Refinable functions with PV dilations

Wayne Lawton

Research output: Chapter in Book/Conference paperConference paper

Abstract

A PV number is an algebraic integer α of degree d ≥ 2 all of whose Galois conjugates other than itself have modulus less than 1. Erdös [8] proved that the Fourier transform (formula presented), of a nonzero compactly supported scalar-valued function satisfying the refinement equation (formula presented) with PV dilation α, does not vanish at infinity so by the Riemann–Lebesgue lemma ϕ is not integrable. Dai, Feng, and Wang [5] extended his result to scalar-valued solutions of (formula presented)s where τ(k) are integers and a has finite support and sums to |α|. In ([22], Conjecture 4.2), we conjectured that their result holds under the weaker assumption that τ has values in the ring of polynomials in α with integer coefficients. This paper formulates a stronger conjecture and provides support for it based on a solenoidal representation of (formula presented), and deep results of Erdös and Mahler [9]; Odoni [26] that gives lower bounds for the asymptotic density of integers represented by integral binary forms of degree > 2; degree = 2, respectively. We also construct an integrable vector-valued refinable function with PV dilation.

Original languageEnglish
Title of host publication15th International Conference on Approximation Theory, 2016
EditorsGregory E. Fasshauer, Larry L. Schumaker
Place of PublicationCham, Switzerland
PublisherSpringer
Pages177-188
Number of pages12
Volume201
ISBN (Print)9783319599113
DOIs
Publication statusPublished - 2017
Event15th International Conference on Approximation Theory, 2016 - San Antonio, United States
Duration: 22 May 201625 May 2016

Publication series

NameSpringer Proceedings in Mathematics & Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

Conference15th International Conference on Approximation Theory, 2016
CountryUnited States
CitySan Antonio
Period22/05/1625/05/16

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