Refinable functions with PV dilations

Wayne Lawton

Research output: Chapter in Book/Conference paperConference paperpeer-review


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
Number of pages12
ISBN (Print)9783319599113
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


Conference15th International Conference on Approximation Theory, 2016
Country/TerritoryUnited States
CitySan Antonio


Dive into the research topics of 'Refinable functions with PV dilations'. Together they form a unique fingerprint.

Cite this