On Equivalence of Moduli of Smoothness

Yingkang Hu

doi:10.1006/jath.1997.3264

On Equivalence of Moduli of Smoothness

Yingkang Hu

Emeriti, Retired & In Memoriam (COSM)

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

Access to Document

10.1006/jath.1997.3264

Cite this

@article{0fe9aaf9e35247ec88c3537fb5fef0a2,

title = "On Equivalence of Moduli of Smoothness",

abstract = "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.",

keywords = "Modulus of smoothness; splines; shift-invariant spaces; wavelets; degree of approximation; convex approximation",

author = "Yingkang Hu",

year = "1999",

month = apr,

doi = "10.1006/jath.1997.3264",

language = "English",

volume = "97",

pages = "282--293",

journal = "Journal of Approximation Theory",

issn = "0021-9045",

publisher = "Academic Press Inc.",

number = "2",

}

TY - JOUR

T1 - On Equivalence of Moduli of Smoothness

AU - Hu, Yingkang

PY - 1999/4

Y1 - 1999/4

N2 - 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.

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.

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

UR - http://www.scopus.com/inward/record.url?scp=0000396694&partnerID=8YFLogxK

U2 - 10.1006/jath.1997.3264

DO - 10.1006/jath.1997.3264

M3 - Article

AN - SCOPUS:0000396694

SN - 0021-9045

VL - 97

SP - 282

EP - 293

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 2

ER -

On Equivalence of Moduli of Smoothness

Abstract

Scopus Subject Areas

Keywords

Access to Document

Other files and links

Fingerprint

Cite this