×

Spline bases in classical function spaces on compact \(C^{\infty}\) manifolds. (English) Zbl 0599.46042

The aim of this paper is to prove Theorem A and B from Part I, published ibid. 76, 1-58 (1983; review above), concerning construction of Schauder bases resp. unconditional Schauder bases in spaces \(C^ k(M)\) resp. \(W^ k_ p(M)\) over a compact, d-dimensional \(C^{\infty}\)-manifold M, and in some other relative spaces. This is done, reducing the problem to construction of suitable bases in Sobolev spaces \(W^ k_ p(Q)_ Z\) and Besov spaces \(B^ s_{p,q}(Q)_ Z\), where Q is the d-dimensional unit cube and Z is the union of some (d-1)-dimensional closed faces of Q, \(k\geq 0\), \(s>0\), \(1\leq p\leq \infty\), \(1\leq q\leq \infty.\)
The paper under review (Part II) is opened by Section 7, containing basic facts on vector-valued splines. In Section 8 there are considered families of orthogonal projections onto increasing subspaces of splines corresponding to various boundary conditions. Constructions of spline bases are given in Section 9 in case of Sobolev spaces and in Section 10 in case of Besov spaces. Section 11 suggests some applications. Reading Part II requires former lecture of Part I which contains the necessary notations. The results of both Parts form a very general solution to a problem which goes back as far as to S. Banach.
Reviewer: J.Musielak

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
41A15 Spline approximation

Citations:

Zbl 0599.46041
PDFBibTeX XMLCite
Full Text: EuDML