On the Rate of Decay of a Meyer Scaling Function

摘要

A function with the following properties is called a Meyer scaling function: φ: ? → ?, its integral shifts φ(· + n), n ∈ ?, are orthonormal in L2(?), and its Fourier transform φ?y=12π∫?φte?iytdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ hat{varphi}(y)=frac{1}{sqrt{2pi }}underset{mathrm{mathbb{R}}}{int}varphi (t){e}^{- iyt} $$end{document}dt has the following properties: φ?documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ hat{varphi} $$end{document} is even, φ?=0documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ hat{varphi}=0 $$end{document} outside ?π ?ε, π +ε, φ?=12πdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ hat{varphi}=frac{1}{sqrt{2pi }} $$end{document} on ?π + ε, π ? ε, where ε∈0π3.documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ varepsilon in left(0,frac{uppi}{3}right. $$end{document} Here is the main result of the paper. Assume thatω:0+∞→0+∞documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ omega :left0,+infty right)to left0,+infty right) $$end{document}and the function ωxxdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ frac{omega (x)}{x} $$end{document} decreases. Then the following assertions are equivalent.1. For every (or, equivalently, for some) ε∈0π3,documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ varepsilon in left(0,frac{uppi}{3}right, $$end{document} there exists x0 > 0 and a Meyer scaling function φ such that φ?=0documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ hat{varphi}=0 $$end{document} outside ?π ? ε, π + ε and φ(x) ≤ e?ω(x) for all x > x0.2. ∫1+∞ωxx2dx<+∞.documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ int_1^{+infty}frac{omega (x)}{x^2} dx<+infty . $$end{document}