### Abstract

Language | English |
---|---|

Pages | 107–124 |

Number of pages | 18 |

Journal | Zeitschrift für Analysis und ihre Anwendungen |

Volume | 38 |

Issue number | 1 |

DOIs | |

State | Published - 7 Jan 2019 |

### Fingerprint

### Keywords

- math.CA
- math.FA
- 41A30, 34C25

### Cite this

*Zeitschrift für Analysis und ihre Anwendungen*,

*38*(1), 107–124. DOI: 10.4171/ZAA/1630

}

*Zeitschrift für Analysis und ihre Anwendungen*, vol. 38, no. 1, pp. 107–124. DOI: 10.4171/ZAA/1630

**A Multi-Term Basis Criterion for Families of Dilated Periodic Functions.** / Boulton, Lyonell; Melkonian, Houry.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Multi-Term Basis Criterion for Families of Dilated Periodic Functions

AU - Boulton,Lyonell

AU - Melkonian,Houry

N1 - The introduction of the updated version now includes an example which is arbitrarily close to the jump function such that the corresponding dilations form a Riesz basis. Section 2 has been substantially changed to reflect the established multi-term criterion in terms of the multipliers instead of the corresponding operators. The paper consists of 26 pages and 4 figures

PY - 2019/1/7

Y1 - 2019/1/7

N2 - In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in $L^2(0,1)$. This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients. Our focus is on the concrete verification of the hypotheses by means of analytical or accurate numerical approximations. We then examine the basis question for profiles in a neighbourhood of a non-basis family generated by periodic jump functions. For one of these profiles, the $p$-sine functions, we determine a threshold for positive answer to the basis question which improves upon those found recently.

AB - In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in $L^2(0,1)$. This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients. Our focus is on the concrete verification of the hypotheses by means of analytical or accurate numerical approximations. We then examine the basis question for profiles in a neighbourhood of a non-basis family generated by periodic jump functions. For one of these profiles, the $p$-sine functions, we determine a threshold for positive answer to the basis question which improves upon those found recently.

KW - math.CA

KW - math.FA

KW - 41A30, 34C25

U2 - 10.4171/ZAA/1630

DO - 10.4171/ZAA/1630

M3 - Article

VL - 38

SP - 107

EP - 124

JO - Zeitschrift für Analysis und ihre Anwendungen

T2 - Zeitschrift für Analysis und ihre Anwendungen

JF - Zeitschrift für Analysis und ihre Anwendungen

SN - 0232-2064

IS - 1

ER -