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 |
|---|---|
| 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 |
|---|---|
| ISSN (Print) | 2194-1009 |
| ISSN (Electronic) | 2194-1017 |
Conference
| Conference | 15th International Conference on Approximation Theory, 2016 |
|---|---|
| Country | United States |
| City | San Antonio |
| Period | 22/05/16 → 25/05/16 |
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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 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 -