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Chebyshevian splines. (English) Zbl 0725.41010

It is well known that an extended complete Chebyshev system (ECT-system) \(U=\{u_ i:i=0,1,...,n\}\) of functions of class \(C^ n\) on the interval \(I=[a,b]\) admits an integral representation of the form \[ (1)\quad u_ 0=w_ 0,\quad u_ i(t)=\int^{t}_{0}w_ 0(t)\int^{t}_{0}w_ 1(s_ 1)ds_ 1...\int^{s_{i-1}}_{0}w_ i(s_ i)ds_ i, \] where \(w_ j\in C^{n+j}(I)\), \(w_ j(t)>0\), for all \(t\in I\) and \(i,j=0,1,...,n\) [see S. Karlin, Total Positivity, (1968; Zbl 0219.47030)]. The aim of this paper is to study canonical complete Chebyshev systems (CCT-systems), defined also by the relations (1), but supposing the functions \(w_ j\) to be only positive and integrable on I, and splines with respect to CCT-systems. The paper is divided into four parts.
In Part I. Canonical complete Chebyshev systems, it is shown that, like in the case of ECT-systems, a basic property of a CCT-system is the positivity of some determinants. Lagrange and Newton interpolation formula by polynomials with respect to a CCT-system are expressed in terms of generalized divided difference and the Markov inequality for generalized polynomials is proved.
Part II. Chebyshevian splines, is concerned with Chebyshevian and B- splines with respect to a CCT-system. Marsden identity, various representation formula and de Boor inequalities are proved. A recurrence relation using generalized divided differences, proved by T. Popoviciu [Mathematica (Cluj) 8, 1-85 (1934; Zbl 0009.05901) and 1, 95-142 (1959; Zbl 0091.246)] in the algebraic case and respectively of ECT-systems, is extended to CCT-systems. In Part III. Spline operators, it is proved that the orthogonal spline projections are \(L_{\infty}\)-bounded and that the biorthogonal spline systems form a Schauder basis in \(L_ p(I)\), \(1\leq p\leq \infty\). In Part IV. Generalized moduli of smoothness, the generalized modulus of smoothness with respect to a CCT-system U is defined such that \(\omega_ U(f,h)=0\), for every linear combination f of functions in U. This part contains also estimates of the best approximation of a function f by Chebyshevian splines, in terms of the modulus of smoothness and a Bernstein type inequality.
In Part V. Applications to approximation of analytic functions, the author proves the existence of Schauder bases formed of biorthogonal systems of spline functions, in the space A(D) of functions analytic in the unit disc D and continuous on \(\bar D\). In the case of continuous \(2\pi\)-periodic functions it is shown that the conjugate system to some biorthogonal periodic spline system is a basis in \(C_{2\pi}\). The results in this part extend previous results of S. V. Bochkarev and of the author.

MSC:

41A15 Spline approximation
46E15 Banach spaces of continuous, differentiable or analytic functions
41A50 Best approximation, Chebyshev systems
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces