Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Paper coordinates
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.
[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).
Dedicated to Professor Iulian Coroian on his 60^("th ")60^{\text {th }} 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 inN,m <= nm, n \in \mathbb{N}, m \leq n, and [a,b][a, b] an interval contained in R\mathbb{R}. Let also
be a fixed partition of the interval [a,b][a, b].
Definition 1 A function s:[a,b]rarrRs:[a, b] \rightarrow \mathbb{R} verifying the conditions 1^(0)s inC^(2m-2)[a,b]1^{0} s \in C^{2 m-2}[a, b], 2^(0)s inP_(m-1)2^{0} s \in \mathcal{P}_{m-1} on ( a,x_(1)a, x_{1} ) and ( x_(n),bx_{n}, b ), 3^(0)s inP_(2m-1)3^{0} s \in \mathcal{P}_{2 m-1} on (x_(i),x_(i+1)),i=1,2,dots,n-1\left(x_{i}, x_{i+1}\right), i=1,2, \ldots, n-1,
where P_(k)\mathcal{P}_{k} stands for the set of polynomials of degree at most k(k inN)k(k \in \mathbb{N}), is called an interpolating natural spline function (associated to the partition Delta_(n)\Delta_{n} ).
Denoting by S_(2m-1)(Delta_(n))\mathcal{S}_{2 m-1}\left(\Delta_{n}\right) the set of all interpolating natural spline functions, one sees that S_(2m-1)(Delta_(n))S_{2 m-1}\left(\Delta_{n}\right) is an nn-dimensional subspace of the linear space C^(2m-2)[a,b]C^{2 m-2}[a, b].
If s inS_(2m-1)(Delta_(n))s \in \mathcal{S}_{2 m-1}\left(\Delta_{n}\right) then ss is of the form
(2)
(see [3]).
If Y=(y_(1),y_(2),dots,y_(n))Y=\left(y_{1}, y_{2}, \ldots, y_{n}\right) is a fixed vector in R^(n)\mathbb{R}^{n} then there exists exactly one function s_(Y)inS_(2m-1)(Delta_(n))s_{Y} \in \mathcal{S}_{2 m-1}\left(\Delta_{n}\right) verifying the equalities:
{:(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*}
see ([3]).
Let
(5) 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. is absolutely continuous and {:f^((m))inL_(2)[a,b]}\left.f^{(m)} \in L_{2}[a, b]\right\}
and
(6)
and the functional J:H_(2,Y)^(m)[a,b]rarrR_(+)J: H_{2, Y}^{m}[a, b] \rightarrow \mathbb{R}_{+}given by
{:(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*}
i.e. the minimum of the L_(2)L_{2}-norms of the derivatives of order mm of the functions in H_(2,Y)^(m)[a,b]H_{2, Y}^{m}[a, b] is attained at the interpolating natural spline function s_(Y)s_{Y} ( "the minimal norm property").
Also, for each f inH_(2,Y)^(m)[a,b]f \in H_{2, Y}^{m}[a, b] the inequality
holds for any s inS_(2m-1)(Delta_(n))s \in \mathcal{S}_{2 m-1}\left(\Delta_{n}\right) ( "the best approximation property") (see [3]).
Now, we shall introduce the derivative-interpolating spline functions of even order 2m2 m, having properties similar to (9) and (10). (see [2]).
Let m,n inN,m <= n+1m, n \in \mathbb{N}, m \leq n+1, and Delta_(n)\Delta_{n} the partition (1) of the interval [a,b][a, b].
Definition 2 ([2]). A function S:[a,b]rarrRS:[a, b] \rightarrow \mathbb{R} is called a natural spline function of order 2m2 m if it verifies the conditions:
{:[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}
The set of all interpolating natural spline functions of order 2m2 m will be denoted by S_(2m)(Delta_(n))\mathcal{S}_{2 m}\left(\Delta_{n}\right). It follows that S_(2m)(Delta_(n))\mathcal{S}_{2 m}\left(\Delta_{n}\right) is an ( n+1n+1 )-dimensional subspace of C^(2m-1)[a,b]C^{2 m-1}[a, b]. (see [6]).
If bar(Y)=(y_(alpha),y_(1),dots,y_(n))\bar{Y}=\left(y_{\alpha}, y_{1}, \ldots, y_{n}\right) is a fixed vector in R^(n+1)\mathbb{R}^{n+1} then there exists only one function S_( bar(Y))inS_(2m)(Delta_(n))S_{\bar{Y}} \in S_{2 m}\left(\Delta_{n}\right) verifying the conditions
where x_(i),i≐ bar(1,n)x_{i}, i \doteq \overline{1, n} are the nodes of the partition Delta_(n)\Delta_{n} given by (??).
The function S_( bar(Y))inS_(2m)(Delta_(n))S_{\bar{Y}} \in S_{2 m}\left(\Delta_{n}\right) verifying the condition (??) is called the derivative-interpolating spline of even order 2m2 m associated to the vector bar(Y)\bar{Y} and to the partition Delta_(n)\Delta_{n}.
Any function S inS_(2m)(Delta_(n))S \in S_{2 m}\left(\Delta_{n}\right) admits the representation
and the functional J_(alpha):H_(2, bar(Y))^(m+1)[a,b]rarrR_(;)J_{\alpha}: H_{2, \bar{Y}}^{m+1}[a, b] \rightarrow \mathbb{R}_{;}defined by
{:(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*}
attains its minimum at the function S_( bar(Y))S_{\bar{Y}} :
holds for any S inS_(2m)(Delta_(m))S \in \mathcal{S}_{2 m}\left(\Delta_{m}\right).
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 2m2 m 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 pp-derivativeinterpolating spline functions for p=2p=2 and m=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 inS_(2m-1)(Delta_(n)),alpha in[a,b]s \in \mathcal{S}_{2 m-1}\left(\Delta_{n}\right), \alpha \in[a, b] fixed and
{:(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*}
Then every S in hat(I)(s)S \in \hat{I}(s) belongs to S_(2m)(Delta_(n))S_{2 m}\left(\Delta_{n}\right).
where 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}, 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, n and 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.
Taking into account (12) and (13) it follows S inS_(2m)(Delta_(n))S \in \mathcal{S}_{2 m}\left(\Delta_{n}\right).
Lemma 4 Let f inH_(2)^(m)[a,b]f \in H_{2}^{m}[a, b] and
{:(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*}
Then g in hat(I)(f)g \in \hat{I}(f) if and only if g inH_(2)^(m+1)[a,b]g \in H_{2}^{m+1}[a, b].
Proof. Obviously that hat(I)(f)subC^(m)[a,b]\hat{I}(f) \subset C^{m}[a, b], and if g in hat(I)(f)g \in \hat{I}(f) then g^((m))=f^((m-1))g^{(m)}= f^{(m-1)} (absolutely continuous on [a,b][a, b] ) and g^((m+1))=f^((m))inL_(2)[a,b]g^{(m+1)}=f^{(m)} \in L_{2}[a, b], showing that g inH_(2)^(m+1)[a,b]g \in H_{2}^{m+1}[a, b].
If g inH_(2)^(m+1)[a,b]g \in H_{2}^{m+1}[a, b] then g^(')inH_(2)^(m)[a,b]g^{\prime} \in H_{2}^{m}[a, b] so that
g(x)=int_(alpha)^(x)g^(')(t)dt+g(alpha)g(x)=\int_{\alpha}^{x} g^{\prime}(t) d t+g(\alpha)
i.e. g in hat(I)(g^('))g \in \hat{I}\left(g^{\prime}\right).
Lemma 5 Let. Y=(y_(1),y_(2),dots,y_(n))inR^(n)Y=\left(y_{1}, y_{2}, \ldots, y_{n}\right) \in \mathbb{R}^{n} and 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}. Then the operator
{:(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*}
showing that I_(alpha)(f)inH_(2, bar(Y))^(m+1)[a,b]I_{\alpha}(f) \in H_{2, \bar{Y}}^{m+1}[a, b], for every f inH_(2,Y)^(m)[a,b]f \in H_{2, Y}^{m}[a, b].
If f_(1),f_(2)inH_(2,Y)^(m+1)[a,b]f_{1}, f_{2} \in H_{2, Y}^{m+1}[a, b] and I_(alpha)(f_(1))=I_(alpha)(f_(2))I_{\alpha}\left(f_{1}\right)=I_{\alpha}\left(f_{2}\right) then
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
for all x in[a,b]x \in[a, b], implying f_(1)(t)=f_(2)(t)f_{1}(t)=f_{2}(t) for all t in[a,b]t \in[a, b], i.e. I_(alpha)I_{\alpha} is injective.
Let g inH_(2, bar(Y))^(m+1)[a,b]g \in H_{2, \bar{Y}}^{m+1}[a, b]. Then g^(')inH_(2,Y)^(m)[a,b]g^{\prime} \in H_{2, Y}^{m}[a, b] and I_(alpha)(f)=gI_{\alpha}(f)=g, for f=g^(')f=g^{\prime}, showing that I_(alpha)I_{\alpha} is surjective, too.
Lemma 6I_(alpha)(s_(Y))=S_( bar(Y))6 I_{\alpha}\left(s_{Y}\right)=S_{\bar{Y}}
Proof. By Lemmas 1 and 2.
Proof. For g inI_(2, bar(Y))^(m+1)[a,b]g \in I_{2, \bar{Y}}^{m+1}[a, b] we have
{:[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}
Theorem 8 a) If the functional J:H_(2,Y)^(m)[a,b]rarrR_(+)J: H_{2, Y}^{m}[a, b] \rightarrow \mathbb{R}_{+}attains its minimal value at the spline function 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) then the functional J_(alpha):H_(2, bar(Y))^(m+1)[a,b]rarrR_(+)J_{\alpha}: H_{2, \bar{Y}}^{m+1}[a, b] \rightarrow \mathbb{R}_{+}attains its minimal value at 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);
b) If f inH_(2,Y^('))^(m)[a,b]f \in H_{2, Y^{\prime}}^{m}[a, b] and
for any S inS_(2m)(Delta_(n))S \in \mathcal{S}_{2 m}\left(\Delta_{n}\right).
Proof. a) For g inH_(2, bar(Y))^(m+1)[a,b]g \in H_{2, \bar{Y}}^{m+1}[a, b] we have
because g^(')inH_(2,Y)^(m)g^{\prime} \in H_{2, Y}^{m}. Also ||f^((m))||_(2)^(2) >= ||s_(Y.)^((m))||_(2)^(2)\left\|f^{(m)}\right\|_{2}^{2} \geq\left\|s_{Y .}^{(m)}\right\|_{2}^{2} for any f inH_(2,Y)^(m)f \in H_{2, Y}^{m} (see (??)). Therefore
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) .
for any SinS_(2m)(Delta_(n))\mathscr{S} \in \mathcal{S}_{2 m}\left(\Delta_{n}\right). quad\quad.
References
[1] Aubin, J-P., 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. "Babeş-Bolyai", Mathematica XXXVIII, No. 2 (1993), 3140.
[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