### 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 language | English |
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Title of host publication | 15th International Conference on Approximation Theory, 2016 |

Editors | Gregory E. Fasshauer, Larry L. Schumaker |

Place of Publication | Cham, Switzerland |

Publisher | Springer |

Pages | 177-188 |

Number of pages | 12 |

Volume | 201 |

ISBN (Print) | 9783319599113 |

DOIs | |

Publication status | Published - 2017 |

Event | 15th International Conference on Approximation Theory, 2016 - San Antonio, United States Duration: 22 May 2016 → 25 May 2016 |

### Publication series

Name | Springer Proceedings in Mathematics & Statistics |
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ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | 15th International Conference on Approximation Theory, 2016 |
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Country | United States |

City | San Antonio |

Period | 22/05/16 → 25/05/16 |

### Fingerprint

### Cite this

*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

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*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.

Research output: Chapter in Book/Conference paper › Conference paper

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

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

ER -