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Quasi-projection operators in weighted \(L_p\) spaces. (English) Zbl 1460.41015

Signals of various forms, in particular band-limited ones, can be well approximated by expansions that are effectively quasi-interpolants, i.e., scales of shifts of a kernel function times function values. Another case that is specialised like that, is the case of projections, where function evaluations are replaced by inner products with the kernels (\(L^2\) projections); altogether these shapes can be replaced by general quasi-interpolants that are no longer restricted to function values and neither to inner products (they could be averages, derivatives, etc; the useful concept of quasi-interpolation is quite general).
The scaling of the arguments of the kernels are usually by a power of \(h\), but in this article, more general scalings by powers of matrices \(M\) are allowed. In that case, the approximation orders are no longer expressed in powers of (or moduli of smoothness of) \(h\) but in reciprocals of smallest diagonal elements in modulus of the matrices and their powers.
One of the essential points of the approximations of this forms is the choice of approximation spaces, and the authors of this paper present an Ansatz using \(L^p\)-spaces with weights. The particular result the authors have in mind are error bounds that represent the approximation power of the projections.
The approximation orders depend on conditions on the kernel’s Fourier transform at zero; they are of the same type as the celebrated Strang and Fix conditions. This usually means that the kernel’s Fourier transform is \(1\) at the origin and certain of its derivatives vanish there, or in the extreme case, that it is identically one almost everywhere in a neighbourhood. (In this paper, these properties are called “weakly compatible” and “strictly compatible”, respectively.)
Depending on these orders of the Strang and Fix type conditions, \(n\) say, the authors derive convergence results of the projectors, written in \(n\)th moduli of continuity of weighted norms of inverse powers of the scaling matrix. The norms are weighted \(L^p\) norms. If the order of the weak compatibility is \(n\), we get order \(n\) (Theorem 9 for example), if there is strict compatibility, it can be any order (Corollary 8 for example).

MSC:

41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
41A25 Rate of convergence, degree of approximation
41A63 Multidimensional problems
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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