TY - GEN

T1 - Refinable functions with PV dilations

AU - Lawton, Wayne

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Integral binary form

KW - Lo-jasiewicz’s structure theorem

KW - PV number

KW - Real analytic

KW - Refinable

UR - http://www.scopus.com/inward/record.url?scp=85028059142&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-59912-0_8

DO - 10.1007/978-3-319-59912-0_8

M3 - Conference paper

AN - SCOPUS:85028059142

SN - 9783319599113

VL - 201

T3 - Springer Proceedings in Mathematics & Statistics

SP - 177

EP - 188

BT - 15th International Conference on Approximation Theory, 2016

A2 - Fasshauer, Gregory E.

A2 - Schumaker, Larry L.

PB - Springer

CY - Cham, Switzerland

T2 - 15th International Conference on Approximation Theory, 2016

Y2 - 22 May 2016 through 25 May 2016

ER -