Interpolation in abstract spaces

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E. Moldovan (Popoviciu)
Institutul de Calcul

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E. Moldovan (Popoviciu), Interpolarea ȋn spaţii abstracte (1959), vol.10, nr.2, p.329-335

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Studii si Cercetari Matematice

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Academy of the Republic of S.R.

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INTERPOLATION IN ABSTRACT SPACES

OFELENA MOLDOVANPaper presented at the session of May 20-22, 1959 of the University"Babeş - Bolyai" - Cluj

  1. In this paper we aim to introduce a general interpolation scheme with the aim of giving an extension of some properties that are related to interpolation by polynomials.
It is considered a normed linear space¹) V V VVVand a subspace S S SSShis/her V V VVV. Either U U UUUa linear operation 2 2 ^(2){ }^{2}2), defined on the space V V VVVand with the values ​​also belonging to the space V V VVV.
Definition 1. We call the subspace S the interpolating subspace relative to the operation U U UUU, if : 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘. whatever f and V ∈ V f and V ∈ V IVF in Vfive \in VfandV∈V, we have U ( V ) ∈ S ; 2 ∘ U ( V ) ∈ S ; 2 ∘ U(v)in S;2^(@)U(v) \in S ; 2^{\circ}U(V)∈S;2∘. whatever f and V ∈ S f and V ∈ S be quad v in Sbe \quad v \in SfandV∈S, we have U ( V ) = V U ( V ) = V U(v)=vU(v)=vU(V)=V.
Whether U U U\mathcal{U}Ua set of linear operations defined on the space V V VVVand with the values ​​in V V VVV.
Definition 2. We call the subspace S an interpolator relative to the set U U U\mathcal{U}U, if it is interpolating relative to each element U ∈ U U ∈ U U inUU \in \mathcal{U}U∈U.
To exemplify the notion of interpolator subspace with respect to an operation U U UUUlet's consider space C C CCCof continuous functions on the interval [ 0 , 1 ] [ 0 , 1 ] [0.1][0.1][0,1]Let us denote by IT n IT n L_(n)L_{n}ITna system of functions φ 1 ( x ) , φ 2 ( x ) , … , φ n ( x ) φ 1 ( x ) , φ 2 ( x ) , … , φ n ( x ) varphi_(1)(x),varphi_(2)(x),dots,varphi_(n)(x)\varphi_{1}(x), \varphi_{2}(x), \ldots, \varphi_{n}(x)φ1(x),φ2(x),…,φn(x), from C C CCC, linearly independent. Then there are at least n n nnndistinct points x 1 , x 2 , … , x n x 1 , x 2 , … , x n x_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n}x1,x2,…,xnin [ 0 , 1 ] [ 0 , 1 ] [0.1][0.1][0,1], so that the determinant
(1) | φ 1 ( x 1 ) φ 2 ( x 1 ) … φ n ( x 1 ) φ 1 ( x 2 ) φ 2 ( x 2 ) … φ n ( x 2 ) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ φ 1 ( x n ) φ 2 ( x n ) … φ n ( x n ) | (1) φ 1 x 1 φ 2 x 1 … φ n x 1 φ 1 x 2 φ 2 x 2 … φ n x 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ φ 1 x n φ 2 x n … φ n x n {:(1)|[varphi_(1)(x_(1)),varphi_(2)(x_(1)),dots,varphi_(n)(x_(1))],[varphi_(1)(x_(2)),varphi_(2)(x_(2)), dots,varphi_(n)(x_(2))],[*,*,*,*],[*,*,*],[varphi_(1)(x_(n)),varphi_(2)(x_(n)),dots,varphi_(n)(x_(n))]|:}\left|\begin{array}{cccc} \varphi_{1}\left(x_{1}\right) & \varphi_{2}\left(x_{1}\right) & \ldots & \varphi_{n}\left(x_{1}\right) \tag{1}\\ \varphi_{1}\left(x_{2}\right) & \varphi_{2}\left(x_{2}\right) & \ldots & \varphi_{n}\left(x_{2}\right) \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \\ \varphi_{1}\left(x_{n}\right) & \varphi_{2}\left(x_{n}\right) & \ldots & \varphi_{n}\left(x_{n}\right) \end{array}\right|(1)|φ1(x1)φ2(x1)…φn(x1)φ1(x2)φ2(x2)…φn(x2)⋅⋅⋅⋅⋅⋅⋅φ1(xn)φ2(xn)…φn(xn)|
to be nonzero. Let us consider the subspace S S SSShis/her C C CCC, formed by all linear combinations ∑ i = 1 n α i φ i ( x ) ∑ i = 1 n   α i φ i ( x ) sum_(i=1)^(n)alpha_(i)varphi_(i)(x)\sum_{i=1}^{n} \alpha_{i} \varphi_{i}(x)∑and=1nαandφand(x)of functions φ i ( x ) φ i ( x ) varphi_(i)(x)\varphi_{i}(x)φand(x)The determinant (1) being assumed to be different from zero, exists in S S SSSone and only one function h ( x ) h ( x ) h(x)h(x)h(x), so that h ( x i ) = y i , i = 1 , 2 , … , n , y i h x i = y i , i = 1 , 2 , … , n , y i h(x_(i))=y_(i),i=1,2,dots,n,y_(i)h\left(x_{i}\right)=y_{i}, i=1,2, \ldots, n, y_{i}h(xand)=yand,and=1,2,…,n,yandbeing any given numbers. 3 3 ^(3){ }^{3}3) Let U U UUUthe operation by which something is made to correspond to a function f ( x ) f ( x ) f(x)f(x)f(x)FROM C C CCC, function U ( f ) = H ( φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ; f ∣ x ) U ( f ) = H φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ; f ∣ x U(f)=H([varphi_(1)","varphi_(2)","dots","varphi_(n)],[x_(1)","x_(2)","dots","x_(n)];f∣x)U(f)=H\left(\begin{array}{l}\varphi_{1}, \varphi_{2}, \ldots, \varphi_{n} \\ x_{1}, x_{2}, \ldots, x_{n}\end{array} ; f \mid x\right)U(f)=H(φ1,φ2,…,φnx1,x2,…,xn;f∣x). Subspace S S SSSconsidered is interpolator to the operation U ( f ) U ( f ) U(f)U(f)U(f)thus defined.
If the functions φ 1 ( x ) , φ 2 ( x ) , … , φ n ( x ) φ 1 ( x ) , φ 2 ( x ) , … , φ n ( x ) varphi_(1)(x),varphi_(2)(x),dots,varphi_(n)(x)\varphi_{1}(x), \varphi_{2}(x), \ldots, \varphi_{n}(x)φ1(x),φ2(x),…,φn(x), forms a Chebyshev system on the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], then the determinant (1) is nonzero, whatever the distinct points are x 1 , x 2 , … , x n x 1 , x 2 , … , x n x_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n}x1,x2,…,xnIt immediately follows that the operation U ( f ) = H ( φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ; f ∣ x ) U ( f ) = H φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ; f ∣ x U(f)=H([varphi_(1)","varphi_(2)","dots","varphi_(n)],[x_(1)","x_(2)","dots","x_(n)];f∣x)U(f)=H\left(\begin{array}{l}\varphi_{1}, \varphi_{2}, \ldots, \varphi_{n} \\ x_{1}, x_{2}, \ldots, x_{n}\end{array} ; f \mid x\right)U(f)=H(φ1,φ2,…,φnx1,x2,…,xn;f∣x)has the above property for any system of distinct points x 1 , x 2 , … , x n x 1 , x 2 , … , x n x_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n}x1,x2,…,xnFROM [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1].
Another example that plays an important role in numerical analysis, ni-1 provides the interpolation scheme of L. Gonciarov [1].
We consider the system of linear functionals
(2) A k ( f ) , k = 0 , 1 , 2 , … , n (2) A k ( f ) , k = 0 , 1 , 2 , … , n {:(2)A_(k)(f)","quad k=0","1","2","dots","n:}\begin{equation*} A_{k}(f), \quad k=0,1,2, \ldots, n \tag{2} \end{equation*}(2)Ak(f),k=0,1,2,…,n
defined on the space C C CCC.
Either P n P n P_(n)\mathcal{P}_{n}Pnthe set of polynomials of degree at most equal to n n nnnLet's note cu n P n ( A 0 , A 1 , … , A n ; f ∣ x ) cu n P n A 0 , A 1 , … , A n ; f ∣ x cu_(n)P_(n)(A_(0),A_(1),dots,A_(n);f∣x)\operatorname{cu}_{n} P_{n}\left(A_{0}, A_{1}, \ldots, A_{n} ; f \mid x\right)withnPn(A0,A1,…,An;f∣x)polynomial P n ( x ) ∈ D n P n ( x ) ∈ D n P_(n)(x)inD_(n)P_{n}(x) \in \mathcal{D}_{n}Pn(x)∈Dn, which satisfies the conditions
(3) A k ( P n ) = A k ( f ) , k = 0 , 1 , 2 , … , n (3) A k P n = A k ( f ) , k = 0 , 1 , 2 , … , n {:(3)A_(k)(P_(n))=A_(k)(f)","quad k=0","1","2","dots","n:}\begin{equation*} A_{k}\left(P_{n}\right)=A_{k}(f), \quad k=0,1,2, \ldots, n \tag{3} \end{equation*}(3)Ak(Pn)=Ak(f),k=0,1,2,…,n
function f ( x ) ∈ C f ( x ) ∈ C f(x)in Cf(x) \in Cf(x)∈Cbeing given. It is clear that if the determinant
(4) | A 0 ( 1 ) A 0 ( x ) … A 0 ( x n ) A 1 ( 1 ) A 1 ( x ) … A 1 ( x n ) ⋅ ⋅ ⋅ ⋅ A n ( 1 ) A n ( x ) … A n ( x n ) | (4) A 0 ( 1 ) A 0 ( x ) … A 0 x n A 1 ( 1 ) A 1 ( x ) … A 1 x n ⋅ ⋅ ⋅ ⋅ A n ( 1 ) A n ( x ) … A n x n {:(4)|[A_(0)(1),A_(0)(x),dots,A_(0)(x^(n))],[A_(1)(1),A_(1)(x),dots,A_(1)(x^(n))],[*,*,*,*],[A_(n)(1),A_(n)(x),dots,A_(n)(x^(n))]|:}\left|\begin{array}{cccc} A_{0}(1) & A_{0}(x) & \ldots & A_{0}\left(x^{n}\right) \tag{4}\\ A_{1}(1) & A_{1}(x) & \ldots & A_{1}\left(x^{n}\right) \\ \cdot & \cdot & \cdot & \cdot \\ A_{n}(1) & A_{n}(x) & \ldots & A_{n}\left(x^{n}\right) \end{array}\right|(4)|A0(1)A0(x)…A0(xn)A1(1)A1(x)…A1(xn)⋅⋅⋅⋅An(1)An(x)…An(xn)|
is nonzero, then for any f ∈ C f ∈ C f in Cf \in Cf∈C, there exists the polynomial P n ( A 0 , A 1 , … , A n ; f ∣ x ) P n A 0 , A 1 , … , A n ; f ∣ x P_(n)(A_(0),A_(1),dots,A_(n);f∣x)P_{n}\left(A_{0}, A_{1}, \ldots, A_{n} ; f \mid x\right)Pn(A0,A1,…,An;f∣x)and it is uniquely determined. Let us consider the operation U ( f ) = P n ( A 0 , A 1 , … , A n ; f ∣ x ) U ( f ) = P n A 0 , A 1 , … , A n ; f ∣ x U(f)=P_(n)(A_(0),A_(1),dots,A_(n);f∣x)U(f)=P_{n}\left(A_{0}, A_{1}, \ldots, A_{n} ; f \mid x\right)U(f)=Pn(A0,A1,…,An;f∣x). Subspace D n D n D_(n)D_{n}Dnhis/her C C CCCis interpolator relative to the operation U U UUUthus defined.
By particularizing the system of functionals (2), we obtain various well-known interpolation procedures. For example, if
(5) A k ( f ) = ∫ 0 1 x k f ( x ) d x , k = 0 , 1 , 2 , … , n (5) A k ( f ) = ∫ 0 1   x k f ( x ) d x , k = 0 , 1 , 2 , … , n {:(5)A_(k)(f)=int_(0)^(1)x^(k)f(x)dx","quad k=0","1","2","dots","n:}\begin{equation*} A_{k}(f)=\int_{0}^{1} x^{k} f(x) d x, \quad k=0,1,2, \ldots, n \tag{5} \end{equation*}(5)Ak(f)=∫01xkf(x)dx,k=0,1,2,…,n
then the determinant (4) becomes
| 1 1 2 1 3 ⋯ 1 n + 1 1 2 1 3 1 4 ⋯ 1 n + 2 ⋯ ⋯ ⋯ ⋯ ⋯ 1 n + 1 1 n + 2 1 n + 3 ⋯ 1 2 n + 1 | 1 1 2 1 3 ⋯ 1 n + 1 1 2 1 3 1 4 ⋯ 1 n + 2 ⋯ ⋯ ⋯ ⋯ ⋯ 1 n + 1 1 n + 2 1 n + 3 ⋯ 1 2 n + 1 |[1,(1)/(2),(1)/(3),cdots,(1)/(n+1)],[(1)/(2),(1)/(3),(1)/(4),cdots,(1)/(n+2)],[cdots,cdots,cdots,cdots,cdots],[(1)/(n+1),(1)/(n+2),(1)/(n+3),cdots,(1)/(2n+1)]|\left|\begin{array}{cccccc} 1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n+1} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \cdots & \frac{1}{n+2} \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ \frac{1}{n+1} & \frac{1}{n+2} & \frac{1}{n+3} & \cdots & \frac{1}{2 n+1} \end{array}\right||11213⋯1n+1121314⋯1n+2⋯⋯⋯⋯⋯1n+11n+21n+3⋯12n+1|
which is known to be different from zero. The operation U ( f ) = P n ( A 0 , A 1 , … , A n ; f ∣ x ) U ( f ) = P n A 0 , A 1 , … , A n ; f ∣ x U(f)=P_(n)(A_(0),A_(1),dots,A_(n);f∣x)U(f)=\mathscr{P}_{n}\left(A_{0}, A_{1}, \ldots, A_{n} ; f \mid x\right)U(f)=Pn(A0,A1,…,An;f∣x)corresponding to system (5) transforms any function 4 ) 4 ) ^(4)){ }^{4)}4)FROM C C CCCin the section on the order n n nnnof its Fourier series relative to Legendre polynomials.
In general, if we consider an orthogonal system of functions in a basis space V V VVV, the linear subspace generated by this system 5 5 ^(5){ }^{5}5) is an interpolator with respect to the operation that transforms a function from V V VVVin the section of a given order, of its Fourier series, relative to the orthogonal system considered.
2. Let us consider again the linear space V V VVVand its subspaces, S 1 ⊂ S 2 S 1 ⊂ S 2 S_(1)subS_(2)S_{1} \subset S_{2}S1⊂S2, which we assume are interpolators with respect to the set U 1 U 1 U_(1)\mathcal{U}_{1}U1respectively U 2 U 2 U_(2)\mathcal{U}_{2}U2of linear operations.
Definition 3. An element v v vvVof space V V VVVwe call it convex with respect to the subspace S 1 S 1 S_(1)S_{1}S1, if for any U ∈ U 2 U ∈ U 2 U inU_(2)U \in \mathcal{U}_{2}U∈U2HAVE U 2 ( v ) ϵ ¯ S 1 U 2 ( v ) ϵ ¯ S 1 U_(2)(v) bar(epsilon)S_(1)U_{2}(v) \bar{\epsilon} S_{1}U2(V)ε¯S1.
If the crowd U 1 U 1 U_(1)\mathcal{U}_{1}U1contains at least two distinct elements, then we can also give the following definition of convexity:
Definition 3*. An element v of the space V is called convex with respect to the subspace S 1 S 1 S_(1)S_{1}S1if for any pair of elements U 1 , U 2 ∈ U 1 U 1 , U 2 ∈ U 1 U_(1),U_(2)inU_(1)U_{1}, U_{2} \in \mathcal{U}_{1}U1,U2∈U1, we have U 1 ( v ) ≠ U 2 ( v ) U 1 ( v ) ≠ U 2 ( v ) U_(1)(v)!=U_(2)(v)U_{1}(v) \neq U_{2}(v)U1(V)≠U2(V).

theorem 1. If:

1 ∘ . V 1 ∘ . V 1^(@).V1^{\circ} . V1∘.Vis the space of continuous functions on a finite and closed interval [ a , b ] [ a , b ] [a,b][a, b][A,b],
2 ∘ . S 2 2 ∘ . S 2 2^(@).S_(2)2^{\circ} . S_{2}2∘.S2is the subspace generated by a Chebyshev system formed by the functions φ 1 ( x ) , φ 2 ( x ) , … , φ n ( x ) , n ⩾ 2 φ 1 ( x ) , φ 2 ( x ) , … , φ n ( x ) , n ⩾ 2 varphi_(1)(x),varphi_(2)(x),dots,varphi_(n)(x),n >= 2\varphi_{1}(x), \varphi_{2}(x), \ldots, \varphi_{n}(x), n \geqslant 2φ1(x),φ2(x),…,φn(x),n⩾2, and S 1 S 1 S_(1)S_{1}S1is the subspace generated by the functions φ 1 ( x ) , … , φ n − 1 ( x ) φ 1 ( x ) , … , φ n − 1 ( x ) varphi_(1)(x),dots,varphi_(n-1)(x)\varphi_{1}(x), \ldots, \varphi_{n-1}(x)φ1(x),…,φn−1(x), which is also supposed to be a system of Chebyshev,
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘the crowd U 1 U 1 U_(1)\mathcal{U}_{1}U1has as elements all operations 6 6 ^(6){ }^{6}6)
U ( f ) = Φ ( x 1 , x 2 , … x n − 1 ; f ∣ x ) U ( f ) = Φ x 1 , x 2 , … x n − 1 ; f ∣ x U(f)=Phi(x_(1),x_(2),dotsx_(n-1);f∣x)U(f)=\Phi\left(x_{1}, x_{2}, \ldots x_{n-1} ; f \mid x\right)U(f)=Φ(x1,x2,…xn−1;f∣x)
x i , i = 1 , 2 , … , n − 1 x i , i = 1 , 2 , … , n − 1 x_(i),quad i=1,2,dots,n-1x_{i}, \quad i=1,2, \ldots, n-1xand,and=1,2,…,n−1being distinct points from [ a , b ] [ a , b ] [a,b][a, b][A,b],
4 ∘ 4 ∘ 4^(@)4^{\circ}4∘the crowd U 2 U 2 U_(2)U_{2}U2has as elements all operations 7 ) U ( f ) = Φ ( x 1 , x 2 , … , x n ; f ∣ x ) 7 U ( f ) = Φ x 1 , x 2 , … , x n ; f ∣ x {:^(7))U(f)=Phi(x_(1),x_(2),dots,x_(n);f∣x)\left.{ }^{7}\right) U(f)=\Phi\left(x_{1}, x_{2}, \ldots, x_{n} ; f \mid x\right)7)U(f)=Φ(x1,x2,…,xn;f∣x), x i , i = 1 , 2 , … , n x i , i = 1 , 2 , … , n x_(i),i=1,2,dots,nx_{i}, i=1,2, \ldots, nxand,and=1,2,…,n, being distinct points in [ a , b ] [ a , b ] [a,b][a, b][A,b], then definition 3 is equivalent to definition 3 ∗ 3 ∗ 3^(**)3^{*}3∗.
To prove Theorem 1, it is sufficient to observe that in the generalized interpolation polynomial Φ ( x 1 , x 2 , … , x n ; f ∣ x ) Φ x 1 , x 2 , … , x n ; f ∣ x Phi(x_(1),x_(2),dots,x_(n);f∣x)\Phi\left(x_{1}, x_{2}, \ldots, x_{n} ; f \mid x\right)Φ(x1,x2,…,xn;f∣x)his coefficient φ n ( x ) φ n ( x ) varphi_(n)(x)\varphi_{n}(x)φn(x)is the generalized divided difference [5]
(6) [ φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ] ⟩ = | φ 1 ( x 1 ) φ 2 ( x 1 ) … φ n − 1 ( x 1 ) f ( x 1 ) φ 1 ( x 2 ) φ 2 ( x 2 ) … φ n − 1 ( x n ) f ( x 2 ) ⋅ ⋅ ⋅ ⋅ ⋅ φ 1 ( x n ) φ 2 ( x n ) … ⋅ ⋅ φ n − 1 ( x n ) ⋅ ⋅ φ 1 ( x 1 ) φ 2 ( x 1 ) … φ n − 1 ( x 1 ) φ n ( x 1 ) φ 1 ( x 2 ) φ 2 ( x 2 ) … φ n − 1 ( x 2 ) φ n ( x 2 ) ⋅ ⋅ ⋅ ⋅ ⋅ φ 1 ( x n ) φ 2 ( x n ) … ⋅ ⋅ φ n − 1 ( x n ) φ n ( x n ) | ∣ (6) φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n = φ 1 x 1 φ 2 x 1 … φ n − 1 x 1 f x 1 φ 1 x 2 φ 2 x 2 … φ n − 1 x n f x 2 ⋅ ⋅ ⋅ ⋅ ⋅ φ 1 x n φ 2 x n … ⋅ ⋅ φ n − 1 x n ⋅ ⋅ φ 1 x 1 φ 2 x 1 … φ n − 1 x 1 φ n x 1 φ 1 x 2 φ 2 x 2 … φ n − 1 x 2 φ n x 2 ⋅ ⋅ ⋅ ⋅ ⋅ φ 1 x n φ 2 x n … ⋅ ⋅ φ n − 1 x n φ n x n ∣ {:(6){:[[varphi_(1)","varphi_(2)","dots","varphi_(n)],[x_(1)","x_(2)","dots","x_(n)]]:)=(|[varphi_(1)(x_(1)),varphi_(2)(x_(1)),dots,varphi_(n-1)(x_(1)),f(x_(1))],[varphi_(1)(x_(2)),varphi_(2)(x_(2)),dots,varphi_(n-1)(x_(n)),f(x_(2))],[*,*,*,*,*],[varphi_(1)(x_(n)),varphi_(2)(x_(n)),dots,*,*],[varphi_(n-1)(x_(n)),*,*],[varphi_(1)(x_(1)),varphi_(2)(x_(1)),dots,varphi_(n-1)(x_(1)),varphi_(n)(x_(1))],[varphi_(1)(x_(2)),varphi_(2)(x_(2)),dots,varphi_(n-1)(x_(2)),varphi_(n)(x_(2))],[*,*,*,*,*],[varphi_(1)(x_(n)),varphi_(2)(x_(n)),dots,*,*],[varphi_(n-1)(x_(n)),varphi_(n)(x_(n))]|)/(∣):}\left.\left[\begin{array}{c} \varphi_{1}, \varphi_{2}, \ldots, \varphi_{n} \tag{6}\\ x_{1}, x_{2}, \ldots, x_{n} \end{array}\right]\right\rangle=\frac{\left|\begin{array}{ccccc} \varphi_{1}\left(x_{1}\right) & \varphi_{2}\left(x_{1}\right) & \ldots & \varphi_{n-1}\left(x_{1}\right) & f\left(x_{1}\right) \\ \varphi_{1}\left(x_{2}\right) & \varphi_{2}\left(x_{2}\right) & \ldots & \varphi_{n-1}\left(x_{n}\right) & f\left(x_{2}\right) \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \varphi_{1}\left(x_{n}\right) & \varphi_{2}\left(x_{n}\right) & \ldots & \cdot & \cdot \\ \varphi_{n-1}\left(x_{n}\right) & \cdot & \cdot \\ \varphi_{1}\left(x_{1}\right) & \varphi_{2}\left(x_{1}\right) & \ldots & \varphi_{n-1}\left(x_{1}\right) & \varphi_{n}\left(x_{1}\right) \\ \varphi_{1}\left(x_{2}\right) & \varphi_{2}\left(x_{2}\right) & \ldots & \varphi_{n-1}\left(x_{2}\right) & \varphi_{n}\left(x_{2}\right) \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \varphi_{1}\left(x_{n}\right) & \varphi_{2}\left(x_{n}\right) & \ldots & \cdot & \cdot \\ \varphi_{n-1}\left(x_{n}\right) & \varphi_{n}\left(x_{n}\right) \end{array}\right|}{\mid}(6)[φ1,φ2,…,φnx1,x2,…,xn]⟩=|φ1(x1)φ2(x1)…φn−1(x1)f(x1)φ1(x2)φ2(x2)…φn−1(xn)f(x2)⋅⋅⋅⋅⋅φ1(xn)φ2(xn)…⋅⋅φn−1(xn)⋅⋅φ1(x1)φ2(x1)…φn−1(x1)φn(x1)φ1(x2)φ2(x2)…φn−1(x2)φn(x2)⋅⋅⋅⋅⋅φ1(xn)φ2(xn)…⋅⋅φn−1(xn)φn(xn)|∣
According to definition 3, a function in V V VVVis convex with respect to S 1 S 1 S_(1)S_{1}S1if [ φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ; f ] ≠ 0 φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ; f ≠ 0 [[varphi_(1)","varphi_(2)","dots","varphi_(n)],[x_(1)","x_(2)","dots","x_(n)];f]!=0\left[\begin{array}{c}\varphi_{1}, \varphi_{2}, \ldots, \varphi_{n} \\ x_{1}, x_{2}, \ldots, x_{n}\end{array} ; f\right] \neq 0[φ1,φ2,…,φnx1,x2,…,xn;f]≠0on any point system x 1 , x 2 , … , x n x 1 , x 2 , … , x n x_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n}x1,x2,…,xn. The condition U 1 ( v ) ≠ U 2 ( v ) U 1 ( v ) ≠ U 2 ( v ) U_(1)(v)!=U_(2)(v)U_{1}(v) \neq U_{2}(v)U1(V)≠U2(V)from the definition 3 ∗ 3 ∗ 3^(**)3^{*}3∗expresses the same property, because it excludes the existence of a system of n n nnnpuncture x 1 , x 2 , … , x n x 1 , x 2 , … , x n x_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n}x1,x2,…,xnon which the divided difference (6) cancels out.
Remark. The notion of convexity introduced by definitions 3 and 3 ∗ 3 ∗ 3^(**)3^{*}3∗does not coincide with the well-known notion of convexity [ 5,3 ], with respect to a system of interpolating functions. The class of convex elements now includes both convex and concave elements in the sense of the definitions in [4] and [5].
In the case of Gonciarov's interpolation scheme, definition 3 is applicable.
In the assumptions made at the beginning of this paragraph, it is clear that there are convex elements in the sense of definition 3. All elements of S 2 S 2 S_(2)S_{2}S2which do not belong to him S 1 S 1 S_(1)S_{1}S1are convex in the sense of definition 3.
It is important to study, in the theory of interpolation procedures, those interpolation schemes - given by definition 2 - for which definitions 3 and 3 ∗ 3 ∗ 3^(**)3^{*}3∗are equivalent.
THEOREM 2. If for V 1 , S , S 2 , U 1 , U 2 V 1 , S , S 2 , U 1 , U 2 V_(1),S,S_(2),U_(1),U_(2)V_{1}, S, S_{2}, U_{1}, U_{2}V1,S,S2,U1,U2given, definitions 3 and 3 ∗ 3 ∗ 3^(**)3^{*}3∗are equivalent, then the property holds: if v ∈ V v ∈ V v in Vv \in VV∈Vand for U 1 U 1 U_(1)U_{1}U1, U 2 ∈ U 1 U 2 ∈ U 1 U_(2)inU_(1)U_{2} \in U_{1}U2∈U1HAVE U 1 ( v ) = U 2 ( v ) U 1 ( v ) = U 2 ( v ) U_(1)(v)=U_(2)(v)U_{1}(v)=U_{2}(v)U1(V)=U2(V), then there is an element U 3 ∈ U 2 U 3 ∈ U 2 U_(3)inU_(2)U_{3} \in U_{2}U3∈U2so that U 3 ( v ) ∈ S 1 U 3 ( v ) ∈ S 1 U_(3)(v)inS_(1)U_{3}(v) \in S_{1}U3(V)∈S1.
The proof of Theorem 2 is immediate. It is contained in it as un i caz un i caz un_(i)caz\mathrm{un}_{\mathrm{i}} \mathrm{caz}aandcasein particular, a property of divided differences that underlies several mean theorems related to interpolation by functions belonging to an interpolating set [ 3 , 4 , 6 ] [ 3 , 4 , 6 ] [3,4,6][3,4,6][3,4,6].
theorem 3. Let A [ v ] A [ v ] A[v]A[v]A[V]a linear functional defined on the space V V VVVin which they are given S 1 , S 2 , U 1 S 1 , S 2 , U 1 S_(1),S_(2),U_(1)S_{1}, S_{2}, \mathcal{U}_{1}S1,S2,U1and U 2 U 2 U_(2)\mathcal{U}_{2}U2If:
1 ∘ . A [ v ] = 0 1 ∘ . A [ v ] = 0 1^(@).A[v]=0quad1^{\circ} . A[v]=0 \quad1∘.A[V]=0whatever f i v ∈ S 1 f i v ∈ S 1 fiv inS_(1)f i v \in S_{1}fandV∈S1,
2 ∘ . A [ v ] ≠ 0 2 ∘ . A [ v ] ≠ 0 2^(@).A[v]!=02^{\circ} . A[v] \neq 02∘.A[V]≠0if v is convex with respect to S 1 S 1 S_(1)S_{1}S1in the sense of definition 3, then for any v ∈ V v ∈ V v in Vv \in VV∈Vthere is an element U ∈ U 2 U ∈ U 2 U inU_(2)U \in U_{2}U∈U2so that A [ v ] == A [ U ( v ) ] A [ v ] == A [ U ( v ) ] A[v]==A[U(v)]A[v]= =A[U(v)]A[V]==A[U(V)].
For demonstration, let us first assume A [ v ] = 0 A [ v ] = 0 A[v]=0A[v]=0A[V]=0. Then the element v v vvVcannot be convex with respect to S 1 S 1 S_(1)S_{1}S1. So there is U ∈ U 2 U ∈ U 2 U inU_(2)U \in U_{2}U∈U2so that U ( v ) ∈ S 1 U ( v ) ∈ S 1 U(v)inS_(1)U(v) \in S_{1}U(V)∈S1, and therefore A [ U ( v ) ] = 0 A [ U ( v ) ] = 0 A[U(v)]=0A[U(v)]=0A[U(V)]=0If A [ v ] ≠ 0 A [ v ] ≠ 0 A[v]!=0A[v] \neq 0A[V]≠0, we consider the element z = v − A [ v ] A [ g ] g z = v − A [ v ] A [ g ] g z=v-(A[v])/(A[g])gz=v-\frac{A[v]}{A[g]} gz=V−A[V]A[g]g, where g ∈ S 2 g ∈ S 2 g inS_(2)g \in S_{2}g∈S2and g ϵ ¯ S 1 g ϵ ¯ S 1 g bar(epsilon)S_(1)g \bar{\epsilon} S_{1}gε¯S1It follows that A [ g ] ≠ 0 A [ g ] ≠ 0 A[g]!=0A[g] \neq 0A[g]≠0and A [ z ] = 0 A [ z ] = 0 A[z]=0A[z]=0A[z]=0There is therefore a U ∈ U 2 U ∈ U 2 U inU_(2)U \in \mathcal{U}_{2}U∈U2so that A [ U ( z ) ] = 0 A [ U ( z ) ] = 0 A[U(z)]=0A[U(z)]=0A[U(z)]=0. But because of linearity, U ( z ) = U ( v ) − A [ v ] A [ g ] U [ g ] U ( z ) = U ( v ) − A [ v ] A [ g ] U [ g ] U(z)=U(v)-(A[v])/(A[g])U[g]U(z)=U(v)-\frac{A[v]}{A[g]} U[g]U(z)=U(V)−A[V]A[g]U[g]The operation U U UUUpreserve the element g ∈ S 2 g ∈ S 2 g inS_(2)g \in S_{2}g∈S2It results A [ U ( v ) ] = A [ v ] A [ U ( v ) ] = A [ v ] A[U(v)]=A[v]A[U(v)]=A[v]A[U(V)]=A[V].
In Theorem 3, a large number of well-known mean theorems are included as particular cases [4,6]. These theorems intervene in the study of the remainder of linear approximation procedures.
3. In the study of generalized interpolation procedures, it is interesting to examine the case when V V VVVis a Banach space. In this case, we can study the continuity properties of the operations involved in the definition of a general interpolation scheme.
theorem 4. If V V VVVis a Banach space and S S SSSis an interpolating subspace with respect to the operation U U UUU, then if any bounded subset of S S SSSis compact, U U UUUit is a continuous operation 8 8 ^(8){ }^{8}8).
The demonstration results from the consequence of the hypothesis made, namely that S S SSSis a subspace with a finite number of dimensions.
4. Let V V VVVa linear space and S S SSS
a subspace of it. Theorem 5 holds. If S S SSSis an n-dimensional subspace, generated by the elements v 1 , v 2 , … , v n v 1 , v 2 , … , v n v_(1),v_(2),dots,v_(n)v_{1}, v_{2}, \ldots, v_{n}V1,V2,…,Vnand there is n n nnnlinear functionals A 1 , A 2 , … , A n A 1 , A 2 , … , A n A_(1),A_(2),dots,A_(n)A_{1}, A_{2}, \ldots, A_{n}A1,A2,…,An, so that
(7) | A 1 [ v 1 ] A 2 [ v 1 ] … . . . A n [ v 1 ] A 1 [ v 2 ] A 2 [ v 2 ] … … . . A n [ v 2 ] . . . … . A 1 [ v n ] A 2 [ v n ] … . . . … A n [ v n ] | ≠ 0 (7) A 1 v 1 A 2 v 1 … . . . A n v 1 A 1 v 2 A 2 v 2 … … . . A n v 2 . . . … . A 1 v n A 2 v n … . . . … A n v n ≠ 0 {:(7)|[A_(1)[v_(1)],A_(2)[v_(1)],dots...,A_(n)[v_(1)]],[A_(1)[v_(2)],A_(2)[v_(2)]dots dots..,A_(n)[v_(2)]],[..,.,dots,.],[A_(1)[v_(n)],A_(2)[v_(n)]dots...,dots,A_(n)[v_(n)]]|!=0:}\left|\begin{array}{cccc} A_{1}\left[v_{1}\right] & A_{2}\left[v_{1}\right] & \ldots . . . & A_{n}\left[v_{1}\right] \tag{7}\\ A_{1}\left[v_{2}\right] & A_{2}\left[v_{2}\right] \ldots \ldots . . & A_{n}\left[v_{2}\right] \\ . . & . & \ldots & . \\ A_{1}\left[v_{n}\right] & A_{2}\left[v_{n}\right] \ldots . . . & \ldots & A_{n}\left[v_{n}\right] \end{array}\right| \neq 0(7)|A1[V1]A2[V1]…...An[V1]A1[V2]A2[V2]……..An[V2]...….A1[Vn]A2[Vn]…...…An[Vn]|≠0
then there is a linear operation U U UUUdefined on V V VVV, compared to which S S SSSis interpolating.
8) lim n → ∞ ‖ U ( v n ) − U ( v ) ‖ = 0 lim n → ∞   U v n − U ( v ) = 0 lim_(n rarr oo)||U(v_(n))-U(v)||=0\lim _{n \rightarrow \infty}\left\|U\left(v_{n}\right)-U(v)\right\|=0limn→∞‖U(Vn)−U(V)‖=0if lim n → ∞ ‖ v − v ‖ = 0 lim n → ∞   ‖ v − v ‖ = 0 lim_(n rarr oo)||v-v||=0\lim _{n \rightarrow \infty}\|v-v\|=0limn→∞‖V−V‖=0.
For the demonstration it is sufficient to construct for the element v v vvVany of V V VVV, the interpolation process
U ( v ) = ∑ k = 1 n α k v k U ( v ) = ∑ k = 1 n   α k v k U(v)=sum_(k=1)^(n)alpha_(k)v_(k)U(v)=\sum_{k=1}^{n} \alpha_{k} v_{k}U(V)=∑k=1nαkVk
where the numerical coefficients α k α k alpha_(k)\alpha_{k}αkis determined by the conditions
A i [ ∑ k = 1 n α k v k ] = A i [ v ] , i = 1 , 2 , … , n A i ∑ k = 1 n   α k v k = A i [ v ] , i = 1 , 2 , … , n A_(i)[sum_(k=1)^(n)alpha_(k)v_(k)]=A_(i)[v],quad i=1,2,dots,nA_{i}\left[\sum_{k=1}^{n} \alpha_{k} v_{k}\right]=A_{i}[v], \quad i=1,2, \ldots, nAand[∑k=1nαkVk]=Aand[V],and=1,2,…,n
and are uniquely determined due to the condition in (7). The operation U ( v ) U ( v ) U(v)U(v)U(V)can be put in the form
U ( v ) = − | 0 v 1 v 2 … … v n A 1 [ v ] A 1 [ v 1 ] A 1 [ v 2 ] … . . A 1 [ v n ] A 2 [ v ] A 2 [ v 1 ] A 2 [ v 2 ] … … . . . . . A n [ v ] A n [ v 1 ] A n [ v 2 ] … . | A 1 [ v 1 ] A 1 [ v 2 ] … … A 2 [ v 1 ] A 1 [ v 2 ] … … . . . . } … . . A n [ v n ] | . U ( v ) = − 0 v 1 v 2 … … v n A 1 [ v ] A 1 v 1 A 1 v 2 … . . A 1 v n A 2 [ v ] A 2 v 1 A 2 v 2 … … . . . . . A n [ v ] A n v 1 A n v 2 … . A 1 v 1 A 1 v 2 … … A 2 v 1 A 1 v 2 … … . . . . … . . A n v n . U(v)=-|[0,v_(1),v_(2),dots dots,v_(n)],[A_(1)[v],A_(1)[v_(1)],A_(1)[v_(2)],dots..,A_(1)[v_(n)]],[A_(2)[v],A_(2)[v_(1)],A_(2)[v_(2)],dots,dots],[.,.,.,.,.],[A_(n)[v],A_(n)[v_(1)],A_(n)[v_(2)],dots,.],[{:|[A_(1)[v_(1)],A_(1)[v_(2)],dots,dots],[A_(2)[v_(1)],A_(1)[v_(2)],dots,dots],[.,.,.,.]}dots..A_(n)[v_(n)]]|.U(v)=-\left|\begin{array}{ccccc} 0 & v_{1} & v_{2} & \ldots \ldots & v_{n} \\ A_{1}[v] & A_{1}\left[v_{1}\right] & A_{1}\left[v_{2}\right] & \ldots . . & A_{1}\left[v_{n}\right] \\ A_{2}[v] & A_{2}\left[v_{1}\right] & A_{2}\left[v_{2}\right] & \ldots & \ldots \\ . & . & . & . & . \\ A_{n}[v] & A_{n}\left[v_{1}\right] & A_{n}\left[v_{2}\right] & \ldots & . \\ \left.\left\lvert\, \begin{array}{cccc} A_{1}\left[v_{1}\right] & A_{1}\left[v_{2}\right] & \ldots & \ldots \\ A_{2}\left[v_{1}\right] & A_{1}\left[v_{2}\right] & \ldots & \ldots \\ . & . & . & . \end{array}\right.\right\} \ldots . . A_{n}\left[v_{n}\right] \end{array}\right| .U(V)=−|0V1V2……VnA1[V]A1[V1]A1[V2]…..A1[Vn]A2[V]A2[V1]A2[V2]…….....An[V]An[V1]An[V2]….|A1[V1]A1[V2]……A2[V1]A1[V2]……....}…..An[Vn]|.
  1. The definitions and theorems given in this paper constitute only the introductory notions in the study of general interpolation procedures that can be defined in a linear space. Theorem 3 has numerous applications in the study of linear approximation. As for the two definitions given for convexity, their usefulness results mainly from the particularization of the space V V VVVand the interpolating subspaces S 1 ∈ S 2 S 1 ∈ S 2 S_(1)inS_(2)S_{1} \in S_{2}S1∈S2chosen.
The notion of interpolation procedure is closely related to certain particular best approximation problems. In this paper we do not deal with these problems. We only give the formulation of one of the fundamental best approximation problems:
Being given V V VVVand the subspace S S SSSinterpolator to the crowd U U U\mathbb{U}Uof linear operations, assuming that V V VVVit is normal, to study the problem of the existence and uniqueness of the operation U ∗ ∈ U U ∗ ∈ U U^(**)inUU^{*} \in \mathcal{U}U∗∈Ufor which
‖ v − U ∗ ( v ) ‖ = inf U ∈ U ‖ v − U ( v ) ‖ v − U ∗ ( v ) = inf U ∈ U   ‖ v − U ( v ) ‖ ||v-U^(**)(v)||=i n f_(U inU)||v-U(v)||\left\|v-U^{*}(v)\right\|=\inf _{U \in \mathcal{U}}\|v-U(v)\|‖V−U∗(V)‖=childU∈U‖V−U(V)‖
v v vvVbeing fixed in V V VVVand not belonging to him S S SSSIf
, for example, V V VVVis the space of integrable quadratic functions and S S SSSis the subspace of trigonometric polynomials of given order n n nnn, it is known that the problem formulated above has a solution and it is unique. In this case U U U\mathcal{U}Uis the set of all linear operations that satisfy the condition required in definition 2.
Of course, a thorough study of the best approximation problem formulated is based on the prior study of the norm defined in V V VVVand on the study of the continuity properties of its elements U U U\mathcal{U}U.

INTERPOLIROVANIE IN ABSTRACT SPACES
(Brief summary)

The work defines the general scheme of interpolation in linear normalized space. The work also contains two convexity definitions relative to the generalized interpolation technique. The specific method of interpolation contains, as special cases, the Goncharov interpolation scheme and other interpolation schemes. The theorem on the average (theorem 3) is given, which applies to the study of the structure of the residual term in linear approximations. The work ends with the formulation of one task of the best approximation.

L'INTERPOLATION DANS DES ESPACES ABSTRAITS

(Summary)

On donne la définition d'un schema général d'interpolation dans un espace linearé normé. Le travail contient aussi deux définitions de la convexité par rapport à un procédé d'interpolation generalisé. The defined interpolation method includes the Gontcharov interpolation scheme as a particular case. On donne un théorème de moyenne (théorème 3) qui a des applications dans l'étude de la structure du reste dans les procédés linéaires d'approximation. Finally we formulate a problème de la meilleure approximation.

BIBLIOGRAPHY

  1. В. L. Goncharov, Theory of interpolation and approximation of functions. Г.И.Т.Т.Л., Moscow, 1954.
  2. L. A. Люстерник, В. И. Sobolev, Elements of functional analysis. Г.И.Т.Т.Л., Moscow, 1951.
  3. E. Mo1dovan, On a generalization of the notion of convexity. Studies and Scientific Research (Cluj), VI, no. 3-4, Series Ia, 65-73 (1955).
    • On the notion of a convex function with respect to a set of interpolating functions. Studii si Cerc. de Mat. (Cluj), IX, 161-224 (1958).
  5. T. Popoviciu, Notes sur les fonctions convexes d'ardre superieur (I). Mathematica, 12, 81-92 (1936).
    • Notes sur les fonctions convexes d'ordre superieur (IX). Bull. Math. Shock. Roumaine des Sci., 43, 85-141 (1941).
  1. 1 1 ^(1){ }^{1}1) Norm of an element x ∈ V x ∈ V x in Vx \in Vx∈Vwill be denoted by the symbol ‖ v ‖ ‖ v ‖ ||v||\|v\|‖V‖.
    2 2 ^(2){ }^{2}2) Additive and homogeneous.
  2. 3 3 ^(3){ }^{3}3) Function h ( x ) h ( x ) h(x)h(x)h(x)we denote it with H ( φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ; y ∣ x ) H φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ; y ∣ x H([varphi_(1)","varphi_(2)","dots","varphi_(n)],[x_(1)","x_(2)","dots","x_(n)];y∣x)H\left(\begin{array}{l}\varphi_{1}, \varphi_{2}, \ldots, \varphi_{n} \\ x_{1}, x_{2}, \ldots, x_{n}\end{array} ; y \mid x\right)H(φ1,φ2,…,φnx1,x2,…,xn;y∣x)or if f ( x i ) = y i , i = 1 , 2 , … , n f x i = y i , i = 1 , 2 , … , n f(x_(i))=y_(i),i=1,2,dots,nf\left(x_{i}\right)=y_{i}, i=1,2, \ldots, nf(xand)=yand,and=1,2,…,n, cu H ( φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ; f ∣ x ) cu H φ 1 , φ 2 , … , φ n x 1 , x 2 , … , x n ; f ∣ x cu H([varphi_(1)","varphi_(2)","dots","varphi_(n)],[x_(1)","x_(2)","dots","x_(n)];f∣x)\operatorname{cu} H\left(\begin{array}{l}\varphi_{1}, \varphi_{2}, \ldots, \varphi_{n} \\ x_{1}, x_{2}, \ldots, x_{n}\end{array} ; f \mid x\right)withH(φ1,φ2,…,φnx1,x2,…,xn;f∣x).
  3. 4 4 ^(4){ }^{4}4) Obviously in this example, the space C C CCCcan be replaced by a more general space containing polynomials.
    5 5 ^(5){ }^{5}5) The set of all linear combinations of the functions that form the system.
    6 ) Φ ( x 1 , x 2 , … , x n − 1 ; f ∣ x ) 6 Φ x 1 , x 2 , … , x n − 1 ; f ∣ x {:^(6))Phi(x_(1),x_(2),dots,x_(n-1);f∣x)\left.{ }^{6}\right) \Phi\left(x_{1}, x_{2}, \ldots, x_{n-1} ; f \mid x\right)6)Φ(x1,x2,…,xn−1;f∣x)is the shape function ∑ i = 1 n − 1 C i φ i ( x ) ∑ i = 1 n − 1   C i φ i ( x ) sum_(i=1)^(n-1)C_(i)varphi_(i)(x)\sum_{i=1}^{n-1} C_{i} \varphi_{i}(x)∑and=1n−1Candφand(x), which on the points x i x i x_(i)x_{i}xandtake the values ​​respectively f ( x i ) , C i f x i , C i f(x_(i)),C_(i)f\left(x_{i}\right), C_{i}f(xand),Candbeing real numbers.
  4. 7 ) Φ ( x 1 , x 2 , … , x n ; f ∣ x ) 7 Φ x 1 , x 2 , … , x n ; f ∣ x {:^(7))Phi(x_(1),x_(2),dots,x_(n);f∣x)\left.{ }^{7}\right) \Phi\left(x_{1}, x_{2}, \ldots, x_{n} ; f \mid x\right)7)Φ(x1,x2,…,xn;f∣x)is the shape function ∑ i = 1 n C i φ i ( x ) ∑ i = 1 n   C i φ i ( x ) sum_(i=1)^(n)C_(i)varphi_(i)(x)\sum_{i=1}^{n} C_{i} \varphi_{i}(x)∑and=1nCandφand(x), which on the points x i x i x_(i)x_{i}xandtake the values ​​respectively f ( x i ) f x i f(x_(i))f\left(x_{i}\right)f(xand).
1959

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