On the derivative-interpolating spline functions of even degree

Costica Mustata
(original), Approximation Theory, html, paper, PDF, splines, works

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Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

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C. Mustăţa, On the derivative-interpolating spline functions of even degree, Bull. Şt. Univ. Baia Mare, Seria B, Fascicola Matematică-informatică, 14 (1998) no. 1, 51-58.

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Buletinul ştiinţific al Universitatii Baia Mare

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https://www.jstor.org/stable/44001718

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[l] Aubin, J-R, Cellina, A., Differential Inclusions, Set-Valued Maps and Viability Theory, Springer- Verlag, 1984.
[2] Blaga. P.. Micula, G.. Polynomial natural spline functions of even degree, Studia Univ. ” Babeą-Bolyai” , Mathematica XXXVIII. No.2 (1993), 31-40.
[3] Greville, T.N.E., Introduction to spline functions. Theory and Applications of Spline Functions , T.N.E. Greville, Ed. Academic Press, New York, 1969. 1-35.
[4] Lyche, T.. Schumaker L.L., Computation of Smoothing and Interpolating Natural Spline via Local Bases, SIAM J. Numer. Math. 10, No.6 (1973), 1027-1038.
[5] Micula, G., Functii spline si aplicatii, Ed. Tehnica, Bucuresti, 1978.
[6] Micula, G., Blaga, P.. Akça, H., The numerical treatment of delay differential equations with constant delay by natural spline functions of even degree, Libertas Mathematica, XVI (1996) 123-131.
[7] Mustata, R., On p-derivative-interpolating spline functions, Rev. Anal. Numér. Théor. Approx. XXVI, No. 1-2 (1997), 149-163.
[8] Mustata, C., Muresan, A., Mustata, R., The approximation by spline functions of the solution of a singular perturbed bilocal problem (to appear).

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1998-Mustata-BAM-On-the-derivative-interpolating-spline-functions
Dedicated to Professor Iulian Coroian on his 60 th 60 th  60^("th ")60^{\text {th }}60th  anniversary

On the derivative-interpolating spline functions of even degree

COSTICĂ MUSTĂTA

The aim of the present note is to show that the derivative-interpolating spline functions, considered in some recent papers ([2],[6]) are in fact primitives, chosen in an appropriate way, of interpolating natural spline functions ([3]).
This fact allows to derive some properties of derivative-interpolating spline functions of even order from the corresponding properties of interpolating natural spline functions.
Let m , n ∈ N , m ≤ n m , n ∈ N , m ≤ n m,n inN,m <= nm, n \in \mathbb{N}, m \leq nm,n∈N,m≤n, and [ a , b ] [ a , b ] [a,b][a, b][a,b] an interval contained in R R R\mathbb{R}R. Let also
(1) Δ n := a < x 1 < x 2 < ⋯ < x n < b (1) Δ n := a < x 1 < x 2 < ⋯ < x n < b {:(1)Delta_(n):=a < x_(1) < x_(2) < cdots < x_(n) < b:}\begin{equation*} \Delta_{n}:=a<x_{1}<x_{2}<\cdots<x_{n}<b \tag{1} \end{equation*}(1)Δn:=a<x1<x2<⋯<xn<b
be a fixed partition of the interval [ a , b ] [ a , b ] [a,b][a, b][a,b].
Definition 1 A function s : [ a , b ] → R s : [ a , b ] → R s:[a,b]rarrRs:[a, b] \rightarrow \mathbb{R}s:[a,b]→R verifying the conditions
1 0 s ∈ C 2 m − 2 [ a , b ] 1 0 s ∈ C 2 m − 2 [ a , b ] 1^(0)s inC^(2m-2)[a,b]1^{0} s \in C^{2 m-2}[a, b]10s∈C2m−2[a,b],
2 0 s ∈ P m − 1 2 0 s ∈ P m − 1 2^(0)s inP_(m-1)2^{0} s \in \mathcal{P}_{m-1}20s∈Pm−1 on ( a , x 1 a , x 1 a,x_(1)a, x_{1}a,x1 ) and ( x n , b x n , b x_(n),bx_{n}, bxn,b ),
3 0 s ∈ P 2 m − 1 3 0 s ∈ P 2 m − 1 3^(0)s inP_(2m-1)3^{0} s \in \mathcal{P}_{2 m-1}30s∈P2m−1 on ( x i , x i + 1 ) , i = 1 , 2 , … , n − 1 x i , x i + 1 , i = 1 , 2 , … , n − 1 (x_(i),x_(i+1)),i=1,2,dots,n-1\left(x_{i}, x_{i+1}\right), i=1,2, \ldots, n-1(xi,xi+1),i=1,2,…,n−1,
where P k P k P_(k)\mathcal{P}_{k}Pk stands for the set of polynomials of degree at most k ( k ∈ N ) k ( k ∈ N ) k(k inN)k(k \in \mathbb{N})k(k∈N), is called an interpolating natural spline function (associated to the partition Δ n Δ n Delta_(n)\Delta_{n}Δn ).
Denoting by S 2 m − 1 ( Δ n ) S 2 m − 1 Δ n S_(2m-1)(Delta_(n))\mathcal{S}_{2 m-1}\left(\Delta_{n}\right)S2m−1(Δn) the set of all interpolating natural spline functions, one sees that S 2 m − 1 ( Δ n ) S 2 m − 1 Δ n S_(2m-1)(Delta_(n))S_{2 m-1}\left(\Delta_{n}\right)S2m−1(Δn) is an n n nnn-dimensional subspace of the linear space C 2 m − 2 [ a , b ] C 2 m − 2 [ a , b ] C^(2m-2)[a,b]C^{2 m-2}[a, b]C2m−2[a,b].
If s ∈ S 2 m − 1 ( Δ n ) s ∈ S 2 m − 1 Δ n s inS_(2m-1)(Delta_(n))s \in \mathcal{S}_{2 m-1}\left(\Delta_{n}\right)s∈S2m−1(Δn) then s s sss is of the form
(2)
s ( x ) = ∑ i = 0 m − 1 B i x i + ∑ k = 1 n b k ( x − x k ) + 2 m − 1 , s ( x ) = ∑ i = 0 m − 1   B i x i + ∑ k = 1 n   b k x − x k + 2 m − 1 , s(x)=sum_(i=0)^(m-1)B_(i)x^(i)+sum_(k=1)^(n)b_(k)(x-x_(k))_(+)^(2m-1),s(x)=\sum_{i=0}^{m-1} B_{i} x^{i}+\sum_{k=1}^{n} b_{k}\left(x-x_{k}\right)_{+}^{2 m-1},s(x)=∑i=0m−1Bixi+∑k=1nbk(x−xk)+2m−1,
where
(3) ∑ k = 1 n b k x k j = 0 , j = 0 , 1 , … , m − 1 (3) ∑ k = 1 n   b k x k j = 0 , j = 0 , 1 , … , m − 1 {:(3)sum_(k=1)^(n)b_(k)x_(k)^(j)=0","j=0","1","dots","m-1:}\begin{equation*} \sum_{k=1}^{n} b_{k} x_{k}^{j}=0, j=0,1, \ldots, m-1 \tag{3} \end{equation*}(3)∑k=1nbkxkj=0,j=0,1,…,m−1
(see [3]).
If Y = ( y 1 , y 2 , … , y n ) Y = y 1 , y 2 , … , y n Y=(y_(1),y_(2),dots,y_(n))Y=\left(y_{1}, y_{2}, \ldots, y_{n}\right)Y=(y1,y2,…,yn) is a fixed vector in R n R n R^(n)\mathbb{R}^{n}Rn then there exists exactly one function s Y ∈ S 2 m − 1 ( Δ n ) s Y ∈ S 2 m − 1 Δ n s_(Y)inS_(2m-1)(Delta_(n))s_{Y} \in \mathcal{S}_{2 m-1}\left(\Delta_{n}\right)sY∈S2m−1(Δn) verifying the equalities:
(4) s Y ( x i ) = y i , i = 1 , 2 , … , n (4) s Y x i = y i , i = 1 , 2 , … , n {:(4)s_(Y)(x_(i))=y_(i)","quad i=1","2","dots","n:}\begin{equation*} s_{Y}\left(x_{i}\right)=y_{i}, \quad i=1,2, \ldots, n \tag{4} \end{equation*}(4)sY(xi)=yi,i=1,2,…,n
see ([3]).
Let
(5) H 2 m [ a , b ] := { f ∈ C m − 1 [ a , b ] : f ( m − 1 ) H 2 m [ a , b ] := f ∈ C m − 1 [ a , b ] : f ( m − 1 ) H_(2)^(m)[a,b]:={f inC^(m-1)[a,b]:f^((m-1)):}H_{2}^{m}[a, b]:=\left\{f \in C^{m-1}[a, b]: f^{(m-1)}\right.H2m[a,b]:={f∈Cm−1[a,b]:f(m−1) is absolutely continuous and f ( m ) ∈ L 2 [ a , b ] } f ( m ) ∈ L 2 [ a , b ] {:f^((m))inL_(2)[a,b]}\left.f^{(m)} \in L_{2}[a, b]\right\}f(m)∈L2[a,b]}
and
(6)
H 2 , Y m [ a , b ] := { f ∈ H 2 m [ a , b ] : f ( x i ) = y i , i = 1 , 2 , … , n } . H 2 , Y m [ a , b ] := f ∈ H 2 m [ a , b ] : f x i = y i , i = 1 , 2 , … , n . H_(2,Y)^(m)[a,b]:={f inH_(2)^(m)[a,b]:f(x_(i))=y_(i),quad i=1,2,dots,n}.H_{2, Y}^{m}[a, b]:=\left\{f \in H_{2}^{m}[a, b]: f\left(x_{i}\right)=y_{i}, \quad i=1,2, \ldots, n\right\} .H2,Ym[a,b]:={f∈H2m[a,b]:f(xi)=yi,i=1,2,…,n}.
Then
(7)
H 2 , Y m [ a , b ] ∩ S 2 m − 1 ( Δ n ) = { s Y } H 2 , Y m [ a , b ] ∩ S 2 m − 1 Δ n = s Y H_(2,Y)^(m)[a,b]nnS_(2m-1)(Delta_(n))={s_(Y)}H_{2, Y}^{m}[a, b] \cap \mathcal{S}_{2 m-1}\left(\Delta_{n}\right)=\left\{s_{Y}\right\}H2,Ym[a,b]∩S2m−1(Δn)={sY}
and the functional J : H 2 , Y m [ a , b ] → R + J : H 2 , Y m [ a , b ] → R + J:H_(2,Y)^(m)[a,b]rarrR_(+)J: H_{2, Y}^{m}[a, b] \rightarrow \mathbb{R}_{+}J:H2,Ym[a,b]→R+given by
(8) J ( f ) = ∫ a b [ f ( m ) ( x ) ] 2 d x , f ∈ H 2 , Y m [ a , b ] (8) J ( f ) = ∫ a b   f ( m ) ( x ) 2 d x , f ∈ H 2 , Y m [ a , b ] {:(8)J(f)=int_(a)^(b)[f^((m))(x)]^(2)dx","quad f inH_(2,Y)^(m)[a","b]:}\begin{equation*} J(f)=\int_{a}^{b}\left[f^{(m)}(x)\right]^{2} d x, \quad f \in H_{2, Y}^{m}[a, b] \tag{8} \end{equation*}(8)J(f)=∫ab[f(m)(x)]2dx,f∈H2,Ym[a,b]
has the property
(9) min { J ( f ) : f ∈ H 2 , Y m [ a , b ] } = J ( s Y ) (9) min J ( f ) : f ∈ H 2 , Y m [ a , b ] = J s Y {:(9)min{J(f):f inH_(2,Y)^(m)[a,b]}=J(s_(Y)):}\begin{equation*} \min \left\{J(f): f \in H_{2, Y}^{m}[a, b]\right\}=J\left(s_{Y}\right) \tag{9} \end{equation*}(9)min{J(f):f∈H2,Ym[a,b]}=J(sY)
i.e. the minimum of the L 2 L 2 L_(2)L_{2}L2-norms of the derivatives of order m m mmm of the functions in H 2 , Y m [ a , b ] H 2 , Y m [ a , b ] H_(2,Y)^(m)[a,b]H_{2, Y}^{m}[a, b]H2,Ym[a,b] is attained at the interpolating natural spline function s Y s Y s_(Y)s_{Y}sY ( "the minimal norm property").
Also, for each f ∈ H 2 , Y m [ a , b ] f ∈ H 2 , Y m [ a , b ] f inH_(2,Y)^(m)[a,b]f \in H_{2, Y}^{m}[a, b]f∈H2,Ym[a,b] the inequality
(10) ‖ f ( m ) − s Y ( m ) ‖ 2 ≤ ‖ f ( m ) − s ‖ 2 (10) f ( m ) − s Y ( m ) 2 ≤ f ( m ) − s 2 {:(10)||f^((m))-s_(Y)^((m))||_(2) <= ||f^((m))-s||_(2):}\begin{equation*} \left\|f^{(m)}-s_{Y}^{(m)}\right\|_{2} \leq\left\|f^{(m)}-s\right\|_{2} \tag{10} \end{equation*}(10)‖f(m)−sY(m)‖2≤‖f(m)−s‖2
holds for any s ∈ S 2 m − 1 ( Δ n ) s ∈ S 2 m − 1 Δ n s inS_(2m-1)(Delta_(n))s \in \mathcal{S}_{2 m-1}\left(\Delta_{n}\right)s∈S2m−1(Δn) ( "the best approximation property") (see [3]).
Now, we shall introduce the derivative-interpolating spline functions of even order 2 m 2 m 2m2 m2m, having properties similar to (9) and (10). (see [2]).
Let m , n ∈ N , m ≤ n + 1 m , n ∈ N , m ≤ n + 1 m,n inN,m <= n+1m, n \in \mathbb{N}, m \leq n+1m,n∈N,m≤n+1, and Δ n Δ n Delta_(n)\Delta_{n}Δn the partition (1) of the interval [ a , b ] [ a , b ] [a,b][a, b][a,b].
Definition 2 ([2]). A function S : [ a , b ] → R S : [ a , b ] → R S:[a,b]rarrRS:[a, b] \rightarrow \mathbb{R}S:[a,b]→R is called a natural spline function of order 2 m 2 m 2m2 m2m if it verifies the conditions:
1 0 S ∈ C 2 m − 1 [ a , b ] 2 0 S ∈ P m on ( a , x 1 ) and ( x n , b ) 3 0 S ∈ P 2 m on ( x i , x i + 1 ) , i = 1 , 2 , … , n − 1 1 0 S ∈ C 2 m − 1 [ a , b ] 2 0 S ∈ P m  on  a , x 1  and  x n , b 3 0 S ∈ P 2 m  on  x i , x i + 1 , i = 1 , 2 , … , n − 1 {:[1^(0)S inC^(2m-1)[a","b]],[2^(0)S inP_(m)" on "(a,x_(1))" and "(x_(n),b)],[3^(0)S inP_(2m)" on "(x_(i),x_(i+1))","i=1","2","dots","n-1]:}\begin{aligned} & 1^{0} S \in C^{2 m-1}[a, b] \\ & 2^{0} S \in \mathcal{P}_{m} \text { on }\left(a, x_{1}\right) \text { and }\left(x_{n}, b\right) \\ & 3^{0} S \in \mathcal{P}_{2 m} \text { on }\left(x_{i}, x_{i+1}\right), i=1,2, \ldots, n-1 \end{aligned}10S∈C2m−1[a,b]20S∈Pm on (a,x1) and (xn,b)30S∈P2m on (xi,xi+1),i=1,2,…,n−1
The set of all interpolating natural spline functions of order 2 m 2 m 2m2 m2m will be denoted by S 2 m ( Δ n ) S 2 m Δ n S_(2m)(Delta_(n))\mathcal{S}_{2 m}\left(\Delta_{n}\right)S2m(Δn). It follows that S 2 m ( Δ n ) S 2 m Δ n S_(2m)(Delta_(n))\mathcal{S}_{2 m}\left(\Delta_{n}\right)S2m(Δn) is an ( n + 1 n + 1 n+1n+1n+1 )-dimensional subspace of C 2 m − 1 [ a , b ] C 2 m − 1 [ a , b ] C^(2m-1)[a,b]C^{2 m-1}[a, b]C2m−1[a,b]. (see [6]).
If Y ¯ = ( y α , y 1 , … , y n ) Y ¯ = y α , y 1 , … , y n bar(Y)=(y_(alpha),y_(1),dots,y_(n))\bar{Y}=\left(y_{\alpha}, y_{1}, \ldots, y_{n}\right)Y¯=(yα,y1,…,yn) is a fixed vector in R n + 1 R n + 1 R^(n+1)\mathbb{R}^{n+1}Rn+1 then there exists only one function S Y ¯ ∈ S 2 m ( Δ n ) S Y ¯ ∈ S 2 m Δ n S_( bar(Y))inS_(2m)(Delta_(n))S_{\bar{Y}} \in S_{2 m}\left(\Delta_{n}\right)SY¯∈S2m(Δn) verifying the conditions
(11) S Y ¯ ( α ) = y α S Y ¯ ( x i ) = y i , i = 1 , 2 , … , n (11) S Y ¯ ( α ) = y α S Y ¯ x i = y i , i = 1 , 2 , … , n {:[(11)S_( bar(Y))(alpha)=y_(alpha)],[S_( bar(Y))(x_(i))=y_(i)","quad i=1","2","dots","n]:}\begin{align*} S_{\bar{Y}}(\alpha) & =y_{\alpha} \tag{11}\\ S_{\bar{Y}}\left(x_{i}\right) & =y_{i}, \quad i=1,2, \ldots, n \end{align*}(11)SY¯(α)=yαSY¯(xi)=yi,i=1,2,…,n
where x i , i ≐ 1 , n ― x i , i ≐ 1 , n ¯ x_(i),i≐ bar(1,n)x_{i}, i \doteq \overline{1, n}xi,i≐1,n― are the nodes of the partition Δ n Δ n Delta_(n)\Delta_{n}Δn given by (??).
The function S Y ¯ ∈ S 2 m ( Δ n ) S Y ¯ ∈ S 2 m Δ n S_( bar(Y))inS_(2m)(Delta_(n))S_{\bar{Y}} \in S_{2 m}\left(\Delta_{n}\right)SY¯∈S2m(Δn) verifying the condition (??) is called the derivative-interpolating spline of even order 2 m 2 m 2m2 m2m associated to the vector Y ¯ Y ¯ bar(Y)\bar{Y}Y¯ and to the partition Δ n Δ n Delta_(n)\Delta_{n}Δn.
Any function S ∈ S 2 m ( Δ n ) S ∈ S 2 m Δ n S inS_(2m)(Delta_(n))S \in S_{2 m}\left(\Delta_{n}\right)S∈S2m(Δn) admits the representation
(12) S ( x ) = ∑ i = 0 m A i x i + ∑ k = 1 n a k ( x − x k ) + 2 m (13) ∑ k = 0 n a k x k j = 0 , j = 0 , 1 , … , m − 1 (12) S ( x ) = ∑ i = 0 m   A i x i + ∑ k = 1 n   a k x − x k + 2 m (13) ∑ k = 0 n   a k x k j = 0 , j = 0 , 1 , … , m − 1 {:[(12)S(x)=sum_(i=0)^(m)A_(i)x^(i)+sum_(k=1)^(n)a_(k)(x-x_(k))_(+)^(2m)],[(13)sum_(k=0)^(n)a_(k)x_(k)^(j)=0","quad j=0","1","dots","m-1]:}\begin{gather*} S(x)=\sum_{i=0}^{m} A_{i} x^{i}+\sum_{k=1}^{n} a_{k}\left(x-x_{k}\right)_{+}^{2 m} \tag{12}\\ \sum_{k=0}^{n} a_{k} x_{k}^{j}=0, \quad j=0,1, \ldots, m-1 \tag{13} \end{gather*}(12)S(x)=∑i=0mAixi+∑k=1nak(x−xk)+2m(13)∑k=0nakxkj=0,j=0,1,…,m−1
see ([2] or [6]).
Let
(14) H 2 m + 1 [ a , b ] : = { f ∈ C m [ a , b ] , f ( m ) is absolutely continuous and f ( m + 1 ) ∈ L 2 [ a , b ] } (14) H 2 m + 1 [ a , b ] : = f ∈ C m [ a , b ] , f ( m )  is absolutely continuous   and  f ( m + 1 ) ∈ L 2 [ a , b ] {:[(14)H_(2)^(m+1)[a","b]:={f inC^(m)[a,b],f^((m)):}" is absolutely continuous "],[" and "{:f^((m+1))inL_(2)[a,b]}]:}\begin{align*} H_{2}^{m+1}[a, b]: & =\left\{f \in C^{m}[a, b], f^{(m)}\right. \text { is absolutely continuous } \tag{14}\\ & \text { and } \left.f^{(m+1)} \in L_{2}[a, b]\right\} \end{align*}(14)H2m+1[a,b]:={f∈Cm[a,b],f(m) is absolutely continuous  and f(m+1)∈L2[a,b]}
and
(15) H 2 , Y ¯ m + 1 := { g ∈ H 2 m + 1 [ a , b ] : g ( α ) = y α and g ′ ( x i ) = y i , i = 1 , 2 , … , n } (15) H 2 , Y ¯ m + 1 := g ∈ H 2 m + 1 [ a , b ] : g ( α ) = y α  and  g ′ x i = y i , i = 1 , 2 , … , n {:[(15)H_(2, bar(Y))^(m+1):={g inH_(2)^(m+1)[a,b]:g(alpha)=y_(alpha):}" and "],[{:g^(')(x_(i))=y_(i),i=1,2,dots,n}]:}\begin{align*} H_{2, \bar{Y}}^{m+1}:= & \left\{g \in H_{2}^{m+1}[a, b]: g(\alpha)=y_{\alpha}\right. \text { and } \tag{15}\\ & \left.g^{\prime}\left(x_{i}\right)=y_{i}, i=1,2, \ldots, n\right\} \end{align*}(15)H2,Y¯m+1:={g∈H2m+1[a,b]:g(α)=yα and g′(xi)=yi,i=1,2,…,n}
Then (see [2])
(16) H 2 , Y ¯ m + 1 [ a , b ] ∩ S 2 m ( Δ n ) = { S Y ¯ } (16) H 2 , Y ¯ m + 1 [ a , b ] ∩ S 2 m Δ n = S Y ¯ {:(16)H_(2, bar(Y))^(m+1)[a","b]nnS_(2m)(Delta_(n))={S_( bar(Y))}:}\begin{equation*} H_{2, \bar{Y}}^{m+1}[a, b] \cap S_{2 m}\left(\Delta_{n}\right)=\left\{S_{\bar{Y}}\right\} \tag{16} \end{equation*}(16)H2,Y¯m+1[a,b]∩S2m(Δn)={SY¯}
and the functional J α : H 2 , Y ¯ m + 1 [ a , b ] → R ; J α : H 2 , Y ¯ m + 1 [ a , b ] → R ; J_(alpha):H_(2, bar(Y))^(m+1)[a,b]rarrR_(;)J_{\alpha}: H_{2, \bar{Y}}^{m+1}[a, b] \rightarrow \mathbb{R}_{;}Jα:H2,Y¯m+1[a,b]→R;defined by
(17) J α ( g ) = ∫ a b [ g ( m + 1 ) ( x ) ] 2 d x (17) J α ( g ) = ∫ a b   g ( m + 1 ) ( x ) 2 d x {:(17)J_(alpha)(g)=int_(a)^(b)[g^((m+1))(x)]^(2)dx:}\begin{equation*} J_{\alpha}(g)=\int_{a}^{b}\left[g^{(m+1)}(x)\right]^{2} d x \tag{17} \end{equation*}(17)Jα(g)=∫ab[g(m+1)(x)]2dx
attains its minimum at the function S Y ¯ S Y ¯ S_( bar(Y))S_{\bar{Y}}SY¯ :
(18) min { J α ( g ) : g ∈ I 2 , Y ¯ m + 1 [ a , b ] } = J α ( S Y ¯ ) (18) min J α ( g ) : g ∈ I 2 , Y ¯ m + 1 [ a , b ] = J α S Y ¯ {:(18)min{J_(alpha)(g):g inI_(2, bar(Y))^(m+1)[a,b]}=J_(alpha)(S_( bar(Y))):}\begin{equation*} \min \left\{J_{\alpha}(g): g \in I_{2, \bar{Y}}^{m+1}[a, b]\right\}=J_{\alpha}\left(S_{\bar{Y}}\right) \tag{18} \end{equation*}(18)min{Jα(g):g∈I2,Y¯m+1[a,b]}=Jα(SY¯)
Also, the inequality
(19) ‖ g ( m + 1 ) − S Y ¯ ( m + 1 ) ‖ 2 ≤ ‖ g m + 1 − S ( m + 1 ) ‖ 2 (19) g ( m + 1 ) − S Y ¯ ( m + 1 ) 2 ≤ g m + 1 − S ( m + 1 ) 2 {:(19)||g^((m+1))-S_( bar(Y))^((m+1))||_(2) <= ||g^(m+1)-S^((m+1))||_(2):}\begin{equation*} \left\|g^{(m+1)}-S_{\bar{Y}}^{(m+1)}\right\|_{2} \leq\left\|g^{m+1}-S^{(m+1)}\right\|_{2} \tag{19} \end{equation*}(19)‖g(m+1)−SY¯(m+1)‖2≤‖gm+1−S(m+1)‖2
holds for any S ∈ S 2 m ( Δ m ) S ∈ S 2 m Δ m S inS_(2m)(Delta_(m))S \in \mathcal{S}_{2 m}\left(\Delta_{m}\right)S∈S2m(Δm).
The relations (18) and (19) (called "the minimal norm property" and "the best approximation property", respectively) are proved in [2], following a way similar to that used to prove the corresponding properties for interpolating natural spline functions (see [6], Theorems 3 and 4).
We mention that the derivative-interpulating spline functions of order 2 m 2 m 2m2 m2m have been successfully used for the numerical solution of boundary value problems (Cauchy problems) for differential equations with modified argument ([6]). Spline functions of degree 5 (particular cases of p p ppp-derivativeinterpolating spline functions for p = 2 p = 2 p=2p=2p=2 and m = 2 m = 2 m=2m=2m=2 ) were used in [8] to solve a singularly perturbed bilocal problem.
In the next we shall show that the functions used in [8] are spline functions obtained by integrating the interpolation natural cubic spline functions.
Lemma 3 Let s ∈ S 2 m − 1 ( Δ n ) , α ∈ [ a , b ] s ∈ S 2 m − 1 Δ n , α ∈ [ a , b ] s inS_(2m-1)(Delta_(n)),alpha in[a,b]s \in \mathcal{S}_{2 m-1}\left(\Delta_{n}\right), \alpha \in[a, b]s∈S2m−1(Δn),α∈[a,b] fixed and
(20) I ^ ( s ) := { ∫ a x s ( t ) d t + C : C ∈ R } (20) I ^ ( s ) := ∫ a x   s ( t ) d t + C : C ∈ R {:(20) hat(I)(s):={int_(a)^(x)s(t)dt+C:C inR}:}\begin{equation*} \hat{I}(s):=\left\{\int_{a}^{x} s(t) d t+C: C \in \mathbb{R}\right\} \tag{20} \end{equation*}(20)I^(s):={∫axs(t)dt+C:C∈R}
Then every S ∈ I ^ ( s ) S ∈ I ^ ( s ) S in hat(I)(s)S \in \hat{I}(s)S∈I^(s) belongs to S 2 m ( Δ n ) S 2 m Δ n S_(2m)(Delta_(n))S_{2 m}\left(\Delta_{n}\right)S2m(Δn).
Proof. By (2)
s ( x ) = ∑ i = 0 m − 1 B i x i + ∑ k = 1 n b k ( x − x k ) + 2 m − 1 s ( x ) = ∑ i = 0 m − 1   B i x i + ∑ k = 1 n   b k x − x k + 2 m − 1 s(x)=sum_(i=0)^(m-1)B_(i)x_(i)+sum_(k=1)^(n)b_(k)(x-x_(k))_(+)^(2m-1)s(x)=\sum_{i=0}^{m-1} B_{i} x_{i}+\sum_{k=1}^{n} b_{k}\left(x-x_{k}\right)_{+}^{2 m-1}s(x)=∑i=0m−1Bixi+∑k=1nbk(x−xk)+2m−1
with
∑ k = 1 n b k x k j = 0 , j = 0 , 1 , 2 , … , m − 1 ∑ k = 1 n   b k x k j = 0 , j = 0 , 1 , 2 , … , m − 1 sum_(k=1)^(n)b_(k)x_(k)^(j)=0,quad j=0,1,2,dots,m-1\sum_{k=1}^{n} b_{k} x_{k}^{j}=0, \quad j=0,1,2, \ldots, m-1∑k=1nbkxkj=0,j=0,1,2,…,m−1
Consequently
S ( x ) = ∫ α x s ( t ) d t + C 0 = C 0 + ∑ i = 0 m − 1 B i i + 1 x i + 1 + ∑ i = 1 n b k 2 f 1 ( x − x k ) + 2 m = = ∑ i = 0 m A i x i + ∑ k = 1 n a k ( x − x k ) + 2 m S ( x ) = ∫ α x   s ( t ) d t + C 0 = C 0 + ∑ i = 0 m − 1   B i i + 1 x i + 1 + ∑ i = 1 n   b k 2 f 1 x − x k + 2 m = = ∑ i = 0 m   A i x i + ∑ k = 1 n   a k x − x k + 2 m {:[S(x)=int_(alpha)^(x)s(t)dt+C_(0)=C_(0)+sum_(i=0)^(m-1)(B_(i))/(i+1)x^(i+1)+sum_(i=1)^(n)(b_(k))/(2f_(1))(x-x_(k))_(+)^(2m)=],[=sum_(i=0)^(m)A_(i)x^(i)+sum_(k=1)^(n)a_(k)(x-x_(k))_(+)^(2m)]:}\begin{aligned} S(x) & =\int_{\alpha}^{x} s(t) d t+C_{0}=C_{0}+\sum_{i=0}^{m-1} \frac{B_{i}}{i+1} x^{i+1}+\sum_{i=1}^{n} \frac{b_{k}}{2 f_{1}}\left(x-x_{k}\right)_{+}^{2 m}= \\ & =\sum_{i=0}^{m} A_{i} x^{i}+\sum_{k=1}^{n} a_{k}\left(x-x_{k}\right)_{+}^{2 m} \end{aligned}S(x)=∫αxs(t)dt+C0=C0+∑i=0m−1Bii+1xi+1+∑i=1nbk2f1(x−xk)+2m==∑i=0mAixi+∑k=1nak(x−xk)+2m
where A 0 = C 1 , C 1 = C 0 − ∑ 1 = 0 m − 1 B i i + 1 α i + 1 − ∑ k = 1 n b k 2 m ( α − x k ) + 2 m , A i = B i − 1 i A 0 = C 1 , C 1 = C 0 − ∑ 1 = 0 m − 1   B i i + 1 α i + 1 − ∑ k = 1 n   b k 2 m α − x k + 2 m , A i = B i − 1 i A_(0)=C_(1),C_(1)=C_(0)-sum_(1=0)^(m-1)(B_(i))/(i+1)alpha^(i+1)-sum_(k=1)^(n)(b_(k))/(2m)(alpha-x_(k))_(+)^(2m),A_(i)=(B_(i-1))/(i)A_{0}=C_{1}, C_{1}=C_{0}-\sum_{1=0}^{m-1} \frac{B_{i}}{i+1} \alpha^{i+1}-\sum_{k=1}^{n} \frac{b_{k}}{2 m}\left(\alpha-x_{k}\right)_{+}^{2 m}, A_{i}=\frac{B_{i-1}}{i}A0=C1,C1=C0−∑1=0m−1Bii+1αi+1−∑k=1nbk2m(α−xk)+2m,Ai=Bi−1i,
i = 1 , 2 , … , m ; a k = b k 2 m , k = 1 , 2 , … , n i = 1 , 2 , … , m ; a k = b k 2 m , k = 1 , 2 , … , n i=1,2,dots,m;a_(k)=(b_(k))/(2m),k=1,2,dots,ni=1,2, \ldots, m ; a_{k}=\frac{b_{k}}{2 m}, k=1,2, \ldots, ni=1,2,…,m;ak=bk2m,k=1,2,…,n and ∑ k = 1 n a k x k j = 0 , j = 0 , 1 , … , m − 1 ∑ k = 1 n   a k x k j = 0 , j = 0 , 1 , … , m − 1 sum_(k=1)^(n)a_(k)x_(k)^(j)=0,j=0,1,dots,m-1\sum_{k=1}^{n} a_{k} x_{k}^{j}=0, j=0,1, \ldots, m-1∑k=1nakxkj=0,j=0,1,…,m−1.
Taking into account (12) and (13) it follows S ∈ S 2 m ( Δ n ) S ∈ S 2 m Δ n S inS_(2m)(Delta_(n))S \in \mathcal{S}_{2 m}\left(\Delta_{n}\right)S∈S2m(Δn).
Lemma 4 Let f ∈ H 2 m [ a , b ] f ∈ H 2 m [ a , b ] f inH_(2)^(m)[a,b]f \in H_{2}^{m}[a, b]f∈H2m[a,b] and
(21) I ^ ( f ) : { ∫ α x f ( t ) d t + c : c ∈ R } (21) I ^ ( f ) : ∫ α x   f ( t ) d t + c : c ∈ R {:(21) hat(I)(f):{int_(alpha)^(x)f(t)dt+c:c inR}:}\begin{equation*} \hat{I}(f):\left\{\int_{\alpha}^{x} f(t) d t+c: c \in \mathbb{R}\right\} \tag{21} \end{equation*}(21)I^(f):{∫αxf(t)dt+c:c∈R}
Then g ∈ I ^ ( f ) g ∈ I ^ ( f ) g in hat(I)(f)g \in \hat{I}(f)g∈I^(f) if and only if g ∈ H 2 m + 1 [ a , b ] g ∈ H 2 m + 1 [ a , b ] g inH_(2)^(m+1)[a,b]g \in H_{2}^{m+1}[a, b]g∈H2m+1[a,b].
Proof. Obviously that I ^ ( f ) ⊂ C m [ a , b ] I ^ ( f ) ⊂ C m [ a , b ] hat(I)(f)subC^(m)[a,b]\hat{I}(f) \subset C^{m}[a, b]I^(f)⊂Cm[a,b], and if g ∈ I ^ ( f ) g ∈ I ^ ( f ) g in hat(I)(f)g \in \hat{I}(f)g∈I^(f) then g ( m ) = f ( m − 1 ) g ( m ) = f ( m − 1 ) g^((m))=f^((m-1))g^{(m)}= f^{(m-1)}g(m)=f(m−1) (absolutely continuous on [ a , b ] [ a , b ] [a,b][a, b][a,b] ) and g ( m + 1 ) = f ( m ) ∈ L 2 [ a , b ] g ( m + 1 ) = f ( m ) ∈ L 2 [ a , b ] g^((m+1))=f^((m))inL_(2)[a,b]g^{(m+1)}=f^{(m)} \in L_{2}[a, b]g(m+1)=f(m)∈L2[a,b], showing that g ∈ H 2 m + 1 [ a , b ] g ∈ H 2 m + 1 [ a , b ] g inH_(2)^(m+1)[a,b]g \in H_{2}^{m+1}[a, b]g∈H2m+1[a,b].
If g ∈ H 2 m + 1 [ a , b ] g ∈ H 2 m + 1 [ a , b ] g inH_(2)^(m+1)[a,b]g \in H_{2}^{m+1}[a, b]g∈H2m+1[a,b] then g ′ ∈ H 2 m [ a , b ] g ′ ∈ H 2 m [ a , b ] g^(')inH_(2)^(m)[a,b]g^{\prime} \in H_{2}^{m}[a, b]g′∈H2m[a,b] so that
g ( x ) = ∫ α x g ′ ( t ) d t + g ( α ) g ( x ) = ∫ α x   g ′ ( t ) d t + g ( α ) g(x)=int_(alpha)^(x)g^(')(t)dt+g(alpha)g(x)=\int_{\alpha}^{x} g^{\prime}(t) d t+g(\alpha)g(x)=∫αxg′(t)dt+g(α)
i.e. g ∈ I ^ ( g ′ ) g ∈ I ^ g ′ g in hat(I)(g^('))g \in \hat{I}\left(g^{\prime}\right)g∈I^(g′).
Lemma 5 Let. Y = ( y 1 , y 2 , … , y n ) ∈ R n Y = y 1 , y 2 , … , y n ∈ R n Y=(y_(1),y_(2),dots,y_(n))inR^(n)Y=\left(y_{1}, y_{2}, \ldots, y_{n}\right) \in \mathbb{R}^{n}Y=(y1,y2,…,yn)∈Rn and Y ¯ = ( y α , Y ) = ( y α , y 1 , y 2 , … , y n ) ∈ R n + 1 Y ¯ = y α , Y = y α , y 1 , y 2 , … , y n ∈ R n + 1 bar(Y)=(y_(alpha),Y)=(y_(alpha),y_(1),y_(2),dots,y_(n))inR^(n+1)\bar{Y}=\left(y_{\alpha}, Y\right)=\left(y_{\alpha}, y_{1}, y_{2}, \ldots, y_{n}\right) \in \mathbf{R}^{n+1}Y¯=(yα,Y)=(yα,y1,y2,…,yn)∈Rn+1. Then the operator
I α : H 2 , Y m [ a , b ] → H 2 , Y m + 1 [ a , b ] I α : H 2 , Y m [ a , b ] → H 2 , Y m + 1 [ a , b ] I_(alpha):H_(2,Y)^(m)[a,b]rarrH_(2,Y)^(m+1)[a,b]I_{\alpha}: H_{2, Y}^{m}[a, b] \rightarrow H_{2, Y}^{m+1}[a, b]Iα:H2,Ym[a,b]→H2,Ym+1[a,b]
defined by
(22) I α ( f ) ( x ) = ∫ α x f ( t ) d t + y α , x ∈ [ a , b ] (22) I α ( f ) ( x ) = ∫ α x   f ( t ) d t + y α , x ∈ [ a , b ] {:(22)I_(alpha)(f)(x)=int_(alpha)^(x)f(t)dt+y_(alpha)","quad x in[a","b]:}\begin{equation*} I_{\alpha}(f)(x)=\int_{\alpha}^{x} f(t) d t+y_{\alpha}, \quad x \in[a, b] \tag{22} \end{equation*}(22)Iα(f)(x)=∫αxf(t)dt+yα,x∈[a,b]
is bijective.
Proof. Obviously that
I α ′ ( f ) ( x i ) = f ( x i ) = y i , i = 1 , 2 , … , n I α ( f ) ( α ) = y α I α ′ ( f ) x i = f x i = y i , i = 1 , 2 , … , n I α ( f ) ( α ) = y α {:[I_(alpha)^(')(f)(x_(i))=f(x_(i))=y_(i)","quad i=1","2","dots","n],[I_(alpha)(f)(alpha)=y_(alpha)]:}\begin{aligned} I_{\alpha}^{\prime}(f)\left(x_{i}\right) & =f\left(x_{i}\right)=y_{i}, \quad i=1,2, \ldots, n \\ I_{\alpha}(f)(\alpha) & =y_{\alpha} \end{aligned}Iα′(f)(xi)=f(xi)=yi,i=1,2,…,nIα(f)(α)=yα
showing that I α ( f ) ∈ H 2 , Y ¯ m + 1 [ a , b ] I α ( f ) ∈ H 2 , Y ¯ m + 1 [ a , b ] I_(alpha)(f)inH_(2, bar(Y))^(m+1)[a,b]I_{\alpha}(f) \in H_{2, \bar{Y}}^{m+1}[a, b]Iα(f)∈H2,Y¯m+1[a,b], for every f ∈ H 2 , Y m [ a , b ] f ∈ H 2 , Y m [ a , b ] f inH_(2,Y)^(m)[a,b]f \in H_{2, Y}^{m}[a, b]f∈H2,Ym[a,b].
If f 1 , f 2 ∈ H 2 , Y m + 1 [ a , b ] f 1 , f 2 ∈ H 2 , Y m + 1 [ a , b ] f_(1),f_(2)inH_(2,Y)^(m+1)[a,b]f_{1}, f_{2} \in H_{2, Y}^{m+1}[a, b]f1,f2∈H2,Ym+1[a,b] and I α ( f 1 ) = I α ( f 2 ) I α f 1 = I α f 2 I_(alpha)(f_(1))=I_(alpha)(f_(2))I_{\alpha}\left(f_{1}\right)=I_{\alpha}\left(f_{2}\right)Iα(f1)=Iα(f2) then
∫ α x f 1 ( t ) d t = ∫ α x f 2 ( t ) d t ∫ α x   f 1 ( t ) d t = ∫ α x   f 2 ( t ) d t int_(alpha)^(x)f_(1)(t)dt=int_(alpha)^(x)f_(2)(t)dt\int_{\alpha}^{x} f_{1}(t) d t=\int_{\alpha}^{x} f_{2}(t) d t∫αxf1(t)dt=∫αxf2(t)dt
for all x ∈ [ a , b ] x ∈ [ a , b ] x in[a,b]x \in[a, b]x∈[a,b], implying f 1 ( t ) = f 2 ( t ) f 1 ( t ) = f 2 ( t ) f_(1)(t)=f_(2)(t)f_{1}(t)=f_{2}(t)f1(t)=f2(t) for all t ∈ [ a , b ] t ∈ [ a , b ] t in[a,b]t \in[a, b]t∈[a,b], i.e.
I α I α I_(alpha)I_{\alpha}Iα is injective.
Let g ∈ H 2 , Y ¯ m + 1 [ a , b ] g ∈ H 2 , Y ¯ m + 1 [ a , b ] g inH_(2, bar(Y))^(m+1)[a,b]g \in H_{2, \bar{Y}}^{m+1}[a, b]g∈H2,Y¯m+1[a,b]. Then g ′ ∈ H 2 , Y m [ a , b ] g ′ ∈ H 2 , Y m [ a , b ] g^(')inH_(2,Y)^(m)[a,b]g^{\prime} \in H_{2, Y}^{m}[a, b]g′∈H2,Ym[a,b] and I α ( f ) = g I α ( f ) = g I_(alpha)(f)=gI_{\alpha}(f)=gIα(f)=g, for f = g ′ f = g ′ f=g^(')f=g^{\prime}f=g′, showing that I α I α I_(alpha)I_{\alpha}Iα is surjective, too.
Lemma 6 I α ( s Y ) = S Y ¯ 6 I α s Y = S Y ¯ 6I_(alpha)(s_(Y))=S_( bar(Y))6 I_{\alpha}\left(s_{Y}\right)=S_{\bar{Y}}6Iα(sY)=SY¯
Proof. By Lemmas 1 and 2.
I a ( s Y ) ∈ S 2 m ( Δ n ) ∩ H 2 m + 1 ( Δ n ) = { S Y ¯ } I a s Y ∈ S 2 m Δ n ∩ H 2 m + 1 Δ n = S Y ¯ I_(a)(s_(Y))inS_(2m)(Delta_(n))nnH_(2)^(m+1)(Delta_(n))={S_( bar(Y))}I_{a}\left(s_{Y}\right) \in \mathcal{S}_{2 m}\left(\Delta_{n}\right) \cap H_{2}^{m+1}\left(\Delta_{n}\right)=\left\{S_{\bar{Y}}\right\}Ia(sY)∈S2m(Δn)∩H2m+1(Δn)={SY¯}
so that
I α ( s Y ) = S Y ¯ I α s Y = S Y ¯ I_(alpha)(s_(Y))=S_( bar(Y))I_{\alpha}\left(s_{Y}\right)=S_{\bar{Y}}Iα(sY)=SY¯
Lemma 7. J α = J ∘ I α − 1 J α = J ∘ I α − 1 J_(alpha)=J@I_(alpha)^(-1)J_{\alpha}=J \circ I_{\alpha}^{-1}Jα=J∘Iα−1
Proof. For g ∈ I 2 , Y ¯ m + 1 [ a , b ] g ∈ I 2 , Y ¯ m + 1 [ a , b ] g inI_(2, bar(Y))^(m+1)[a,b]g \in I_{2, \bar{Y}}^{m+1}[a, b]g∈I2,Y¯m+1[a,b] we have
J α ( g ) = ∫ a b [ g ( m + 1 ) ( x ) ] 2 d x = ∫ a b [ ( g ′ ) ( m ) ( x ) ] 2 d x = = ∫ a b ( [ I α − 1 ( g ) ( x ) ] ( m ) ) 2 d x = J ( I α − 1 ( g ) ) = ( J ∘ I α − 1 ) ( g ) J α ( g ) = ∫ a b   g ( m + 1 ) ( x ) 2 d x = ∫ a b   g ′ ( m ) ( x ) 2 d x = = ∫ a b   I α − 1 ( g ) ( x ) ( m ) 2 d x = J I α − 1 ( g ) = J ∘ I α − 1 ( g ) {:[J_(alpha)(g)=int_(a)^(b)[g^((m+1))(x)]^(2)dx=int_(a)^(b)[(g^('))^((m))(x)]^(2)dx=],[=int_(a)^(b)([I_(alpha)^(-1)(g)(x)]^((m)))^(2)dx=J(I_(alpha)^(-1)(g))=(J@I_(alpha)^(-1))(g)]:}\begin{aligned} J_{\alpha}(g) & =\int_{a}^{b}\left[g^{(m+1)}(x)\right]^{2} d x=\int_{a}^{b}\left[\left(g^{\prime}\right)^{(m)}(x)\right]^{2} d x= \\ & =\int_{a}^{b}\left(\left[I_{\alpha}^{-1}(g)(x)\right]^{(m)}\right)^{2} d x=J\left(I_{\alpha}^{-1}(g)\right)=\left(J \circ I_{\alpha}^{-1}\right)(g) \end{aligned}Jα(g)=∫ab[g(m+1)(x)]2dx=∫ab[(g′)(m)(x)]2dx==∫ab([Iα−1(g)(x)](m))2dx=J(Iα−1(g))=(J∘Iα−1)(g)
Theorem 8 a) If the functional J : H 2 , Y m [ a , b ] → R + J : H 2 , Y m [ a , b ] → R + J:H_(2,Y)^(m)[a,b]rarrR_(+)J: H_{2, Y}^{m}[a, b] \rightarrow \mathbb{R}_{+}J:H2,Ym[a,b]→R+attains its minimal value at the spline function s Y ∈ H 2 , Y m [ a , b ] ∩ S 2 m − 1 ( Δ n ) s Y ∈ H 2 , Y m [ a , b ] ∩ S 2 m − 1 Δ n s_(Y)inH_(2,Y)^(m)[a,b]nnS_(2m-1)(Delta_(n))s_{Y} \in H_{2, Y}^{m}[a, b] \cap \mathcal{S}_{2 m-1}\left(\Delta_{n}\right)sY∈H2,Ym[a,b]∩S2m−1(Δn) then the functional J α : H 2 , Y ¯ m + 1 [ a , b ] → R + J α : H 2 , Y ¯ m + 1 [ a , b ] → R + J_(alpha):H_(2, bar(Y))^(m+1)[a,b]rarrR_(+)J_{\alpha}: H_{2, \bar{Y}}^{m+1}[a, b] \rightarrow \mathbb{R}_{+}Jα:H2,Y¯m+1[a,b]→R+attains its minimal value at S Y ¯ ∈ H 2 , Y ¯ m + 1 [ a , b ] ∩ S 2 m ( Δ n ) S Y ¯ ∈ H 2 , Y ¯ m + 1 [ a , b ] ∩ S 2 m Δ n S_( bar(Y))inH_(2, bar(Y))^(m+1)[a,b]nnS_(2m)(Delta_(n))S_{\bar{Y}} \in H_{2, \bar{Y}}^{m+1}[a, b] \cap \mathcal{S}_{2 m}\left(\Delta_{n}\right)SY¯∈H2,Y¯m+1[a,b]∩S2m(Δn);
b) If f ∈ H 2 , Y ′ m [ a , b ] f ∈ H 2 , Y ′ m [ a , b ] f inH_(2,Y^('))^(m)[a,b]f \in H_{2, Y^{\prime}}^{m}[a, b]f∈H2,Y′m[a,b] and
‖ f ( m ) − s Y ( m ) ‖ 2 ≤ ‖ f ( m ) − s ( m ) ‖ 2 f ( m ) − s Y ( m ) 2 ≤ f ( m ) − s ( m ) 2 ||f^((m))-s_(Y)^((m))||_(2) <= ||f^((m))-s^((m))||_(2)\left\|f^{(m)}-s_{Y}^{(m)}\right\|_{2} \leq\left\|f^{(m)}-s^{(m)}\right\|_{2}‖f(m)−sY(m)‖2≤‖f(m)−s(m)‖2
for any s ∈ S 2 m − 1 ( Δ n ) s ∈ S 2 m − 1 Δ n s inS_(2m-1)(Delta_(n))s \in \mathcal{S}_{2 m-1}\left(\Delta_{n}\right)s∈S2m−1(Δn) then
‖ I α ( m + 1 ) ( f ) − S Y ¯ ( m + 1 ) ‖ 2 ≤ ‖ I α ( m + 1 ) − S ( m + 1 ) ‖ 2 I α ( m + 1 ) ( f ) − S Y ¯ ( m + 1 ) 2 ≤ I α ( m + 1 ) − S ( m + 1 ) 2 ||I_(alpha)^((m+1))(f)-S_( bar(Y))^((m+1))||_(2) <= ||I_(alpha)^((m+1))-S^((m+1))||_(2)\left\|I_{\alpha}^{(m+1)}(f)-S_{\bar{Y}}^{(m+1)}\right\|_{2} \leq\left\|I_{\alpha}^{(m+1)}-S^{(m+1)}\right\|_{2}‖Iα(m+1)(f)−SY¯(m+1)‖2≤‖Iα(m+1)−S(m+1)‖2
for any S ∈ S 2 m ( Δ n ) S ∈ S 2 m Δ n S inS_(2m)(Delta_(n))S \in \mathcal{S}_{2 m}\left(\Delta_{n}\right)S∈S2m(Δn).
Proof. a) For g ∈ H 2 , Y ¯ m + 1 [ a , b ] g ∈ H 2 , Y ¯ m + 1 [ a , b ] g inH_(2, bar(Y))^(m+1)[a,b]g \in H_{2, \bar{Y}}^{m+1}[a, b]g∈H2,Y¯m+1[a,b] we have
‖ g ( m + 1 ) ‖ 2 2 − ‖ S Y ¯ ( m + 1 ) ‖ 2 2 = ‖ ( g ′ ) ( m ) ‖ 2 2 − ‖ s Y ( m ) ‖ 2 2 ≥ 0 g ( m + 1 ) 2 2 − S Y ¯ ( m + 1 ) 2 2 = g ′ ( m ) 2 2 − s Y ( m ) 2 2 ≥ 0 ||g^((m+1))||_(2)^(2)-||S_( bar(Y))^((m+1))||_(2)^(2)=||(g^('))^((m))||_(2)^(2)-||s_(Y)^((m))||_(2)^(2) >= 0\left\|g^{(m+1)}\right\|_{2}^{2}-\left\|S_{\bar{Y}}^{(m+1)}\right\|_{2}^{2}=\left\|\left(g^{\prime}\right)^{(m)}\right\|_{2}^{2}-\left\|s_{Y}^{(m)}\right\|_{2}^{2} \geq 0‖g(m+1)‖22−‖SY¯(m+1)‖22=‖(g′)(m)‖22−‖sY(m)‖22≥0
because g ′ ∈ H 2 , Y m g ′ ∈ H 2 , Y m g^(')inH_(2,Y)^(m)g^{\prime} \in H_{2, Y}^{m}g′∈H2,Ym. Also ‖ f ( m ) ‖ 2 2 ≥ ‖ s Y . ( m ) ‖ 2 2 f ( m ) 2 2 ≥ s Y . ( m ) 2 2 ||f^((m))||_(2)^(2) >= ||s_(Y.)^((m))||_(2)^(2)\left\|f^{(m)}\right\|_{2}^{2} \geq\left\|s_{Y .}^{(m)}\right\|_{2}^{2}‖f(m)‖22≥‖sY.(m)‖22 for any f ∈ H 2 , Y m f ∈ H 2 , Y m f inH_(2,Y)^(m)f \in H_{2, Y}^{m}f∈H2,Ym (see (??)). Therefore
min { J k ( g ) : g ∈ H 2 , Y ¯ m + 1 } = I a ( S Y ¯ ) . min J k ( g ) : g ∈ H 2 , Y ¯ m + 1 = I a S Y ¯ . min{J_(k)(g):g inH_(2, bar(Y))^(m+1)}=I_(a)(S_( bar(Y))).\min \left\{J_{k}(g): g \in H_{2, \bar{Y}}^{m+1}\right\}=I_{a}\left(S_{\bar{Y}}\right) .min{Jk(g):g∈H2,Y¯m+1}=Ia(SY¯).
b)
‖ I α ( m + 1 ) ( f ) − S Y ¯ ( m + 1 ) ‖ 2 = ‖ ( I α ′ ) ( m ) ( f ) − s Y ( m ) ‖ 2 = ‖ f ( m ) − s Y ( m ) ‖ 2 ≤ ≤ ‖ f ( m ) − s ( m ) ‖ 2 , I α ( m + 1 ) ( f ) − S Y ¯ ( m + 1 ) 2 = I α ′ ( m ) ( f ) − s Y ( m ) 2 = f ( m ) − s Y ( m ) 2 ≤ ≤ f ( m ) − s ( m ) 2 , {:[||I_(alpha)^((m+1))(f)-S_( bar(Y))^((m+1))||_(2)=||(I_(alpha)^('))^((m))(f)-s_(Y)^((m))||_(2)=||f^((m))-s_(Y)^((m))||_(2) <= ],[ <= ||f^((m))-s^((m))||_(2)","]:}\begin{aligned} \left\|I_{\alpha}^{(m+1)}(f)-S_{\bar{Y}}^{(m+1)}\right\|_{2} & =\left\|\left(I_{\alpha}^{\prime}\right)^{(m)}(f)-s_{Y}^{(m)}\right\|_{2}=\left\|f^{(m)}-s_{Y}^{(m)}\right\|_{2} \leq \\ & \leq\left\|f^{(m)}-s^{(m)}\right\|_{2}, \end{aligned}‖Iα(m+1)(f)−SY¯(m+1)‖2=‖(Iα′)(m)(f)−sY(m)‖2=‖f(m)−sY(m)‖2≤≤‖f(m)−s(m)‖2,
for any s ∈ S 2 m − 1 ( Δ n ) s ∈ S 2 m − 1 Δ n s inS_(2m-1)(Delta_(n))s \in \mathcal{S}_{2 m-1}\left(\Delta_{n}\right)s∈S2m−1(Δn). Since, by Lemma 1,
S ( x ) = ∫ α x s ( t ) d t + C ∈ S 2 m ( Δ n ) S ( x ) = ∫ α x   s ( t ) d t + C ∈ S 2 m Δ n S(x)=int_(alpha)^(x)s(t)dt+C inS_(2m)(Delta_(n))S(x)=\int_{\alpha}^{x} s(t) d t+C \in \mathcal{S}_{2 m}\left(\Delta_{n}\right)S(x)=∫αxs(t)dt+C∈S2m(Δn)
it follows s ( m ) = S ( m + 1 ) s ( m ) = S ( m + 1 ) s^((m))=S^((m+1))s^{(m)}=S^{(m+1)}s(m)=S(m+1) and f ( m ) = ( I a ( f ) ) ( m + 1 ) f ( m ) = I a ( f ) ( m + 1 ) f^((m))=(I_(a)(f))^((m+1))f^{(m)}=\left(I_{a}(f)\right)^{(m+1)}f(m)=(Ia(f))(m+1) so that
‖ I α ( m + 1 ) ( f ) − s Y ¯ ( m + 1 ) ‖ 2 ≤ ‖ ( I α ( f ) ) ( m + 1 ) − S ( m + 1 ‖ 2 I α ( m + 1 ) ( f ) − s Y ¯ ( m + 1 ) 2 ≤ I α ( f ) ( m + 1 ) − S ( m + 1 2 ||I_(alpha)^((m+1))(f)-s_( bar(Y))^((m+1))||_(2) <= ||(I_(alpha)(f))^((m+1))-S^((m+1)||_(2)\left\|I_{\alpha}^{(m+1)}(f)-s_{\bar{Y}}^{(m+1)}\right\|_{2} \leq\left\|\left(I_{\alpha}(f)\right)^{(m+1)}-S^{(m+1}\right\|_{2}‖Iα(m+1)(f)−sY¯(m+1)‖2≤‖(Iα(f))(m+1)−S(m+1‖2
for any S ∈ S 2 m ( Δ n ) S ∈ S 2 m Δ n SinS_(2m)(Delta_(n))\mathscr{S} \in \mathcal{S}_{2 m}\left(\Delta_{n}\right)S∈S2m(Δn). quad\quad.

References

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[3] Greville, T.N.E., Introduction to spline functions. Theory and Applications of Spline Functions, T.N.E. Greville, Ed. Academic Press, New York, 1969. 1-35.
[4] Lyche, T., Schumaker L.L., Computation of Smoothing and Interpolating Natural Spline via Local Bases, SIAM J. Numer.Math. 10, No. 6 (1973), 1027-1038.
[5] Micula, G., Funcţii spline şi aplicaţii, Ed. Tehnică, Bucureşti, 1978.
[6] Micula, G., Blaga, P., Akça, H., The numerical treatment of delay differential equations with constant delay by natural spline functions of even degree, Libertas Mathematica, XVI (1996) 123-131.
[7] Mustăţa, R., On p-derivative-interpolating spline functions, Revue d'Analyse Numérique et de Théorie de l'Approx. XXVI, No.1-2 (1997), 149-163.
[8] Mustăţa, C., Mureşan, A., Mustăţa, R., The approximation by spline functions of the solution of a singular perturbed bilocal problem (to appear).
Received 15.07.1998
"T.Popoviciu" Institute of Numerical Analysis str. Republicii Nr. 37 3400 Cluj-Napoca Romania
1998

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