On Equivalence of Moduli of Smoothness

Yingkang Hu

doi:10.1006/jath.1997.3264

On Equivalence of Moduli of Smoothness

Yingkang Hu

Mathematical Sciences

Georgia Southern University

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

It is known that iff∈W^k_p, thenω_m(f,t)_p≤tω _m-1(f′,t)_p≤.... Its inverse with any constants independent offis not true in general. Hu and Yu proved that the inverse holds true for splinesSwith equally spaced knots, thusω_m(S,t)_p~tω_m-1(S′,t) _p~t²ω_m-2(S″,t)_p.... In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper.

Original language	English
Pages (from-to)	282-293
Number of pages	12
Journal	Journal of Approximation Theory
Volume	97
Issue number	2
DOIs	https://doi.org/10.1006/jath.1997.3264
State	Published - Apr 1999

Scopus Subject Areas

Analysis
Numerical Analysis
General Mathematics
Applied Mathematics

Keywords

Modulus of smoothness; splines; shift-invariant spaces; wavelets; degree of approximation; convex approximation

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10.1006/jath.1997.3264

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AB - It is known that iff∈Wkp, thenωm(f,t)p≤tω m-1(f′,t)p≤.... Its inverse with any constants independent offis not true in general. Hu and Yu proved that the inverse holds true for splinesSwith equally spaced knots, thusωm(S,t)p~tωm-1(S′,t) p~t2ωm-2(S″,t)p.... In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper.

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ER -

On Equivalence of Moduli of Smoothness

Abstract

Scopus Subject Areas

Keywords

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