By Anatoly Yu. Bezhaev, Vladimir A. Vasilenko
This booklet is a scientific description of the variational idea of splines in Hilbert areas. All relevant elements are mentioned within the basic shape: lifestyles, specialty, characterization through reproducing mappings and kernels, convergence, errors estimations, vector and tensor hybrids in splines, dimensional decreasing (traces of splines onto manifolds), and so forth. All concerns are illustrated by way of useful examples. In each case the numerical algorithms for the development of splines are proven.
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Q. Using R(A) = AN(T) we have q Au = ')' AjAnj , j=1 N(T); . This 20 VARIATIONAL THEORY OF SPLINES where Aj ar e any constants . Hence t he linear algebraic syst em arises with respect to the coeffi cients Aj : q ) ' Aj (A nj , Ani )t = 0, i = 1,2, . . , q. j= l The elements Ani , i = 1,2, ... , n are linear ind epend ent and the syst em has only zero solut ion. Thus u E N(A ). Conversely if u E N (A ), then we have o and u E N(T); . 44) min . It is natural that (T , A ) is the spline-pair , and (T , A) is any maximal spline-pair with resp ect to (T , A ).
PROOF We have to prove that L (u ) = (L( G(s, ·)),u (s )x . In the proof of t he preceding theorem we noted t hat G(s, t) = 7r(kt ) . Since G(s, t ) is a symmet ric function, then 7r(k t )(s) = 7r(ks )(t ). 13) is reduced to the equality 1r(L)(s) the latter is proved in the following way: = L(1r(ks ) ) , and This completes the proof of the theorem. o Let P be a closed subspace in X(D) , and let us equip the space X(D) with a semi-Hilbert structure (X(D) , I· Ip). 6 The function G p(s , t) is said to be the reproducing kernel of the semi-Hilbert space (X (D), I.
0, -1 , I f . 26) for all I = 1, . . , n - 1. We have 2Gel = ° ts - t2 . tl - ti - tl tl - tl+l t2 - tl t2 - tl+l ti - l . - 1 1 tl t2 - tl tl+l ° tl - t n and, thus, () Gel Gel H I - () r tl - tl+l l ti - tl+1 i = rtl+l°- tl , , if i = I ot herwise. 26) and t he statement is proved. 4. Space W;n [O ,27r] of Periodic Functions and Bernoulli Functions Let us consider the integer m 2': 1 and the Sobolev space Wr [O , 27r] of periodic funct ions. More exactly, t he functions u E Wr [O , 27r] are from 40 VARIATIONAL THEORY OF SPLINES the space e m- 1 [0, 21f] and uCr)(O) = u Cr) (21f), 'ir = 0, .