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

Fingerprint

Refinable Functions
Dilation
Integer
Scalar
Refinement Equation
Binary Forms
Asymptotic Density
Algebraic integer
Vector-valued Functions
Galois
Lemma
Vanish
Fourier transform
Modulus
Infinity
Lower bound
Ring
Polynomial
Coefficient

Cite this

Lawton, W. (2017). Refinable functions with PV dilations. In G. E. Fasshauer, & L. L. Schumaker (Eds.), 15th International Conference on Approximation Theory, 2016 (Vol. 201, pp. 177-188). (Springer Proceedings in Mathematics & Statistics). Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-319-59912-0_8
Lawton, Wayne. / Refinable functions with PV dilations. 15th International Conference on Approximation Theory, 2016. editor / Gregory E. Fasshauer ; Larry L. Schumaker. Vol. 201 Cham, Switzerland : Springer, 2017. pp. 177-188 (Springer Proceedings in Mathematics & Statistics).
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Lawton, W 2017, Refinable functions with PV dilations. in GE Fasshauer & LL Schumaker (eds), 15th International Conference on Approximation Theory, 2016. vol. 201, Springer Proceedings in Mathematics & Statistics, Springer, Cham, Switzerland, pp. 177-188, 15th International Conference on Approximation Theory, 2016, San Antonio, United States, 22/05/16. https://doi.org/10.1007/978-3-319-59912-0_8

Refinable functions with PV dilations. / Lawton, Wayne.

15th International Conference on Approximation Theory, 2016. ed. / Gregory E. Fasshauer; Larry L. Schumaker. Vol. 201 Cham, Switzerland : Springer, 2017. p. 177-188 (Springer Proceedings in Mathematics & Statistics).

Research output: Chapter in Book/Conference paperConference paper

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Lawton W. Refinable functions with PV dilations. In Fasshauer GE, Schumaker LL, editors, 15th International Conference on Approximation Theory, 2016. Vol. 201. Cham, Switzerland: Springer. 2017. p. 177-188. (Springer Proceedings in Mathematics & Statistics). https://doi.org/10.1007/978-3-319-59912-0_8