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Published in:
2017 | OriginalPaper | Chapter
Refinable Functions with PV Dilations
Author : Wayne Lawton
Published in: Approximation Theory XV: San Antonio 2016
Publisher: Springer International Publishing
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Abstract
A PV number is an algebraic integer \(\alpha \) of degree \(d \ge 2\) all of whose Galois conjugates other than itself have modulus less than 1. Erdös [8] proved that the Fourier transform \(\widehat{\varphi },\) of a nonzero compactly supported scalar-valued function satisfying the refinement equation \(\varphi (x) = \frac{|\alpha |}{2}\varphi (\alpha x) + \frac{|\alpha |}{2}\varphi (\alpha x-1)\) with PV dilation \(\alpha ,\) does not vanish at infinity so by the Riemann–Lebesgue lemma \(\varphi \) is not integrable. Dai, Feng, and Wang [5] extended his result to scalar-valued solutions of \(\varphi (x) = \sum _k a(k) \varphi (\alpha x - \tau (k))\) where \(\tau (k)\) are integers and a has finite support and sums to \(|\alpha |\). In ([22], Conjecture 4.2), we conjectured that their result holds under the weaker assumption that \(\tau \) has values in the ring of polynomials in \(\alpha \) with integer coefficients. This paper formulates a stronger conjecture and provides support for it based on a solenoidal representation of \(\widehat{\varphi },\) 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.