formula for the vector function Φ ( t ) Φ ( t ) Phi(t)\Phi(t)Φ(t) in t 0 t 0 t_(0)t_{0}t0 until the term l − 1 l − 1 l-1l-1l−1 :
Φ ( t ) = Φ ( t 0 ) + t − t 0 1 ! Φ ′ ( t 0 ) + … + ( t − t 0 ) s − 1 ( s − 1 ) ! Φ ( s − 1 ) ( t 0 ) + + ( t − t 0 ) s s ! Φ ( s ) ( t 0 ) + … + ( t − t 0 ) l − 1 ( l − 1 ) ! Φ ( l − 1 ) ( t 0 ) + o 0 ( t − t 0 ) l − 1 Φ ( t ) = Φ t 0 + t − t 0 1 ! Φ ′ t 0 + … + t − t 0 s − 1 ( s − 1 ) ! Φ ( s − 1 ) t 0 + + t − t 0 s s ! Φ ( s ) t 0 + … + t − t 0 l − 1 ( l − 1 ) ! Φ ( l − 1 ) t 0 + o 0 t − t 0 l − 1 {:[Phi(t)=Phi(t_(0))+(t-t_(0))/(1!)Phi^(')(t_(0))+dots+((t-t_(0))^(s-1))/((s-1)!)Phi^((s-1))(t_(0))+],[+((t-t_(0))^(s))/(s!)Phi^((s))(t_(0))+dots+((t-t_(0))^(l-1))/((l-1)!)Phi^((l-1))(t_(0))+o_(0)(t-t_(0))^(l-1)]:}\begin{gathered} \Phi(t)=\Phi\left(t_{0}\right)+\frac{t-t_{0}}{1!} \Phi^{\prime}\left(t_{0}\right)+\ldots+\frac{\left(t-t_{0}\right)^{s-1}}{(s-1)!} \Phi^{(s-1)}\left(t_{0}\right)+ \\ +\frac{\left(t-t_{0}\right)^{s}}{s!} \Phi^{(s)}\left(t_{0}\right)+\ldots+\frac{\left(t-t_{0}\right)^{l-1}}{(l-1)!} \Phi^{(l-1)}\left(t_{0}\right)+o_{0}\left(t-t_{0}\right)^{l-1} \end{gathered}Φ(t)=Φ(t0)+t−t01!Φ′(t0)+…+(t−t0)s−1(s−1)!Φ(s−1)(t0)++(t−t0)ss!Φ(s)(t0)+…+(t−t0)l−1(l−1)!Φ(l−1)(t0)+o0(t−t0)l−1
where o 0 ( t − t 0 ) l − 1 o 0 t − t 0 l − 1 o_(0)(t-t_(0))^(l-1)o_{0}\left(t-t_{0}\right)^{l-1}o0(t−t0)l−1 denotes a vector with all the components functions of orders o ( t − t 0 ) l − 1 o t − t 0 l − 1 o(t-t_(0))^(l-1)o\left(t-t_{0}\right)^{l-1}o(t−t0)l−1. After the application of the projector p p ppp we get
p Φ ( t ) = ( t − t 0 ) s s ! p Φ ( s ) ( t 0 ) + … + ( t − t 0 ) l − 1 ( l − 1 ) ! p Φ ( l − 1 ) ( t 0 ) + p o 0 ( t − t 0 ) l − 1 p Φ ( t ) = t − t 0 s s ! p Φ ( s ) t 0 + … + t − t 0 l − 1 ( l − 1 ) ! p Φ ( l − 1 ) t 0 + p o 0 t − t 0 l − 1 p Phi(t)=((t-t_(0))^(s))/(s!)pPhi^((s))(t_(0))+dots+((t-t_(0))^(l-1))/((l-1)!)pPhi^((l-1))(t_(0))+po_(0)(t-t_(0))^(l-1)p \Phi(t)=\frac{\left(t-t_{0}\right)^{s}}{s!} p \Phi^{(s)}\left(t_{0}\right)+\ldots+\frac{\left(t-t_{0}\right)^{l-1}}{(l-1)!} p \Phi^{(l-1)}\left(t_{0}\right)+p o_{0}\left(t-t_{0}\right)^{l-1}pΦ(t)=(t−t0)ss!pΦ(s)(t0)+…+(t−t0)l−1(l−1)!pΦ(l−1)(t0)+po0(t−t0)l−1
From this formula it follows that the arc p ( Φ ( [ 0 , 1 ] ) ) p ( Φ ( [ 0 , 1 ] ) ) p(Phi([0,1]))p(\Phi([0,1]))p(Φ([0,1])) has at t = t 0 t = t 0 t=t_(0)t=t_{0}t=t0 (1n the point ()) a nonessential singular point. The tangent vector to this are in the point 0 is the first derivative vector which is different from zero, i.e., in our case will be the vector p Φ ( s ) ( t 0 ) p Φ ( s ) t 0 pPhi^((s))(t_(0))p \Phi^{(s)}\left(t_{0}\right)pΦ(s)(t0). Because Φ ( s ) ( t 0 ) ∈ R h Φ ( s ) t 0 ∈ R h Phi^((s))(t_(0))inR^(h)\Phi^{(s)}\left(t_{0}\right) \in \mathbf{R}^{h}Φ(s)(t0)∈Rh we have p Φ ( s ) ( t 0 ) ∈ p ( R k ) = R k − s p Φ ( s ) t 0 ∈ p R k = R k − s pPhi^((s))(t_(0))in p(R^(k))=R^(k-s)p \Phi^{(s)}\left(t_{0}\right) \in p\left(\mathbb{R}^{k}\right)=\mathbb{R}^{k-s}pΦ(s)(t0)∈p(Rk)=Rk−s. We observe also that p Φ ( s ) ( t 0 ) p Φ ( s ) t 0 pPhi^((s))(t_(0))p \Phi^{(s)}\left(t_{0}\right)pΦ(s)(t0) cannot be the zero vector.
4. We shall say that a sequence of subspaces in R n R n R^(n)\mathbf{R}^{n}Rn, of the dimension m m mmm-tends to a subspace R m R m R^(m)\mathbf{R}^{m}Rm of the same dimension, if there exist bases of each subspace in the sequence such that the sequence of the corresponding elements of the bases are tending to the elements of a basis in R m R m R^(m)\mathbf{R}^{m}Rm. The same terminology will be used, when the notion of the convergence of sequences is changed in the notion of convergence of functions. Using this terminology we have the assertion:
Let be t 0 , t 1 , … , t r , t i ≠ t j , i ≠ j , i , j = 1 , … , r , r < n − 1 t 0 , t 1 , … , t r , t i ≠ t j , i ≠ j , i , j = 1 , … , r , r < n − 1 t_(0),t_(1),dots,t_(r),t_(i)!=t_(j),i!=j,i,j=1,dots,r,r < n-1t_{0}, t_{1}, \ldots, t_{r}, t_{i} \neq t_{j}, i \neq j, i, j=1, \ldots, r, r<n-1t0,t1,…,tr,ti≠tj,i≠j,i,j=1,…,r,r<n−1 points in [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]. Then the space
sp ( Φ ( t 0 ) , Φ ( t 1 ) , … , Φ ( t r ) ) sp Φ t 0 , Φ t 1 , … , Φ t r sp(Phi(t_(0)),Phi(t_(1)),dots,Phi(t_(r)))\operatorname{sp}\left(\Phi\left(t_{0}\right), \Phi\left(t_{1}\right), \ldots, \Phi\left(t_{r}\right)\right)sp(Φ(t0),Φ(t1),…,Φ(tr))
is tending to the space
sp ( Φ ( t 0 ) , Φ ′ ( t 0 ) , … , Φ ( r ) ( t 0 ) ) sp Φ t 0 , Φ ′ t 0 , … , Φ ( r ) t 0 sp(Phi(t_(0)),Phi^(')(t_(0)),dots,Phi^((r))(t_(0)))\operatorname{sp}\left(\Phi\left(t_{0}\right), \Phi^{\prime}\left(t_{0}\right), \ldots, \Phi^{(r)}\left(t_{0}\right)\right)sp(Φ(t0),Φ′(t0),…,Φ(r)(t0))
as sup 1 ⩽ i ⩽ r | t i − t 0 | → 0 sup 1 ⩽ i ⩽ r   t i − t 0 → 0 s u p_(1 <= i <= r)|t_(i)-t_(0)|rarr0\sup _{1 \leqslant i \leqslant r}\left|t_{i}-t_{0}\right| \rightarrow 0sup1⩽i⩽r|ti−t0|→0.
For verification let us consider the Taylor formula for Φ ( t ) Φ ( t ) Phi(t)\Phi(t)Φ(t)
Φ ( t i ) − o i ( t i − t 0 ) r = Φ ( t 0 ) + t i − t 0 1 ! Φ ′ ( t 0 ) + … + ( t i − t 0 ) r r ! Φ ( r ) Φ t i − o i t i − t 0 r = Φ t 0 + t i − t 0 1 ! Φ ′ t 0 + … + t i − t 0 r r ! Φ ( r ) Phi(t_(i))-o_(i)(t_(i)-t_(0))^(r)=Phi(t_(0))+(t_(i)-t_(0))/(1!)Phi^(')(t_(0))+dots+((t_(i)-t_(0))^(r))/(r!)Phi(r)\Phi\left(t_{i}\right)-o_{i}\left(t_{i}-t_{0}\right)^{r}=\Phi\left(t_{0}\right)+\frac{t_{i}-t_{0}}{1!} \Phi^{\prime}\left(t_{0}\right)+\ldots+\frac{\left(t_{i}-t_{0}\right)^{r}}{r!} \Phi(r)Φ(ti)−oi(ti−t0)r=Φ(t0)+ti−t01!Φ′(t0)+…+(ti−t0)rr!Φ(r)
i = 1 , … , r i = 1 , … , r i=1,dots,ri=1, \ldots, ri=1,…,r, where o i ( t i − t 0 ) o i t i − t 0 o_(i)(t_(i)-t_(0))o_{i}\left(t_{i}-t_{0}\right)oi(ti−t0) denotes a vector with all the components functions of order o ( t i − t 0 ) r o t i − t 0 r o(t_(i)-t_(0))^(r)o\left(t_{i}-t_{0}\right)^{r}o(ti−t0)r. Considering Φ ( j ) ( t ) Φ ( j ) ( t ) Phi^((j))(t)\Phi^{(j)}(t)Φ(j)(t) column vectors, we have the identity
‖ Φ ( t 0 ) , Φ ( t 1 ) − o 1 ( t 1 − t 0 ) r , … , Φ ( t r ) − o r ( t r − t 0 ) r ‖ × Φ t 0 , Φ t 1 − o 1 t 1 − t 0 r , … , Φ t r − o r t r − t 0 r × ||Phi(t_(0)),Phi(t_(1))-o_(1)(t_(1)-t_(0))^(r),dots,Phi(t_(r))-o_(r)(t_(r)-t_(0))^(r)||xx\left\|\Phi\left(t_{0}\right), \Phi\left(t_{1}\right)-o_{1}\left(t_{1}-t_{0}\right)^{r}, \ldots, \Phi\left(t_{r}\right)-o_{r}\left(t_{r}-t_{0}\right)^{r}\right\| \times‖Φ(t0),Φ(t1)−o1(t1−t0)r,…,Φ(tr)−or(tr−t0)r‖×
× ‖ 1 1 1 … 1 0 t 1 − t 0 1 ∣ t 2 − t 0 1 ∣ … t r − t 0 1 ! ⋮ ⋮ ⋮ ⋮ 0 ( t 2 − t 0 ) r r ! ( t 2 − t 0 ) r r ∣ … ( t r − t 0 ) r r ! ‖ = = ‖ Φ ( t 0 ) , Φ ′ ( t 0 ) , … , Φ ( r ) ( t 0 ) ‖ . × 1 1 1 … 1 0 t 1 − t 0 1 ∣ t 2 − t 0 1 ∣ … t r − t 0 1 ! ⋮ ⋮ ⋮ ⋮ 0 t 2 − t 0 r r ! t 2 − t 0 r r ∣ … t r − t 0 r r ! = = Φ t 0 , Φ ′ t 0 , … , Φ ( r ) t 0 . {:[xx||[1,1,1,dots,1],[0,(t_(1)-t_(0))/(1∣),(t_(2)-t_(0))/(1∣),dots,(t_(r)-t_(0))/(1!)],[vdots,vdots,vdots,,vdots],[0,((t_(2)-t_(0))^(r))/(r!)((t_(2)-t_(0))^(r))/(r∣),dots,((t_(r)-t_(0))^(r))/(r!)]||=],[=||Phi(t_(0)),Phi^(')(t_(0)),dots,Phi^((r))(t_(0))||.]:}\begin{gathered} \times\left\|\begin{array}{ccccc} 1 & 1 & 1 & \ldots & 1 \\ 0 & \frac{t_{1}-t_{0}}{1 \mid} & \frac{t_{2}-t_{0}}{1 \mid} & \ldots & \frac{t_{r}-t_{0}}{1!} \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & \frac{\left(t_{2}-t_{0}\right)^{r}}{r!} \frac{\left(t_{2}-t_{0}\right)^{r}}{r \mid} & \ldots & \frac{\left(t_{r}-t_{0}\right)^{r}}{r!} \end{array}\right\|= \\ =\left\|\Phi\left(t_{0}\right), \Phi^{\prime}\left(t_{0}\right), \ldots, \Phi^{(r)}\left(t_{0}\right)\right\| . \end{gathered}×‖111…10t1−t01∣t2−t01∣…tr−t01!⋮⋮⋮⋮0(t2−t0)rr!(t2−t0)rr∣…(tr−t0)rr!‖==‖Φ(t0),Φ′(t0),…,Φ(r)(t0)‖.
This means that
sp ( Φ ( t 0 ) , Φ ( t 1 ) − o 1 ( t 1 − t 0 ) r , … , Φ ( t r ) − o r ( t r − t 0 ) r ) = = sp ( Φ ( t 0 ) , Φ ′ ( t 0 ) , … , Φ ( r ) ( t 0 ) ) sp Φ t 0 , Φ t 1 − o 1 t 1 − t 0 r , … , Φ t r − o r t r − t 0 r = = sp Φ t 0 , Φ ′ t 0 , … , Φ ( r ) t 0 {:[sp(Phi(t_(0)),Phi(t_(1))-o_(1)(t_(1)-t_(0))^(r),dots,Phi(t_(r))-o_(r)(t_(r)-t_(0))^(r))=],[=sp(Phi(t_(0)),Phi^(')(t_(0)),dots,Phi^((r))(t_(0)))]:}\begin{gathered} \operatorname{sp}\left(\Phi\left(t_{0}\right), \Phi\left(t_{1}\right)-o_{1}\left(t_{1}-t_{0}\right)^{r}, \ldots, \Phi\left(t_{r}\right)-o_{r}\left(t_{r}-t_{0}\right)^{r}\right)= \\ =\operatorname{sp}\left(\Phi\left(t_{0}\right), \Phi^{\prime}\left(t_{0}\right), \ldots, \Phi^{(r)}\left(t_{0}\right)\right) \end{gathered}sp(Φ(t0),Φ(t1)−o1(t1−t0)r,…,Φ(tr)−or(tr−t0)r)==sp(Φ(t0),Φ′(t0),…,Φ(r)(t0))
which proves our assertion.
5. Suppose that the subspaces R v n t R v n t R_(v)^(nt)\mathbf{R}_{v}^{n t}Rvnt tend for v → ∞ v → ∞ v rarr oov \rightarrow \inftyv→∞ to the subspace R 0 ′ ′ R 0 ′ ′ R_(0)^('')\mathbf{R}_{0}^{\prime \prime}R0′′ in the above sense, and denote by R v n − m , v = 0 , 1 , … R v n − m , v = 0 , 1 , … R_(v)^(n-m),v=0,1,dots\mathbf{R}_{v}^{n-m}, v=0,1, \ldotsRvn−m,v=0,1,… the respective orthogonal complements. Then R v μ − μ R v μ − μ R_(v)^(mu-mu)\mathbf{R}_{v}{ }^{\mu-\mu}Rvμ−μ tends to R 0 n − μ R 0 n − μ R_(0)^(n-mu)\mathbf{R}_{0}^{n-\mu}R0n−μ. If p ν p ν p_(nu)p_{\nu}pν denotes the orthogonal projection onto R v n − m R v n − m R_(v)^(n-m)\mathbf{R}_{v}^{n-m}Rvn−m, then for any a ∈ R n a ∈ R n a inR^(n)a \in \mathbf{R}^{n}a∈Rn we have p v a → p 0 a p v a → p 0 a p_(v)a rarrp_(0)ap_{v} a \rightarrow p_{0} apva→p0a for v → ∞ v → ∞ v rarr oov \rightarrow \inftyv→∞.
Let be R v m = sp ( a v 1 , … , a v m ) , v = 0 , 1 , … R v m = sp a v 1 , … , a v m , v = 0 , 1 , … R_(v)^(m)=sp(a_(v1),dots,a_(vm)),v=0,1,dots\mathbb{R}_{v}^{m}=\operatorname{sp}\left(a_{v 1}, \ldots, a_{v m}\right), v=0,1, \ldotsRvm=sp(av1,…,avm),v=0,1,…, and a v i → a 0 i a v i → a 0 i a_(vi)rarra_(0i)a_{v i} \rightarrow a_{0 i}avi→a0i for v → ∞ v → ∞ v rarr oov \rightarrow \inftyv→∞, i = 1 , … , m i = 1 , … , m i=1,dots,mi=1, \ldots, mi=1,…,m. Suppose that a m + 1 , … , a n a m + 1 , … , a n a_(m+1),dots,a_(n)a_{m+1}, \ldots, a_{n}am+1,…,an are vectors in R n R n R^(n)\mathbf{R}^{n}Rn such that R n = sp ( a 01 , … , a 0 m , a m : 1 , … , a n ) R n = sp a 01 , … , a 0 m , a m : 1 , … , a n R^(n)=sp(a_(01),dots,a_(0m),a_(m:1),dots,a_(n))\mathbf{R}^{n}=\operatorname{sp}\left(a_{01}, \ldots, a_{0 m}, a_{m: 1}, \ldots, a_{n}\right)Rn=sp(a01,…,a0m,am:1,…,an). Then for sufficiently great v v vvv we have also
(9)
H n = sp ( a v 1 , … , a v m , a m − 1 , … , a n ) H n = sp a v 1 , … , a v m , a m − 1 , … , a n H^(n)=sp(a_(v1),dots,a_(vm),a_(m-1),dots,a_(n))\mathbf{H}^{n}=\operatorname{sp}\left(a_{v 1}, \ldots, a_{v m}, a_{m-1}, \ldots, a_{n}\right)Hn=sp(av1,…,avm,am−1,…,an)
Denote by p ν p ν p^(nu)p^{\nu}pν the orthogonal projection onto R ν m , ν = 0 , 1 , … R ν m , ν = 0 , 1 , … R_(nu)^(m),nu=0,1,dots\mathbf{R}_{\nu}^{m}, \nu=0,1, \ldotsRνm,ν=0,1,…. Then p ν a → p 0 a p ν a → p 0 a p^(nu)a rarrp_(0)ap^{\nu} a \rightarrow p_{0} apνa→p0a, if v → ∞ v → ∞ v rarr oov \rightarrow \inftyv→∞. We have
(10) i d R n = p v + p v , v = 0 , 1 , … (10) i d R n = p v + p v , v = 0 , 1 , … {:(10)id_(R^(n))=p^(v)+p_(v)","quad v=0","1","dots:}\begin{equation*} i d_{\mathbf{R}^{n}}=p^{v}+p_{v}, \quad v=0,1, \ldots \tag{10} \end{equation*}(10)idRn=pv+pv,v=0,1,…
from which it follows that
(11) p v a → p 0 a for v → ∞ (11) p v a → p 0 a  for  v → ∞ {:(11)p_(v)a rarrp_(0)a" for "v rarr oo:}\begin{equation*} p_{v} a \rightarrow p_{0} a \text { for } v \rightarrow \infty \tag{11} \end{equation*}(11)pva→p0a for v→∞
For v v vvv sfficiently great p v a i , i = m + 1 , … , n p v a i , i = m + 1 , … , n p_(v)a_(i),i=m+1,dots,np_{v} a_{i}, i=m+1, \ldots, npvai,i=m+1,…,n will be a basis of R v n − m R v n − m R_(v)^(n-m)\mathbf{R}_{v}^{n-m}Rvn−m according (9) and (10), which, together with (11) proves that R v n − m R v n − m R_(v)^(n-m)\mathbf{R}_{v}^{n-m}Rvn−m tends to R 0 n − m R 0 n − m R_(0)^(n-m)\mathbf{R}_{0}^{n-m}R0n−m as ν → ∞ ν → ∞ nu rarr oo\nu \rightarrow \inftyν→∞.

2. Two dimensional Chebyshev spaces with special properties

Consider the functions x 1 x 1 x_(1)x_{1}x1 and x 2 x 2 x_(2)x_{2}x2 in C 2 [ 0 , 1 ] C 2 [ 0 , 1 ] C^(2)[0,1]C^{2}[0,1]C2[0,1]. and
  1. Suppose that L 2 = sp ( x 1 , x 2 ) L 2 = sp x 1 , x 2 L_(2)=sp(x_(1),x_(2))L_{2}=\mathrm{sp}\left(x_{1}, x_{2}\right)L2=sp(x1,x2) is a CSp of dimension 2 on ( 0 , 1 ] ( 0 , 1 ] (0,1](0,1](0,1],
    (i) x 1 ( 0 ) = x 2 ( 0 ) = 0 x 1 ( 0 ) = x 2 ( 0 ) = 0 x_(1)(0)=x_(2)(0)=0x_{1}(0)=x_{2}(0)=0x1(0)=x2(0)=0;
    (ii) the tangent line in the point 0 to the characteristic curve of L 2 L 2 L_(2)L_{2}L2 coincides with the axis 0 x 2 0 x 2 0x^(2)0 x^{2}0x2;
    (iii) x 2 ( 1 ) ≠ 0 , x 1 ( 1 ) = 0 x 2 ( 1 ) ≠ 0 , x 1 ( 1 ) = 0 x_(2)(1)!=0,x_(1)(1)=0x_{2}(1) \neq 0, x_{1}(1)=0x2(1)≠0,x1(1)=0, i.e., the characteristic curve of L 2 L 2 L_(2)L_{2}L2 meets 0 x 2 0 x 2 0x^(2)0 x^{2}0x2 for t = 1 t = 1 t=1t=1t=1.
A CSp with the properties (i), (ii) and (iii) above has the property that its domain cannot be extended with any point.
Suppose that x 2 ( 1 ) > 0 x 2 ( 1 ) > 0 x_(2)(1) > 0x_{2}(1)>0x2(1)>0 and x 1 ( t ) > 0 x 1 ( t ) > 0 x_(1)(t) > 0x_{1}(t)>0x1(t)>0 for t ∈ ( 0 , 1 ) t ∈ ( 0 , 1 ) t in(0,1)t \in(0,1)t∈(0,1). Consider the function φ ( t ) = arctan ⁡ x 2 ( t ) / x 1 ( t ) φ ( t ) = arctan ⁡ x 2 ( t ) / x 1 ( t ) varphi(t)=arctan x_(2)(t)//x_(1)(t)\varphi(t)=\arctan x_{2}(t) / x_{1}(t)φ(t)=arctan⁡x2(t)/x1(t). Then we have
(12) φ ( 1 ) = π / 2 and φ ( 0 ) = lim t → 0 φ ( t ) = − π / 2 (12) φ ( 1 ) = π / 2  and  φ ( 0 ) = lim t → 0   φ ( t ) = − π / 2 {:(12)varphi(1)=pi//2" and "varphi(0)=lim_(t rarr0)varphi(t)=-pi//2:}\begin{equation*} \varphi(1)=\pi / 2 \text { and } \varphi(0)=\lim _{t \rightarrow 0} \varphi(t)=-\pi / 2 \tag{12} \end{equation*}(12)φ(1)=π/2 and φ(0)=limt→0φ(t)=−π/2
The first relation in (12) is obvious. To prove the second, we observe that by (ii) only the cases φ ( 0 ) = ± π / 2 φ ( 0 ) = ± π / 2 varphi(0)=+-pi//2\varphi(0)= \pm \pi / 2φ(0)=±π/2 are possible. From x 1 ( t ) ≧ 0 x 1 ( t ) ≧ 0 x_(1)(t) >= 0x_{1}(t) \geqq 0x1(t)≧0 it follows also that − π / 2 ≦ φ ( t ) ≦ π / 2 − π / 2 ≦ φ ( t ) ≦ π / 2 -pi//2 <= varphi(t) <= pi//2-\pi / 2 \leqq \varphi(t) \leqq \pi / 2−π/2≦φ(t)≦π/2 for any t t ttt in [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]. If φ ( 0 ) = π / 2 φ ( 0 ) = π / 2 varphi(0)=pi//2\varphi(0)=\pi / 2φ(0)=π/2, suppose that t 0 t 0 t_(0)t_{0}t0 is the minimum point for φ φ varphi\varphiφ. We have − π / 2 < φ ( t 0 ) < π / 2 − π / 2 < φ t 0 < π / 2 -pi//2 < varphi(t_(0)) < pi//2-\pi / 2<\varphi\left(t_{0}\right)<\pi / 2−π/2<φ(t0)<π/2, because in the case of φ ( t 0 ) = ± π / 2 φ t 0 = ± π / 2 varphi(t_(0))=+-pi//2\varphi\left(t_{0}\right)= \pm \pi / 2φ(t0)=±π/2 it would follow that the vectors ( x 1 ( 1 ) , x 2 ( 1 ) x 1 ( 1 ) , x 2 ( 1 ) x_(1)(1),x_(2)(1)x_{1}(1), x_{2}(1)x1(1),x2(1) ) and ( x 1 ( t 0 ) , x 2 ( t 0 ) x 1 t 0 , x 2 t 0 x_(1)(t_(0)),x_(2)(t_(0))x_{1}\left(t_{0}\right), x_{2}\left(t_{0}\right)x1(t0),x2(t0) ) are colinear, which contradicts the fact that x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2 form a CS on ( 0,1 ]. From the continuity of φ ( t ) φ ( t ) varphi(t)\varphi(t)φ(t) it follows that for any t 1 ∈ ( 0 , t 0 ) t 1 ∈ 0 , t 0 t_(1)in(0,t_(0))t_{1} \in\left(0, t_{0}\right)t1∈(0,t0) there exists a t 2 ∈ [ t 0 , 1 ) t 2 ∈ t 0 , 1 t_(2)in[t_(0),1)t_{2} \in\left[t_{0}, 1\right)t2∈[t0,1) such that φ ( t 1 ) = φ ( t 2 ) φ t 1 = φ t 2 varphi(t_(1))=varphi(t_(2))\varphi\left(t_{1}\right)=\varphi\left(t_{2}\right)φ(t1)=φ(t2). This means that the vectors ( x 1 ( t 1 ) , x 2 ( t 1 ) x 1 t 1 , x 2 t 1 x_(1)(t_(1)),x_(2)(t_(1))x_{1}\left(t_{1}\right), x_{2}\left(t_{1}\right)x1(t1),x2(t1) ) and ( x 1 ( t 2 ) , x 2 ( t 2 ) x 1 t 2 , x 2 t 2 x_(1)(t_(2)),x_(2)(t_(2))x_{1}\left(t_{2}\right), x_{2}\left(t_{2}\right)x1(t2),x2(t2) ) are colinear for t 1 ≠ t 2 , t 1 , t 2 ∈ ( 0 , 1 ] t 1 ≠ t 2 , t 1 , t 2 ∈ ( 0 , 1 ] t_(1)!=t_(2),t_(1),t_(2)in(0,1]t_{1} \neq t_{2}, t_{1}, t_{2} \in(0,1]t1≠t2,t1,t2∈(0,1], which is a contradiction (see Fig. 1. a). This proves the second relation in (12), i.e., the characteristic curve has the form b b bbb in Fig. 1.
Fig. 1
  1. Suppose that L 2 = sp ( x 1 , x 2 ) L 2 = sp x 1 , x 2 L_(2)=sp(x_(1),x_(2))L_{2}=\mathrm{sp}\left(x_{1}, x_{2}\right)L2=sp(x1,x2) is a CSp on ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1) and
    (i) x 1 ( 0 ) = x 2 ( 0 ) = x 1 ( 1 ) = x 2 ( 1 ) = 0 x 1 ( 0 ) = x 2 ( 0 ) = x 1 ( 1 ) = x 2 ( 1 ) = 0 x_(1)(0)=x_(2)(0)=x_(1)(1)=x_(2)(1)=0x_{1}(0)=x_{2}(0)=x_{1}(1)=x_{2}(1)=0x1(0)=x2(0)=x1(1)=x2(1)=0;
    (ii) the tangent lines for the characteristic curve for t = 0 t = 0 t=0t=0t=0 and t = 1 t = 1 t=1t=1t=1 coincide with 0 x 2 0 x 2 0x^(2)0 x^{2}0x2.
A CSp with the properties (i) and (ii) above has the property that its domain of definition can be extended with a single point α α alpha\alphaα and as extensions of x 1 x 1 x_(1)x_{1}x1 and x 2 x 2 x_(2)x_{2}x2 can be set x 1 ( α ) = 0 , x 2 ( α ) = 1 x 1 ( α ) = 0 , x 2 ( α ) = 1 x_(1)(alpha)=0,x_(2)(alpha)=1x_{1}(\alpha)=0, x_{2}(\alpha)=1x1(α)=0,x2(α)=1.
Suppose that x 1 ( t ′ ) = 0 x 1 t ′ = 0 x_(1)(t^('))=0x_{1}\left(t^{\prime}\right)=0x1(t′)=0 for some t ′ t ′ t^(')t^{\prime}t′ in ( 0,1 ) and that t ′ t ′ t^(')t^{\prime}t′ is the minimal value of t t ttt with this property. We have then x 2 ( t ′ ) ≠ 0 x 2 t ′ ≠ 0 x_(2)(t^('))!=0x_{2}\left(t^{\prime}\right) \neq 0x2(t′)≠0 and by 2.1 above
Fig. 2
φ ( t ) = arctan ⁡ x 2 ( t ) / x 1 ( t ) φ ( t ) = arctan ⁡ x 2 ( t ) / x 1 ( t ) varphi(t)=arctan x_(2)(t)//x_(1)(t)\varphi(t)=\arctan x_{2}(t) / x_{1}(t)φ(t)=arctan⁡x2(t)/x1(t)
is then well defined and continuous on ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1) and − π / 2 ≦ φ ( t ) ≦ π / 2 t ∈ ( 0 , 1 ) − π / 2 ≦ φ ( t ) ≦ π / 2 t ∈ ( 0 , 1 ) -pi//2 <= varphi(t) <= pi//2t in(0,1)-\pi / 2 \leqq \varphi(t) \leqq \pi / 2 t \in(0,1)−π/2≦φ(t)≦π/2t∈(0,1). By a similar argument as in 2.1 we deduce that
φ ( 0 ) = lim t → 0 φ ( t ) = − lim t → 1 φ ( t ) = − φ ( 1 ) = ± π / 2 . φ ( 0 ) = lim t → 0   φ ( t ) = − lim t → 1   φ ( t ) = − φ ( 1 ) = ± π / 2 . varphi(0)=lim_(t rarr0)varphi(t)=-lim_(t rarr1)varphi(t)=-varphi(1)=+-pi//2.\varphi(0)=\lim _{t \rightarrow 0} \varphi(t)=-\lim _{t \rightarrow 1} \varphi(t)=-\varphi(1)= \pm \pi / 2 .φ(0)=limt→0φ(t)=−limt→1φ(t)=−φ(1)=±π/2.
From the continuity of φ ( t ) φ ( t ) varphi(t)\varphi(t)φ(t) it follows that any straight line passing through the origin, except 0 x 2 0 x 2 0x^(2)0 x^{2}0x2 intersects the characteristic curve of L 2 L 2 L_(2)L_{2}L2 in a point (see Fig. 2). It follows also that we may extend the domain of definition of L 2 L 2 L_(2)L_{2}L2 setting for x 1 x 1 x_(1)x_{1}x1 and x 2 x 2 x_(2)x_{2}x2 in the point α ∉ [ 0 , 1 ] α ∉ [ 0 , 1 ] alpha!in[0,1]\alpha \notin[0,1]α∉[0,1] values such that ( x 1 ( α ) , x 2 ( α ) x 1 ( α ) , x 2 ( α ) x_(1)(alpha),x_(2)(alpha)x_{1}(\alpha), x_{2}(\alpha)x1(α),x2(α) ) be on 0 x 2 0 x 2 0x^(2)0 x^{2}0x2, and the domain of the CSp obtained in this form cannot be extended.
Since φ ( 0 ) = − π / 2 φ ( 0 ) = − π / 2 varphi(0)=-pi//2\varphi(0)=-\pi / 2φ(0)=−π/2, it follows that any straight line passing through the origin intersects the characteristic curve of L 2 L 2 L_(2)L_{2}L2 in a point.
Consider any extension of the domain of definition of L 2 L 2 L_(2)L_{2}L2 with a point, and the characteristic curve of the space with the extended domain. From the above conclusion it follows that there exists a line passing through the origin, which intersects the characteristic curve for two distinct values of t t ttt, i.e., the extended space cannot form a CSp. CSp on ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1) and
;
(ii) the tangent lines for the characteristic curve for t = 0 t = 0 t=0t=0t=0 and t = 1 t = 1 t=1t=1t=1 concide with 0 x 2 0 x 2 0x^(2)0 x^{2}0x2.

3. Preparatory lemmas

the differe equation (1). Suppoints of type k k kkk for with a zero of multiplicity i , i ≦ n − k i , i ≦ n − k i,i <= n-ki, i \leqq n-ki,i≦n−k at the point 0 and a zero of multiplicity j ≦ k j ≦ k j <= kj \leqq kj≦k at 1 , and has m m mmm distinct zeros at t 1 , … , t m ∈ ( 0 , 1 ) t 1 , … , t m ∈ ( 0 , 1 ) t_(1),dots,t_(m)in(0,1)t_{1}, \ldots, t_{m} \in(0,1)t1,…,tm∈(0,1), then there multiplicity j j jjj at 1 , which has x 2 x 2 x_(2)x_{2}x2 with zero of multiplicily at 0 and zero of its sign passing through these zeros.
Proof. By a result of A. YU. LEVIN (see [5], Proposition 11, p. 99), there exists an element x 0 x 0 x_(0)x_{0}x0 of 1 , which is positive in ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1). Suppose that t … , t q l t … , t q l t dots,t_(ql)t \ldots, t_{q l}t…,tql are the zeros at which x 1 x 1 x_(1)x_{1}x1 does not change the sign. Then there exist [ ( l + 1 ) / 2 ] [ ( l + 1 ) / 2 ] [(l+1)//2][(l+1) / 2][(l+1)/2] zeros 111 the neighbotthood of which x 1 x 1 x_(1)x_{1}x1 is of the same sufficiently small x x xxx will have m − l + 2 [ ( l + 1 ) / 2 ] ≥ m 0 m − l + 2 [ ( l + 1 ) / 2 ] ≥ m 0 m-l+2[(l+1)//2] >= m_(0)m-l+2[(l+1) / 2] \geq m_{0}m−l+2[(l+1)/2]≥m0 zeros in ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1) at which it changes the sign.
the vectors
(13) Φ ( 0 ) , … , Φ ( n − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 1 ) ( 1 ) (13) Φ ( 0 ) , … , Φ ( n − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 1 ) ( 1 ) {:(13)Phi(0)","dots","Phi^((n-k-1))(0)","Phi(1)","dots","Phi^((k-1))(1):}\begin{equation*} \Phi(0), \ldots, \Phi^{(n-k-1)}(0), \Phi(1), \ldots, \Phi^{(k-1)}(1) \tag{13} \end{equation*}(13)Φ(0),…,Φ(n−k−1)(0),Φ(1),…,Φ(k−1)(1)
are linearly dependent, while the vectors
(14) Φ ( 0 ) , … , Φ ( n − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 2 ) ( 1 ) (14) Φ ( 0 ) , … , Φ ( n − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 2 ) ( 1 ) {:(14)Phi(0)","dots","Phi^((n-k-1))(0)","Phi(1)","dots","Phi^((k-2))(1):}\begin{equation*} \Phi(0), \ldots, \Phi^{(n-k-1)}(0), \Phi(1), \ldots, \Phi^{(k-2)}(1) \tag{14} \end{equation*}(14)Φ(0),…,Φ(n−k−1)(0),Φ(1),…,Φ(k−2)(1)
are lineraly independent, where Φ ( j ) ( t ) = ( x 1 ( j ) ( t ) , … , x n ( j ) ( t ) ) , x 1 , … , x n Φ ( j ) ( t ) = x 1 ( j ) ( t ) , … , x n ( j ) ( t ) , x 1 , … , x n Phi^((j))(t)=(x_(1)^((j))(t),dots,x_(n)^((j))(t)),x_(1),dots,x_(n)\Phi^{(j)}(t)=\left(x_{1}^{(j)}(t), \ldots, x_{n}^{(j)}(t)\right), x_{1}, \ldots, x_{n}Φ(j)(t)=(x1(j)(t),…,xn(j)(t)),x1,…,xn being a fundamental system of solutions of (1).
Proof. From the definition of the conjugate points of type k k kkk, there exists an element x ≠ 0 x ≠ 0 x!=0x \neq 0x≠0 in L n L n L_(n)L_{n}Ln with zero of multiplicity ≧ n − k ≧ n − k >= n-k\geqq n-k≧n−k at 0 and with zero of multiplicity ≧ k ≧ k >= k\geqq k≧k at 1 and k k kkk is the minimal number for which there exists a such x x xxx.
Let be
(15) x = a 1 x 1 + ⋯ + a n x n (15) x = a 1 x 1 + ⋯ + a n x n {:(15)x=a^(1)x_(1)+cdots+a^(n)x_(n):}\begin{equation*} x=a^{1} x_{1}+\cdots+a^{n} x_{n} \tag{15} \end{equation*}(15)x=a1x1+⋯+anxn
By out geometrical interpretation (see 1.1) it follows that the vectors (13) are all orthogonal to a = ( a 1 , … , a n ) ≠ 0 a = a 1 , … , a n ≠ 0 a=(a^(1),dots,a^(n))!=0a=\left(a^{1}, \ldots, a^{n}\right) \neq 0a=(a1,…,an)≠0, which proves the first part of the lemma.
If the vectors (14) would be linearly dependent, then the system of vectors obtained adding to (14) the vector Φ ( n − k ) ( 0 ) Φ ( n − k ) ( 0 ) Phi^((n-k))(0)\Phi^{(n-k)}(0)Φ(n−k)(0) would be also linearly dependent and would have a span H H HHH, a space of dimension ≦ n − 1 ≦ n − 1 <= n-1\leqq n-1≦n−1. Let
a = ( a 1 , … , a n ) a = a 1 , … , a n a=(a^(1),dots,a^(n))a=\left(a^{1}, \ldots, a^{n}\right)a=(a1,…,an) be a nonzero vector which is orthogonal to H H HHH. Then the element x x xxx of the form (15) would have a zero of multiplicity ≧ n − k + 1 ≧ n − k + 1 >= n-k+1\geqq n-k+1≧n−k+1 at 0 and a zero of multiplicity k − 1 k − 1 k-1k-1k−1 at 1 , which is a contradiction.
L , e m m L , e m m L,emm\mathrm{L}, \mathrm{e} \mathrm{m} \mathrm{m}L,emm a 3. Suppose that 0 and 1 are conjugate points of type k k kkk for the differential equation (1). Then the set of functions x x xxx in L n L n L_(n)L_{n}Ln, the space of solutions of (1), with the property that
(16) x ( 0 ) = … = x ( i − 1 ) ( 0 ) = x ( 1 ) = … = x ( j − 1 ) ( 1 ) = 0 , (16) x ( 0 ) = … = x ( i − 1 ) ( 0 ) = x ( 1 ) = … = x ( j − 1 ) ( 1 ) = 0 , {:(16)x(0)=dots=x^((i-1))(0)=x(1)=dots=x^((j-1))(1)=0",":}\begin{equation*} x(0)=\ldots=x^{(i-1)}(0)=x(1)=\ldots=x^{(j-1)}(1)=0, \tag{16} \end{equation*}(16)x(0)=…=x(i−1)(0)=x(1)=…=x(j−1)(1)=0,
with i ≦ n − k , j < k i ≦ n − k , j < k i <= n-k,j < ki \leqq n-k, j<ki≦n−k,j<k form a CSp L n − i − j CSp L n − i − j CSpL^(n-i-j)\operatorname{CSp} L^{n-i-j}CSpLn−i−j of the dimension n − i − j n − i − j n-i-jn-i-jn−i−j on the interval ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1).
Proof. We prove the lemma by contradiction. Suppose that i , i ≦ n − k i , i ≦ n − k i,i <= n-ki, i \leqq n-ki,i≦n−k is the minimal number for which there exists a j < k j < k j < kj<kj<k and the distinct points t 1 , … , t n − i − j t 1 , … , t n − i − j t_(1),dots,t_(n-i-j)t_{1}, \ldots, t_{n-i-j}t1,…,tn−i−j in ( 0,1 ), such that there exists a nonzero solution x = a 1 x 1 + … + a n x n x = a 1 x 1 + … + a n x n x=a^(1)x_(1)+dots+a^(n)x_(n)x=a^{1} x_{1}+\ldots+a^{n} x_{n}x=a1x1+…+anxn, which has a zero of multiplicity i i iii at 0 , a zero of multiplicity j j jjj at 1 and zeros at the distinct points t 1 , … , t n − i − j t 1 , … , t n − i − j t_(1),dots,t_(n-i-j)t_{1}, \ldots, t_{n-i-j}t1,…,tn−i−j in ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1). By the Lemma 1 we may suppose that x x xxx changes the sign passing through t 1 , … , t n − i − j t 1 , … , t n − i − j t_(1),dots,t_(n-i-j)t_{1}, \ldots, t_{n-i-j}t1,…,tn−i−j. Geometrically the existence of an x x xxx with these properties (see 1.1) means that: the vectors
(17) Φ ( 0 ) , … , Φ ( i − 1 ) ( 0 ) (18) Φ ( 1 ) , … , Φ ( j − 1 ) ( 1 ) (19) Φ ( t 1 ) , … , Φ ( t n − i − j ) (17) Φ ( 0 ) , … , Φ ( i − 1 ) ( 0 ) (18) Φ ( 1 ) , … , Φ ( j − 1 ) ( 1 ) (19) Φ t 1 , … , Φ t n − i − j {:[(17)Phi(0)","dots","Phi^((i-1))(0)],[(18)Phi(1)","dots","Phi^((j-1))(1)],[(19)Phi(t_(1))","dots","Phi(t_(n-i-j))]:}\begin{align*} & \Phi(0), \ldots, \Phi^{(i-1)}(0) \tag{17}\\ & \Phi(1), \ldots, \Phi^{(j-1)}(1) \tag{18}\\ & \Phi\left(t_{1}\right), \ldots, \Phi\left(t_{n-i-j}\right) \tag{19} \end{align*}(17)Φ(0),…,Φ(i−1)(0)(18)Φ(1),…,Φ(j−1)(1)(19)Φ(t1),…,Φ(tn−i−j)
are linearly dependent and the curve Φ ( [ 0 , 1 ] ) Φ ( [ 0 , 1 ] ) Phi([0,1])\Phi([0,1])Φ([0,1]) is passing trough the hyperplane
(20) R n − 1 = { ξ ∈ R n : a 1 ξ 1 + … + a n ξ n = 0 } (20) R n − 1 = ξ ∈ R n : a 1 ξ 1 + … + a n ξ n = 0 {:(20)R^(n-1)={xi inR^(n):a^(1)xi^(1)+dots+a^(n)xi^(n)=0}:}\begin{equation*} \mathbf{R}^{n-1}=\left\{\xi \in \mathbf{R}^{n}: a^{1} \xi^{1}+\ldots+a^{n} \xi^{n}=0\right\} \tag{20} \end{equation*}(20)Rn−1={ξ∈Rn:a1ξ1+…+anξn=0}
at the points (19).
We shall prove that for any m , 1 ≦ m ≦ n − i − j m , 1 ≦ m ≦ n − i − j m,1 <= m <= n-i-jm, 1 \leqq m \leqq n-i-jm,1≦m≦n−i−j the system of the vectors (17), (18) and
(21) Φ ( t 1 ) , … , Φ ( t m ^ ) , … , Φ ( t n − i − j ) (21) Φ t 1 , … , Φ t m ^ , … , Φ t n − i − j {:(21)Phi(t_(1))","dots","Phi(( hat(t_(m))))","dots","Phi(t_(n-i-j)):}\begin{equation*} \Phi\left(t_{1}\right), \ldots, \Phi\left(\hat{t_{m}}\right), \ldots, \Phi\left(t_{n-i-j}\right) \tag{21} \end{equation*}(21)Φ(t1),…,Φ(tm^),…,Φ(tn−i−j)
(the symbol ∧ ∧ ^^\wedge∧ above a term of a sequence means that the respective term is omitted) cannot be linearly independent. Suppose the contrary. Then for t ′ t ′ t^(')t^{\prime}t′ close to 1 , t ′ ≠ t q , q = 1 , … , n − i − j 1 , t ′ ≠ t q , q = 1 , … , n − i − j 1,t^(')!=t_(q),q=1,dots,n-i-j1, t^{\prime} \neq t_{q}, q=1, \ldots, n-i-j1,t′≠tq,q=1,…,n−i−j the system of vectors
(22) Φ ( t ′ ) , … , Φ ( j − 1 ) ( t ′ ) (22) Φ t ′ , … , Φ ( j − 1 ) t ′ {:(22)Phi(t^('))","dots","Phi^((j-1))(t^(')):}\begin{equation*} \Phi\left(t^{\prime}\right), \ldots, \Phi^{(j-1)}\left(t^{\prime}\right) \tag{22} \end{equation*}(22)Φ(t′),…,Φ(j−1)(t′)
can be arbitrarily close to the system of vectors (18) and the hyperplane determined by (17), (22), (21) is arbitrarily close to the hyperplane (20)
(see 1.4 for this notion), which according our hypothesis is spanned by (17), (18) and (21). Because the hyperplane (20) has an intersection point with the are Φ ( [ 0 , 1 ] ) Φ ( [ 0 , 1 ] ) Phi([0,1])\Phi([0,1])Φ([0,1]) at Φ ( ℓ m ) Φ ℓ m Phi(ℓ_(m))\Phi\left(\ell_{m}\right)Φ(ℓm) and this are pass through (20) in this point, it follows that the hyperplane through the origin spanned by (17), (22), (21) will have, for t ′ t ′ t^(')t^{\prime}t′ sufficiently close to 1 , an intursection point Φ ( t ′ ′ ) Φ t ′ ′ Phi(t^(''))\Phi\left(t^{\prime \prime}\right)Φ(t′′) with the curve Φ ( [ 0 , 1 ] ) Φ ( [ 0 , 1 ] ) Phi([0,1])\Phi([0,1])Φ([0,1]) such that t ′ ′ ∈ ( 0 , 1 ) ∖ { t 1 , … , t ^ m , … , t n − i − j } t ′ ′ ∈ ( 0 , 1 ) ∖ t 1 , … , t ^ m , … , t n − i − j t^('')in(0,1)\\{t_(1),dots, hat(t)_(m),dots,t_(n-i-j)}t^{\prime \prime} \in(0,1) \backslash\left\{t_{1}, \ldots, \hat{t}_{m}, \ldots, t_{n-i-j}\right\}t′′∈(0,1)∖{t1,…,t^m,…,tn−i−j}. But this means that the element
x 1 = b 1 x 1 + … + b n x , x 1 = b 1 x 1 + … + b n x , x_(1)=b^(1)x_(1)+dots+b^(n)x,x_{1}=b^{1} x_{1}+\ldots+b^{n} x,x1=b1x1+…+bnx,
where b = ( b 1 , … , b n ) b = b 1 , … , b n b=(b^(1),dots,b^(n))b=\left(b^{1}, \ldots, b^{n}\right)b=(b1,…,bn) is a normal vector of the hyperplane containing the vectors (17), (22) and (21), will have a zero of the multiplicity i i iii at 0 , a zero of multiplicity j j jjj at t ′ t ′ t^(')t^{\prime}t′ and zeros at the distinct points t 1 , … , t m ^ , … … . t n − i − j , t ′ ′ t 1 , … , t m ^ , … … . t n − i − j , t ′ ′ t_(1),dots, widehat(t_(m)),dots dots.t_(n-i-j),t^('')t_{1}, \ldots, \widehat{t_{m}}, \ldots \ldots . t_{n-i-j}, t^{\prime \prime}t1,…,tm^,…….tn−i−j,t′′, that is, n n nnn zeros in [ 0,1 ), which contradicts the fact that 0 and 1 are conjugate points.
We have proved that the system of vectors (17), (18) and (21) is linearly dependent for any m , 1 ≦ m ≦ n − i − j m , 1 ≦ m ≦ n − i − j m,1 <= m <= n-i-jm, 1 \leqq m \leqq n-i-jm,1≦m≦n−i−j. For n − i − j = 1 n − i − j = 1 n-i-j=1n-i-j=1n−i−j=1 this contradicts the hypothesis. Suppose n − i − j ≧ 2 n − i − j ≧ 2 n-i-j >= 2n-i-j \geqq 2n−i−j≧2 and let be 1 ≦ m 1 << m 2 ≦ n − i − j 1 ≦ m 1 << m 2 ≦ n − i − j 1 <= m_(1)<<m_(2) <= n-i-j1 \leqq m_{1}< <m_{2} \leqq n-i-j1≦m1<<m2≦n−i−j. Then from the linear dependence of the respective systems (17), (18), (21) of the vectors for m = m 1 m = m 1 m=m_(1)m=m_{1}m=m1 and m = m 2 m = m 2 m=m_(2)m=m_{2}m=m2, it follows that there exist the constants c q , γ γ = 1 , 2 c q , γ γ = 1 , 2 c_(q,)^(gamma)gamma=1,2c_{q,}^{\gamma} \gamma=1,2cq,γγ=1,2 such that
(23,)
∑ q = 0 i − 1 c q r Φ ( q ) ( 0 ) + ∑ q = 1 m r − 1 c q + i − 1 r Φ ( t q ) + ∑ s = m r + 1 n − i − j c q + i − 2 r Φ ( t q ) + ∑ q = n − j − 1 n − 2 c q r Φ ( q − n + j + 1 ) ( 1 ) = 0 ∑ q = 0 i − 1   c q r Φ ( q ) ( 0 ) + ∑ q = 1 m r − 1   c q + i − 1 r Φ t q + ∑ s = m r + 1 n − i − j   c q + i − 2 r Φ t q + ∑ q = n − j − 1 n − 2   c q r Φ ( q − n + j + 1 ) ( 1 ) = 0 sum_(q=0)^(i-1)c_(q)^(r)Phi^((q))(0)+sum_(q=1)^(m_(r)-1)c_(q+i-1)^(r)Phi(t_(q))+sum_(s=m_(r)+1)^(n-i-j)c_(q+i-2)^(r)Phi(t_(q))+sum_(q=n-j-1)^(n-2)c_(q)^(r)Phi^((q-n+j+1))(1)=0\sum_{q=0}^{i-1} c_{q}^{r} \Phi^{(q)}(0)+\sum_{q=1}^{m_{r}-1} c_{q+i-1}^{r} \Phi\left(t_{q}\right)+\sum_{s=m_{r}+1}^{n-i-j} c_{q+i-2}^{r} \Phi\left(t_{q}\right)+\sum_{q=n-j-1}^{n-2} c_{q}^{r} \Phi^{(q-n+j+1)}(1)=0∑q=0i−1cqrΦ(q)(0)+∑q=1mr−1cq+i−1rΦ(tq)+∑s=mr+1n−i−jcq+i−2rΦ(tq)+∑q=n−j−1n−2cqrΦ(q−n+j+1)(1)=0, where ∑ q = 0 n − 1 | c q ∗ | ≠ 0 , r = 1 , 2 ∑ q = 0 n − 1   c q ∗ ≠ 0 , r = 1 , 2 sum_(q=0)^(n-1)|c_(q)^(**)|!=0,r=1,2\sum_{q=0}^{n-1}\left|c_{q}^{*}\right| \neq 0, r=1,2∑q=0n−1|cq∗|≠0,r=1,2. We observe that c i − 1 ∗ ≠ 0 , r = 1 , 2 c i − 1 ∗ ≠ 0 , r = 1 , 2 c_(i-1)^(**)!=0,r=1,2c_{i-1}^{*} \neq 0, r=1,2ci−1∗≠0,r=1,2, because the minimality of i i iii. Really, if contrary, say c i − 1 1 = 0 c i − 1 1 = 0 c_(i-1)^(1)=0c_{i-1}^{1}=0ci−11=0, we would have that the system of the n − 2 n − 2 n-2n-2n−2 vectors
(24) Φ ( 0 ) , … , Φ ( i − 2 ) ( 0 ) , ( 18 ) and ( 21 ) (24) Φ ( 0 ) , … , Φ ( i − 2 ) ( 0 ) , ( 18 )  and  ( 21 ) {:(24)Phi(0)","dots","Phi^((i-2))(0)","(18)" and "(21):}\begin{equation*} \Phi(0), \ldots, \Phi^{(i-2)}(0),(18) \text { and }(21) \tag{24} \end{equation*}(24)Φ(0),…,Φ(i−2)(0),(18) and (21)
is linearly dependent, i.e., it span a subspace of the dimension ≦ n − 3 ≦ n − 3 <= n-3\leqq n-3≦n−3. Complete this system by two vectors: Φ ( t ′ ) Φ t ′ Phi(t^('))\Phi\left(t^{\prime}\right)Φ(t′) and Φ ( t ′ ′ ) , t ′ , t ′ ′ ≠ t q , q == 1 , … , m , … , n − i − j Φ t ′ ′ , t ′ , t ′ ′ ≠ t q , q == 1 , … , m , … , n − i − j Phi(t^('')),t^('),t^('')!=t_(q),q==1,dots,m,dots,n-i-j\Phi\left(t^{\prime \prime}\right), t^{\prime}, t^{\prime \prime} \neq t_{q}, q= =1, \ldots, m, \ldots, n-i-jΦ(t′′),t′,t′′≠tq,q==1,…,m,…,n−i−j. The obtained system of vectors is contained in a subspace R n − 1 R n − 1 R^(n-1)\mathbf{R}^{n-1}Rn−1 of dimension n − 1 n − 1 n-1n-1n−1. Let c = ( c 1 , … , c n ) c = c 1 , … , c n c=(c^(1),dots,c^(n))c=\left(c^{1}, \ldots, c^{n}\right)c=(c1,…,cn) be a normal vector to R n − 1 R n − 1 R^(n-1)\mathbf{R}^{n-1}Rn−1. Then the element
x 2 = c 1 x 1 + … + c 11 x n x 2 = c 1 x 1 + … + c 11 x n x_(2)=c^(1)x_(1)+dots+c^(11)x_(n)x_{2}=c^{1} x_{1}+\ldots+c^{11} x_{n}x2=c1x1+…+c11xn
will have a zero of the multiplicity i − 1 i − 1 i-1i-1i−1 at 0 , a zero of multiplicity j j jjj at 1 and n − i − j + 1 n − i − j + 1 n-i-j+1n-i-j+1n−i−j+1 zeros in ( 0,1 ), which is the desired contradiction,
Now, multiplying ( 23 2 ) 23 2 (23_(2))\left(23_{2}\right)(232) by − c i − 1 1 / c i − 1 2 − c i − 1 1 / c i − 1 2 -c_(i-1)^(1)//c_(i-1)^(2)-c_{i-1}^{1} / c_{i-1}^{2}−ci−11/ci−12 and adding to ( 23 1 ) 23 1 (23_(1))\left(23_{1}\right)(231), in the case of | c i − 1 + m 1 2 | + | c i − 2 + m 2 1 | ≠ 0 c i − 1 + m 1 2 + c i − 2 + m 2 1 ≠ 0 |c_(i-1+m_(1))^(2)|+|c_(i-2+m_(2))^(1)|!=0\left|c_{i-1+m_{1}}^{2}\right|+\left|c_{i-2+m_{2}}^{1}\right| \neq 0|ci−1+m12|+|ci−2+m21|≠0 we conclude that the system (24) of vectors
is linearly dependent which yield a contradiction as above. Hence c i − 1 + m 1 2 = 0 c i − 1 + m 1 2 = 0 c_(i-1+m_(1))^(2)=0c_{i-1+m_{1}}^{2}=0ci−1+m12=0 for any m 1 , 1 ≦ m 1 < n − i − j m 1 , 1 ≦ m 1 < n − i − j m_(1),1 <= m_(1) < n-i-jm_{1}, 1 \leqq m_{1}<n-i-jm1,1≦m1<n−i−j. This, together with ( 23 2 23 2 23_(2)23_{2}232 ) means that the system of vectors (17) and (18) is linearly dependent, i ≦ n − k , j < k i ≦ n − k , j < k i <= n-k,j < ki \leqq n-k, j<ki≦n−k,j<k. But this contradicts the Lemma 2. This last contradiction proves the lemma.

4. Proof of the theorems

In the conditions of the theorem the vectors
(25) Φ ( 0 ) , … , Φ ( n − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 1 ) ( 1 ) (25) Φ ( 0 ) , … , Φ ( n − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 1 ) ( 1 ) {:(25)Phi(0)","dots","Phi^((n-k-1))(0)","Phi(1)","dots","Phi^((k-1))(1):}\begin{equation*} \Phi(0), \ldots, \Phi^{(n-k-1)}(0), \Phi(1), \ldots, \Phi^{(k-1)}(1) \tag{25} \end{equation*}(25)Φ(0),…,Φ(n−k−1)(0),Φ(1),…,Φ(k−1)(1)
are linearly dependent and the vectors
(26) Φ ( 0 ) , … , Φ ( k − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 2 ) ( 1 ) (26) Φ ( 0 ) , … , Φ ( k − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 2 ) ( 1 ) {:(26)Phi(0)","dots","Phi^((k-k-1))(0)","Phi(1)","dots","Phi^((k-2))(1):}\begin{equation*} \Phi(0), \ldots, \Phi^{(k-k-1)}(0), \Phi(1), \ldots, \Phi^{(k-2)}(1) \tag{26} \end{equation*}(26)Φ(0),…,Φ(k−k−1)(0),Φ(1),…,Φ(k−2)(1)
are linearly independent (Lemma 2). From the condition that there exists a single function (up to a scalar factor) in L n L n L_(n)L_{n}Ln with zero of multiplicity n − k − 1 n − k − 1 n-k-1n-k-1n−k−1 at 0 and a zero of multiplicity k k kkk at 1 , it follows also that
(27) Φ ( 0 ) , … , Φ ( n − k − 2 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 1 ) ( 1 ) (27) Φ ( 0 ) , … , Φ ( n − k − 2 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 1 ) ( 1 ) {:(27)Phi(0)","dots","Phi^((n-k-2))(0)","Phi(1)","dots","Phi^((k-1))(1):}\begin{equation*} \Phi(0), \ldots, \Phi^{(n-k-2)}(0), \Phi(1), \ldots, \Phi^{(k-1)}(1) \tag{27} \end{equation*}(27)Φ(0),…,Φ(n−k−2)(0),Φ(1),…,Φ(k−1)(1)
are linearly independent vectors.
We observe that according Proposition 1 in [14] or Theorem 1 in [13], it follows by the linear independence of the system of vectors (26) or (27) that L n L n L_(n)L_{n}Ln is actually a Chebyshev space on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1].
In what follows we shall consider two cases.
  1. The case k ≧ 2 k ≧ 2 k >= 2k \geqq 2k≧2. Denote
(28) R n − 2 = sp ( Φ ( 0 ) , … , Φ ( n − k − 2 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 2 ) ( 1 ) ) (28) R n − 2 = sp Φ ( 0 ) , … , Φ ( n − k − 2 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 2 ) ( 1 ) {:(28)R^(n-2)=sp(Phi(0),dots,Phi^((n-k-2))(0),Phi(1),dots,Phi^((k-2))(1)):}\begin{equation*} \mathbf{R}^{n-2}=\operatorname{sp}\left(\Phi(0), \ldots, \Phi^{(n-k-2)}(0), \Phi(1), \ldots, \Phi^{(k-2)}(1)\right) \tag{28} \end{equation*}(28)Rn−2=sp(Φ(0),…,Φ(n−k−2)(0),Φ(1),…,Φ(k−2)(1))
and let be R 2 R 2 R^(2)\mathbf{R}^{2}R2 the orthogonal complement of R n − 2 R n − 2 R^(n-2)\mathbf{R}^{n-2}Rn−2 in R n R n R^(n)\mathbf{R}^{n}Rn. The set of the elements in L n L n L_(n)L_{n}Ln which have a zero of multiplicity n − k − 1 n − k − 1 n-k-1n-k-1n−k−1 at 0 and a zero of multiplicity k − 1 k − 1 k-1k-1k−1 at 1 form, according Lemma 3, a 2-dimensional CSp on ( 0,1 ). The characteristic curve of this CSp may be obtained according 1.2 by projection of Φ ( ( 0 , 1 ) ) Φ ( ( 0 , 1 ) ) Phi((0,1))\Phi((0,1))Φ((0,1)) into R 2 R 2 R^(2)\mathbf{R}^{2}R2. Let us consider the space
(29) R n − 1 = sp ( Φ ( 0 ) , … , Φ ( n − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 1 ) ( 1 ) ) . (29) R n − 1 = sp Φ ( 0 ) , … , Φ ( n − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k − 1 ) ( 1 ) . {:(29)R^(n-1)=sp(Phi(0),dots,Phi^((n-k-1))(0),Phi(1),dots,Phi^((k-1))(1)).:}\begin{equation*} \mathbf{R}^{n-1}=\operatorname{sp}\left(\Phi(0), \ldots, \Phi^{(n-k-1)}(0), \Phi(1), \ldots, \Phi^{(k-1)}(1)\right) . \tag{29} \end{equation*}(29)Rn−1=sp(Φ(0),…,Φ(n−k−1)(0),Φ(1),…,Φ(k−1)(1)).
This space will be projected by the projection p p ppp on R 2 R 2 R^(2)\mathbb{R}^{2}R2 in the line R 1 R 1 R^(1)\mathbb{R}^{1}R1. Because of the linear independence of the vectors (26) and (27), Φ ( n − k − 1 ) ( 0 ) Φ ( n − k − 1 ) ( 0 ) Phi^((n-k-1))(0)\Phi^{(n-k-1)}(0)Φ(n−k−1)(0) and Φ ( k − 1 ) ( 1 ) Φ ( k − 1 ) ( 1 ) Phi^((k-1))(1)\Phi^{(k-1)}(1)Φ(k−1)(1) will be projected in nonzero vectors in R 1 R 1 R^(1)\mathbb{R}^{1}R1. This means by 1.3 that the projected curve p Φ ( ( 0 , 1 ) ) p Φ ( ( 0 , 1 ) ) p Phi((0,1))p \Phi((0,1))pΦ((0,1)) has the line R 1 R 1 R^(1)\mathbb{R}^{1}R1 as tangent line for t = 0 t = 0 t=0t=0t=0 and t = 1 t = 1 t=1t=1t=1. Because p Φ ( 0 ) = p Φ ( 1 ) = 0 p Φ ( 0 ) = p Φ ( 1 ) = 0 p Phi(0)=p Phi(1)=0p \Phi(0)=p \Phi(1)=0pΦ(0)=pΦ(1)=0, it follows that L 2 L 2 L_(2)L_{2}L2, the
CSp CSp CSp\operatorname{CSp}CSp on ( 0,1 ) of the functions in L n L n L_(n)L_{n}Ln having zero of multiplicity n − k − 1 n − k − 1 n-k-1n-k-1n−k−1 at 0 and zero of multiplicity k − 1 k − 1 k-1k-1k−1 at 1 , is in fact a CSp of the type 2.2.
Suppose that the domain of definition of the CSp defined L n L n L_(n)L_{n}Ln on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] may be extended with n − 2 n − 2 n-2n-2n−2 distinct points, say α 1 , … , α n − 2 α 1 , … , α n − 2 alpha_(1),dots,alpha_(n-2)\alpha_{1}, \ldots, \alpha_{n-2}α1,…,αn−2. Then the vectors Φ ( 0 ) , Φ ( 1 ) , Φ ( α 1 ) , … , Φ ( α n − 2 ) Φ ( 0 ) , Φ ( 1 ) , Φ α 1 , … , Φ α n − 2 Phi(0),Phi(1),Phi(alpha_(1)),dots,Phi(alpha_(n-2))\Phi(0), \Phi(1), \Phi\left(\alpha_{1}\right), \ldots, \Phi\left(\alpha_{n-2}\right)Φ(0),Φ(1),Φ(α1),…,Φ(αn−2) must be linearly independent and therefore at least one of them, say Φ ( α 1 ) Φ α 1 Phi(alpha_(1))\Phi\left(\alpha_{1}\right)Φ(α1) cannot be contained in the space R n − 1 R n − 1 R^(n-1)\mathbf{R}^{n-1}Rn−1 defined by (29). Then the projection p Φ ( α 1 ) p Φ α 1 p Phi(alpha_(1))p \Phi\left(\alpha_{1}\right)pΦ(α1) cannot be in R 1 R 1 R^(1)\mathbb{R}^{1}R1, the projection in R 2 R 2 R^(2)\mathbf{R}^{2}R2 of R n − 1 R n − 1 R^(n-1)\mathbf{R}^{n-1}Rn−1. Then sp ( p Φ ( α 1 ) ) sp p Φ α 1 sp(p Phi(alpha_(1)))\mathrm{sp}\left(p \Phi\left(\alpha_{1}\right)\right)sp(pΦ(α1)) will intersect the characteristic curve Ψ ( ( 0 , 1 ) ) = p Φ ( ( 0 , 1 ) ) Ψ ( ( 0 , 1 ) ) = p Φ ( ( 0 , 1 ) ) Psi((0,1))=p Phi((0,1))\Psi((0,1))=p \Phi((0,1))Ψ((0,1))=pΦ((0,1)) of the subspace L 2 L 2 L_(2)L_{2}L2 in a point Ψ ( t 0 ) , t 0 ∈ ( 0 , 1 ) Ψ t 0 , t 0 ∈ ( 0 , 1 ) Psi(t_(0)),t_(0)in(0,1)\Psi\left(t_{0}\right), t_{0} \in(0,1)Ψ(t0),t0∈(0,1). According 1.4 we may choose the distinct points t 1 ′ , … , t n − k − 2 ′ t 1 ′ , … , t n − k − 2 ′ t_(1)^('),dots,t_(n-k-2)^(')t_{1}^{\prime}, \ldots, t_{n-k-2}^{\prime}t1′,…,tn−k−2′ in ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1) in the neighbourhood of 0 and the distinct points t 1 ′ ′ , … , t k − 2 ′ ′ t 1 ′ ′ , … , t k − 2 ′ ′ t_(1)^(''),dots,t_(k-2)^('')t_{1}^{\prime \prime}, \ldots, t_{k-2}^{\prime \prime}t1′′,…,tk−2′′ in ( 0,1 ) in the neighbourhood of 1 such that the subspace R 1 n − 2 = sp ( Φ ( 0 ) , Φ ( t 1 ′ ) , … R 1 n − 2 = sp Φ ( 0 ) , Φ t 1 ′ , … R_(1)^(n-2)=sp(Phi(0),Phi(t_(1)^(')),dots:}\mathbb{R}_{1}^{n-2}=\mathrm{sp}\left(\Phi(0), \Phi\left(t_{1}^{\prime}\right), \ldots\right.R1n−2=sp(Φ(0),Φ(t1′),…
. , Φ ( t n − k − 2 ′ ) , Φ ( 1 ) , Φ ( t 1 ′ ′ ) , … , Φ ( t k − 2 ′ ′ ) ) Φ t n − k − 2 ′ , Φ ( 1 ) , Φ t 1 ′ ′ , … , Φ t k − 2 ′ ′ {: Phi(t_(n-k-2)^(')),Phi(1),Phi(t_(1)^('')),dots,Phi(t_(k-2)^('')))\left.\Phi\left(t_{n-k-2}^{\prime}\right), \Phi(1), \Phi\left(t_{1}^{\prime \prime}\right), \ldots, \Phi\left(t_{k-2}^{\prime \prime}\right)\right)Φ(tn−k−2′),Φ(1),Φ(t1′′),…,Φ(tk−2′′)) be arbitrarily close to the subspace R n − 2 R n − 2 R^(n-2)\mathbf{R}^{n-2}Rn−2 given by (28). Suppose t 0 ′ > t i ′ , i = 1 , … , n − k − 2 , t 0 ′ ′ < t i ′ ′ , i == 1 , … , k − 2 t 0 ′ > t i ′ , i = 1 , … , n − k − 2 , t 0 ′ ′ < t i ′ ′ , i == 1 , … , k − 2 t_(0)^(') > t_(i)^('),i=1,dots,n-k-2,t_(0)^('') < t_(i)^(''),i==1,dots,k-2t_{0}^{\prime}>t_{i}^{\prime}, i=1, \ldots, n-k-2, t_{0}^{\prime \prime}<t_{i}^{\prime \prime}, i= =1, \ldots, k-2t0′>ti′,i=1,…,n−k−2,t0′′<ti′′,i==1,…,k−2, and t 0 ∈ ( t 0 ′ , t 0 ′ ′ ) t 0 ∈ t 0 ′ , t 0 ′ ′ t_(0)in(t_(0)^('),t_(0)^(''))t_{0} \in\left(t_{0}^{\prime}, t_{0}^{\prime \prime}\right)t0∈(t0′,t0′′). It follows from 1.5 that the projection of Φ ( ( t 0 ′ , t 0 ′ ′ ) ) Φ t 0 ′ , t 0 ′ ′ Phi((t_(0)^('),t_(0)^('')))\Phi\left(\left(t_{0}^{\prime}, t_{0}^{\prime \prime}\right)\right)Φ((t0′,t0′′)) into R 1 2 R 1 2 R_(1)^(2)\mathbf{R}_{1}^{2}R12, the ortogonal complement of R 1 n − 2 R 1 n − 2 R_(1)^(n-2)\mathbf{R}_{1}^{n-2}R1n−2 will be arbitrarily Φ ( ( t 0 ′ , t 0 ′ ′ ) ) Φ t 0 ′ , t 0 ′ ′ Phi((t_(0)^('),t_(0)^('')))\Phi\left(\left(t_{0}^{\prime}, t_{0}^{\prime \prime}\right)\right)Φ((t0′,t0′′)) in R 2 R 2 R^(2)\mathbf{R}^{2}R2 and the same is projection of close to ( α 1 ) ) α 1 {:(alpha_(1)))\left.\left(\alpha_{1}\right)\right)(α1)). This means that we may realise that (he line sp ( Φ ( α 1 ) sp Φ α 1 sp(Phi(alpha_(1)):}\operatorname{sp}\left(\Phi\left(\alpha_{1}\right)\right.sp(Φ(α1) ts the projection ϕ ′ Φ ( t ′ t ′ ′ ) ) ϕ ′ Φ t ′ t ′ ′ {:phi^(')Phi(t^(')t^('')))\left.\phi^{\prime} \Phi\left(t^{\prime} t^{\prime \prime}\right)\right)ϕ′Φ(t′t′′)) in ( Φ ( α 1 ) ) Φ α 1 (Phi(alpha_(1)))\left(\Phi\left(\alpha_{1}\right)\right)(Φ(α1)) pro1jected in R 1 2 R 1 2 R_(1)^(2)\mathbf{R}_{1}^{2}R12 intersects the projection p ′ Φ ( ( t 0 ′ , t 0 ′ ′ ) ) p ′ Φ t 0 ′ , t 0 ′ ′ p^(')Phi((t_(0)^('),t_(0)^('')))p^{\prime} \Phi\left(\left(t_{0}^{\prime}, t_{0}^{\prime \prime}\right)\right)p′Φ((t0′,t0′′)) in R 1 2 R 1 2 R_(1)^(2)\mathbf{R}_{1}^{2}R12 in a point p ′ Φ ( t ) p ′ Φ ( t ) p^(')Phi(t)p^{\prime} \Phi(t)p′Φ(t). But then R n − 1 = p ′ − 1 ( p ′ sp ( Φ ( α 1 ) ) ) R n − 1 = p ′ − 1 p ′ sp Φ α 1 R^(n-1)=p^('-1)(p^(')sp(Phi(alpha_(1))))\mathbf{R}^{n-1}=p^{\prime-1}\left(p^{\prime} \mathrm{sp}\left(\Phi\left(\alpha_{1}\right)\right)\right)Rn−1=p′−1(p′sp(Φ(α1))) will contain the vectors Φ ( t ¯ ) , Φ ( α 1 ) , Φ ( 0 ) , Φ ( t 1 ′ ) , … … , Φ ( t n − k − 2 ′ ) , Φ ( 1 ) , Φ ( t 1 ′ ′ ) , … , Φ ( t k − 2 ′ ′ ) Φ ( t ¯ ) , Φ α 1 , Φ ( 0 ) , Φ t 1 ′ , … … , Φ t n − k − 2 ′ , Φ ( 1 ) , Φ t 1 ′ ′ , … , Φ t k − 2 ′ ′ Phi( bar(t)),Phi(alpha_(1)),Phi(0),Phi(t_(1)^(')),dots dots,Phi(t_(n-k-2)^(')),Phi(1),Phi(t_(1)^('')),dots,Phi(t_(k-2)^(''))\Phi(\bar{t}), \Phi\left(\alpha_{1}\right), \Phi(0), \Phi\left(t_{1}^{\prime}\right), \ldots \ldots, \Phi\left(t_{n-k-2}^{\prime}\right), \Phi(1), \Phi\left(t_{1}^{\prime \prime}\right), \ldots, \Phi\left(t_{k-2}^{\prime \prime}\right)Φ(t¯),Φ(α1),Φ(0),Φ(t1′),……,Φ(tn−k−2′),Φ(1),Φ(t1′′),…,Φ(tk−2′′), that is, n n nnn vectors. But this contradicts the fact that L n L n L_(n)L_{n}Ln extended to [ 0 , 1 ] ∪ { α 1 } [ 0 , 1 ] ∪ α 1 [0,1]uu{alpha_(1)}[0,1] \cup\left\{\alpha_{1}\right\}[0,1]∪{α1} is a CSp.
2. The case k = 1 k = 1 k=1k=1k=1. Let us denote R n − 2 = sp ( Φ ( 0 ) , … , Φ n − 3 ( 0 ) ) R n − 2 = sp Φ ( 0 ) , … , Φ n − 3 ( 0 ) R^(n-2)=sp(Phi(0),dots,Phi^(n-3)(0))\mathbb{R}^{n-2}=\operatorname{sp}\left(\Phi(0), \ldots, \Phi^{n-3}(0)\right)Rn−2=sp(Φ(0),…,Φn−3(0)) and let be R 2 R 2 R^(2)\mathbb{R}^{2}R2 the orthogonal complement of R n − 2 R n − 2 R^(n-2)\mathbb{R}^{n-2}Rn−2 in R 21 R 21 R^(21)\mathbb{R}^{21}R21. The space L 2 L 2 L_(2)L_{2}L2 of all solutions of (1) with zero of multiplicity n − 2 n − 2 n-2n-2n−2 at 0 form a CSp of dimension 2 on ( 0,1 ). By the linear independence of the vectors (27) (for k = 1 k = 1 k=1k=1k=1 ), it follows that the solutions having a zero of multiplicity n = 2 n = 2 n=2n=2n=2 at 0 cannot all vanish in the point 1 . Then by Theorem 2 in [13], L 2 L 2 L_(2)L_{2}L2 forms a CSp also on ( 0,1 ] ] ]]]. According 1.2 the characteristic curve of L 2 L 2 L_(2)L_{2}L2 can be obtained by projection of the curve Φ ( ( 0 , 1 ] ) Φ ( ( 0 , 1 ] ) Phi((0,1])\Phi((0,1])Φ((0,1]) into R 2 R 2 R^(2)\mathbf{R}^{2}R2. Let be
(30) R n − 1 = sp ( Φ ( 0 ) , … , Φ ( n − 2 ) ( 0 ) , Φ ( 1 ) ) . (30) R n − 1 = sp Φ ( 0 ) , … , Φ ( n − 2 ) ( 0 ) , Φ ( 1 ) . {:(30)R^(n-1)=sp(Phi(0),dots,Phi^((n-2))(0),Phi(1)).:}\begin{equation*} \mathbf{R}^{n-1}=\operatorname{sp}\left(\Phi(0), \ldots, \Phi^{(n-2)}(0), \Phi(1)\right) . \tag{30} \end{equation*}(30)Rn−1=sp(Φ(0),…,Φ(n−2)(0),Φ(1)).
From 1.3. and from the linear independence of the vectors Φ ( 0 ) , … , Φ ( n − 2 ) ( 0 ) Φ ( 0 ) , … , Φ ( n − 2 ) ( 0 ) Phi(0),dots,Phi^((n-2))(0)\Phi(0), \ldots, \Phi^{(n-2)}(0)Φ(0),…,Φ(n−2)(0), it follows that by the projection p p ppp onto R 2 R 2 R^(2)\mathbf{R}^{2}R2 the vector Φ ( n − 2 ) ( 0 ) Φ ( n − 2 ) ( 0 ) Phi^((n-2))(0)\Phi^{(n-2)}(0)Φ(n−2)(0) becomes a tangent vector to p ( Φ ( ( 0 , 1 ] ) p ( Φ ( ( 0 , 1 ] ) p(Phi((0,1])p(\Phi((0,1])p(Φ((0,1]) in the point 0 . The support of this tangent vector is R 1 = p ( R n − 1 ) R 1 = p R n − 1 R^(1)=p(R^(n-1))\mathbb{R}^{1}=p\left(\mathbb{R}^{n-1}\right)R1=p(Rn−1). But R 1 R 1 R^(1)\mathbb{R}^{1}R1 contains also the vector p Φ ( 1 ) p Φ ( 1 ) p Phi(1)p \Phi(1)pΦ(1) which cannot be zero by the linear independence of the vectors (27) for k = 1 k = 1 k=1k=1k=1. This means that the space L 2 L 2 L_(2)L_{2}L2 defined on ( 0,1 ] is a Chebyshev space of the type 2.1.
Suppose that the domain of definition of the CSp L n CSp L n CSpL_(n)\operatorname{CSp} L_{n}CSpLn defined on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] can be extended with n − 2 n − 2 n-2n-2n−2 distinct points, say α 1 , … , α n − 2 α 1 , … , α n − 2 alpha_(1),dots,alpha_(n-2)\alpha_{1}, \ldots, \alpha_{n-2}α1,…,αn−2. Then the vectors Φ ( 0 ) , Φ ( 1 ) , Φ ( α 1 ) , … , Φ ( α n − 2 ) Φ ( 0 ) , Φ ( 1 ) , Φ α 1 , … , Φ α n − 2 Phi(0),Phi(1),Phi(alpha_(1)),dots,Phi(alpha_(n-2))\Phi(0), \Phi(1), \Phi\left(\alpha_{1}\right), \ldots, \Phi\left(\alpha_{n-2}\right)Φ(0),Φ(1),Φ(α1),…,Φ(αn−2) must be linearly independent, and therefore at least one of them, say Φ ( α 1 ) Φ α 1 Phi(alpha_(1))\Phi\left(\alpha_{1}\right)Φ(α1) cannot be contained in R n − 1 R n − 1 R^(n-1)\mathbf{R}^{n-1}Rn−1 given by (30). This means that sp ( p Φ ( α 1 ) p Φ α 1 p Phi(alpha_(1))p \Phi\left(\alpha_{1}\right)pΦ(α1) ) will be a line passing through the origin, which is different from R 1 = p ( R n − 1 ) R 1 = p R n − 1 R^(1)=p(R^(n-1))\mathbf{R}^{1}=p\left(\mathbf{R}^{n-1}\right)R1=p(Rn−1). According 2.1 this line
will intersect the characteristic curve Ψ ( 0 , 1 ) = p Φ ( ( 0 , 1 ] ) Ψ ( 0 , 1 ) = p Φ ( ( 0 , 1 ] ) Psi(0,1)=p Phi((0,1])\Psi(0,1)=p \Phi((0,1])Ψ(0,1)=pΦ((0,1]) in a point Ψ ( t 0 ) Ψ t 0 Psi(t_(0))\Psi\left(t_{0}\right)Ψ(t0) for t 0 ∈ ( 0 , 1 ) , Ψ ( t 0 ) = p Φ ( t 0 ) t 0 ∈ ( 0 , 1 ) , Ψ t 0 = p Φ t 0 t_(0)in(0,1),Psi(t_(0))=p Phi(t_(0))t_{0} \in(0,1), \Psi\left(t_{0}\right)=p \Phi\left(t_{0}\right)t0∈(0,1),Ψ(t0)=pΦ(t0). Repeating a similar argument as in the case k ≧ 2 k ≧ 2 k >= 2k \geqq 2k≧2 we obtain a contradiction with the hypothesis that L n L n L_(n)L_{n}Ln is a CSp on [ 0 , 1 ] ∪ { α 1 } [ 0 , 1 ] ∪ α 1 [0,1]uu{alpha_(1)}[0,1] \cup\left\{\alpha_{1}\right\}[0,1]∪{α1}. This completes the proof.
We observe that the above method of proof works also for the following generalised form of our theorem:
TEOREMA 2. Suppose that 0 and 1 are conjugate points of type k k kkk for (1) and that Φ ( 0 ) Φ ( 0 ) Phi(0)\Phi(0)Φ(0) and Φ ( 1 ) Φ ( 1 ) Phi(1)\Phi(1)Φ(1) are linearly independent. Suppose that there exists an s , s ≧ 0 s , s ≧ 0 s,s >= 0s, s \geqq 0s,s≧0 such that
rank ‖ Φ ( 0 ) , … , Φ ( n − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k + s − 1 ) ( 1 ) ‖ = rank ‖ Φ ( 0 ) , … , Φ ( n − k − 2 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k + s − 1 ) ( 1 ) ‖ = n − 1 rank Φ ( 0 ) , … , Φ ( n − k − 1 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k + s − 1 ) ( 1 ) = rank Φ ( 0 ) , … , Φ ( n − k − 2 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k + s − 1 ) ( 1 ) = n − 1 {:[rank||Phi(0),dots,Phi^((n-k-1))(0),Phi(1),dots,Phi^((k+s-1))(1)||=],[rank||Phi(0),dots,Phi^((n-k-2))(0),Phi(1),dots,Phi^((k+s-1))(1)||=n-1]:}\begin{aligned} & \operatorname{rank}\left\|\Phi(0), \ldots, \Phi^{(n-k-1)}(0), \Phi(1), \ldots, \Phi^{(k+s-1)}(1)\right\|= \\ & \operatorname{rank}\left\|\Phi(0), \ldots, \Phi^{(n-k-2)}(0), \Phi(1), \ldots, \Phi^{(k+s-1)}(1)\right\|=n-1 \end{aligned}rank‖Φ(0),…,Φ(n−k−1)(0),Φ(1),…,Φ(k+s−1)(1)‖=rank‖Φ(0),…,Φ(n−k−2)(0),Φ(1),…,Φ(k+s−1)(1)‖=n−1
and
rank ‖ Φ ( 0 ) , … , Φ ( n − k − 2 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k + s − 2 ) ( 0 ) ‖ = n − 2  rank  Φ ( 0 ) , … , Φ ( n − k − 2 ) ( 0 ) , Φ ( 1 ) , … , Φ ( k + s − 2 ) ( 0 ) = n − 2 " rank "||Phi(0),dots,Phi^((n-k-2))(0),Phi(1),dots,Phi^((k+s-2))(0)||=n-2\text { rank }\left\|\Phi(0), \ldots, \Phi^{(n-k-2)}(0), \Phi(1), \ldots, \Phi^{(k+s-2)}(0)\right\|=n-2 rank ‖Φ(0),…,Φ(n−k−2)(0),Φ(1),…,Φ(k+s−2)(0)‖=n−2
Then the space L n L n L_(n)L_{n}Ln of the solutions of (1) forms a CSp on [0,1], whose domain of definition can be extended with at most n − 3 n − 3 n-3n-3n−3 distinct points.

5. Examples

The difficulty to give concrete examples of differential equations verifying the conditions in Theorem 1 or 2 have two aspects: (i) the theorems are not of qualitative character and (ii) even in the case when we have the explicite form of the solutions, the determination of the conjugate points may be difficult. In what follows we shall give examples only in the class of equations with constant coefficients and shall illustrate how is possible in some cases to evit the concrete determination of the conjugate points.
  1. Let us consider the differential equation
(31) d 2 d t 2 ( d 2 d t 2 + 1 2 ) ( d 2 d t 2 + 2 2 ) ⋯ ( d 2 d t 2 + m 2 ) x = 0 (31) d 2 d t 2 d 2 d t 2 + 1 2 d 2 d t 2 + 2 2 ⋯ d 2 d t 2 + m 2 x = 0 {:(31)(d^(2))/(dt^(2))((d^(2))/(dt^(2))+1^(2))((d^(2))/(dt^(2))+2^(2))cdots((d^(2))/(dt^(2))+m^(2))x=0:}\begin{equation*} \frac{d^{2}}{d t^{2}}\left(\frac{d^{2}}{d t^{2}}+1^{2}\right)\left(\frac{d^{2}}{d t^{2}}+2^{2}\right) \cdots\left(\frac{d^{2}}{d t^{2}}+m^{2}\right) x=0 \tag{31} \end{equation*}(31)d2dt2(d2dt2+12)(d2dt2+22)⋯(d2dt2+m2)x=0
A fundamental system of this differential equation is the following:
(32) 1 , t , sin ⁡ t , cos ⁡ t , … , sin ⁡ m t , cos ⁡ m t (32) 1 , t , sin ⁡ t , cos ⁡ t , … , sin ⁡ m t , cos ⁡ m t {:(32)1","t","sin t","cos t","dots","sin mt","cos mt:}\begin{equation*} 1, t, \sin t, \cos t, \ldots, \sin m t, \cos m t \tag{32} \end{equation*}(32)1,t,sin⁡t,cos⁡t,…,sin⁡mt,cos⁡mt
By a result of V. I. ANDREEV [3] (see also [10], problem II. 4.1, p. 67) the system of functions (32) form a CS on [0,2 π π pi\piπ ] whose domain of definition cannot be extended to an interval containing this closed interval as a proper subset. From this it follows that 0 and 2 π 2 π 2pi2 \pi2π are conjugate points
for (31). A direct verification gives that the type of these conjugate points is k = 2 k = 2 k=2k=2k=2.
For m = 1 m = 1 m=1m=1m=1 we are in the conditions of the Theorem 1. Then it follows that L 4 = sp ( 1 , t , sin ⁡ t , cos ⁡ t ) L 4 = sp ( 1 , t , sin ⁡ t , cos ⁡ t ) L_(4)=sp(1,t,sin t,cos t)L_{\mathbf{4}}=\mathrm{sp}(1, t, \sin t, \cos t)L4=sp(1,t,sin⁡t,cos⁡t) form CSp on [ 0 , 2 π ] [ 0 , 2 π ] [0,2pi][0,2 \pi][0,2π], whose domain of definition can be extended with at most a single point. In [14] we have shown that this extension is actually possible.
For m > 1 m > 1 m > 1m>1m>1 we are in the conditions of the Theorem 2. Really, we have Φ ( j ) ( 0 ) = Φ ( j ) ( 2 π ) Φ ( j ) ( 0 ) = Φ ( j ) ( 2 π ) Phi^((j))(0)=Phi^((j))(2pi)\Phi^{(j)}(0)=\Phi^{(j)}(2 \pi)Φ(j)(0)=Φ(j)(2π) for j = 1 , 2 , … j = 1 , 2 , … j=1,2,dotsj=1,2, \ldotsj=1,2,…, and therefore
2 m + 1 = rank ‖ Φ ( 0 ) , Φ ′ ( 0 ) , … , Φ ( 2 m − 1 ) ( 0 ) , Φ ( 2 π ) ‖ = 2 m + 1 = rank Φ ( 0 ) , Φ ′ ( 0 ) , … , Φ ( 2 m − 1 ) ( 0 ) , Φ ( 2 π ) = 2m+1=rank||Phi(0),Phi^(')(0),dots,Phi^((2m-1))(0),Phi(2pi)||=2 m+1=\operatorname{rank}\left\|\Phi(0), \Phi^{\prime}(0), \ldots, \Phi^{(2 m-1)}(0), \Phi(2 \pi)\right\|=2m+1=rank‖Φ(0),Φ′(0),…,Φ(2m−1)(0),Φ(2π)‖=
= rank ‖ Φ ( 0 ) , Φ ′ ( 0 ) , … , Φ ( 2 m − 1 ) ( 0 ) , Φ ( 2 π ) , Φ ′ ( 2 π ) , … , Φ ( 2 n − 1 ) ( 2 π ) ‖ = = rank Φ ( 0 ) , Φ ′ ( 0 ) , … , Φ ( 2 m − 1 ) ( 0 ) , Φ ( 2 π ) , Φ ′ ( 2 π ) , … , Φ ( 2 n − 1 ) ( 2 π ) = =rank||Phi(0),Phi^(')(0),dots,Phi^((2m-1))(0),Phi(2pi),Phi^(')(2pi),dots,Phi^((2n-1))(2pi)||==\operatorname{rank}\left\|\Phi(0), \Phi^{\prime}(0), \ldots, \Phi^{(2 m-1)}(0), \Phi(2 \pi), \Phi^{\prime}(2 \pi), \ldots, \Phi^{(2 n-1)}(2 \pi)\right\|==rank‖Φ(0),Φ′(0),…,Φ(2m−1)(0),Φ(2π),Φ′(2π),…,Φ(2n−1)(2π)‖=
= rank ‖ Φ ( 0 ) , Φ ′ ( 0 ) , … , Φ ( 2 m − 2 ) ( 0 ) , Φ ( 2 π ) , Φ ′ ( 2 π ) , … , Φ ( 2 m − 1 ) ( 2 π ) ‖ = rank Φ ( 0 ) , Φ ′ ( 0 ) , … , Φ ( 2 m − 2 ) ( 0 ) , Φ ( 2 π ) , Φ ′ ( 2 π ) , … , Φ ( 2 m − 1 ) ( 2 π ) =rank||Phi(0),Phi^(')(0),dots,Phi^((2m-2))(0),Phi(2pi),Phi^(')(2pi),dots,Phi^((2m-1))(2pi)||=\operatorname{rank}\left\|\Phi(0), \Phi^{\prime}(0), \ldots, \Phi^{(2 m-2)}(0), \Phi(2 \pi), \Phi^{\prime}(2 \pi), \ldots, \Phi^{(2 m-1)}(2 \pi)\right\|=rank‖Φ(0),Φ′(0),…,Φ(2m−2)(0),Φ(2π),Φ′(2π),…,Φ(2m−1)(2π)‖, and
rank ‖ Φ ( 0 ) , Φ ′ ( 0 ) , … , Φ ( 2 n − 2 ) ( 0 ) , Φ ( 2 π ) , Φ ′ ( 2 π ) , … , Φ ( 2 n − 2 ) ( 2 π ) ‖ = 2 m Φ ( 0 ) , Φ ′ ( 0 ) , … , Φ ( 2 n − 2 ) ( 0 ) , Φ ( 2 π ) , Φ ′ ( 2 π ) , … , Φ ( 2 n − 2 ) ( 2 π ) = 2 m ||Phi(0),Phi^(')(0),dots,Phi^((2n-2))(0),Phi(2pi),Phi^(')(2pi),dots,Phi^((2n-2))(2pi)||=2m\left\|\Phi(0), \Phi^{\prime}(0), \ldots, \Phi^{(2 n-2)}(0), \Phi(2 \pi), \Phi^{\prime}(2 \pi), \ldots, \Phi^{(2 n-2)}(2 \pi)\right\|=2 m‖Φ(0),Φ′(0),…,Φ(2n−2)(0),Φ(2π),Φ′(2π),…,Φ(2n−2)(2π)‖=2m,
i.e., we have the conditions in Theorem 2 for n = 2 m + 2 , k = 2 , s = 2 m − 2 n = 2 m + 2 , k = 2 , s = 2 m − 2 n=2m+2,k=2,s=2m-2n=2 m+2, k=2, s=2 m-2n=2m+2,k=2,s=2m−2. We conclude then that:
The space L 2 m + 2 = sp ( 1 , t , sin ⁡ t , cos ⁡ t , … , sin ⁡ m t , cos ⁡ m t ) , m ≧ 1 L 2 m + 2 = sp ( 1 , t , sin ⁡ t , cos ⁡ t , … , sin ⁡ m t , cos ⁡ m t ) , m ≧ 1 L_(2m+2)=sp(1,t,sin t,cos t,dots,sin mt,cos mt),m >= 1L_{2 m+2}=\operatorname{sp}(1, t, \sin t, \cos t, \ldots, \sin m t, \cos m t), m \geqq 1L2m+2=sp(1,t,sin⁡t,cos⁡t,…,sin⁡mt,cos⁡mt),m≧1, is a CSp on [ 0 , 2 π ] [ 0 , 2 π ] [0,2pi][0,2 \pi][0,2π] whose domain of definition can be extended with 2 m − 1 2 m − 1 2m-12 m-12m−1 points at most. (For m = 1 m = 1 m=1m=1m=1 this extension is effectively possible.)
2. Suppose that we have a differential equation (1) defined on [ 0 , ∞ ) [ 0 , ∞ ) [0,oo)[0, \infty)[0,∞) for which we know that 0 has a conjugate point < ∞ < ∞ < oo<\infty<∞. Then, if we can verify that for any t t ttt in some neighbourhood of this conjugate point (the exact value of which isn't known) we have
(33) rank ‖ Φ ( 0 ) , … , Φ ( i − 1 ) ( 0 ) , Φ ( t ) , … , Φ ( n − i − 2 ) ( t ) ‖ = n − 1 (33) rank Φ ( 0 ) , … , Φ ( i − 1 ) ( 0 ) , Φ ( t ) , … , Φ ( n − i − 2 ) ( t ) = n − 1 {:(33)rank||Phi(0),dots,Phi^((i-1))(0),Phi(t),dots,Phi^((n-i-2))(t)||=n-1:}\begin{equation*} \operatorname{rank}\left\|\Phi(0), \ldots, \Phi^{(i-1)}(0), \Phi(t), \ldots, \Phi^{(n-i-2)}(t)\right\|=n-1 \tag{33} \end{equation*}(33)rank‖Φ(0),…,Φ(i−1)(0),Φ(t),…,Φ(n−i−2)(t)‖=n−1
for i = 1 , … , n − 2 i = 1 , … , n − 2 i=1,dots,n-2i=1, \ldots, n-2i=1,…,n−2, then the conditions in Theorem 1 are automatically verified.
For illustration we consider the differential equation
(34) d 2 d t 2 ( d 2 d t 2 + 1 ) ( d d t − 1 ) x = 0 (34) d 2 d t 2 d 2 d t 2 + 1 d d t − 1 x = 0 {:(34)(d^(2))/(dt^(2))((d^(2))/(dt^(2))+1)((d)/(dt)-1)x=0:}\begin{equation*} \frac{d^{2}}{d t^{2}}\left(\frac{d^{2}}{d t^{2}}+1\right)\left(\frac{d}{d t}-1\right) x=0 \tag{34} \end{equation*}(34)d2dt2(d2dt2+1)(ddt−1)x=0
The differential equation which corresponds to the first two factors in (34) is in fact (31) for m = 1 m = 1 m=1m=1m=1, and has 0 and 2 π 2 π 2pi2 \pi2π as conjugate points. The differential equation corresponding to the third factor is disconjugate on the whole real line. This means according Proposition 8 p. 94 in [5], that (34) has two conjugate points : 0 and a point η ≧ 2 π η ≧ 2 π eta >= 2pi\eta \geqq 2 \piη≧2π. Because 1 , t , e t , sin ⁡ t , cos ⁡ t 1 , t , e t , sin ⁡ t , cos ⁡ t 1,t,e^(t),sin t,cos t1, t, e^{t}, \sin t, \cos t1,t,et,sin⁡t,cos⁡t is a fundamental system of solutions, it follows that η < ∞ η < ∞ eta < oo\eta<\inftyη<∞. For 2 π ≦ t < ∞ 2 π ≦ t < ∞ 2pi <= t < oo2 \pi \leqq t<\infty2π≦t<∞ we can verify the conditions of the type (33).
Then from Theorem 1 it follows that there exists an η , 2 π ≦ η < ∞ η , 2 π ≦ η < ∞ eta,2pi <= eta < oo\eta, 2 \pi \leqq \eta<\inftyη,2π≦η<∞ such that L 5 = sp ( 1 , t , e t , sin ⁡ t , cos ⁡ t ) L 5 = sp 1 , t , e t , sin ⁡ t , cos ⁡ t L_(5)=sp(1,t,e^(t),sin t,cos t)L_{5}=\mathrm{sp}\left(1, t, e^{t}, \sin t, \cos t\right)L5=sp(1,t,et,sin⁡t,cos⁡t) is a CSp on [ 0 , η ] [ 0 , η ] [0,eta][0, \eta][0,η] whose domain of definition can be extended with two distinct points at most.

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