### Abstract

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L_{2}. In the present article the so-called θ-summability (with a function parameter θ) is considered which induces norm convergence for a large class of function spaces. Under some conditions on θ we prove that the summation of the short-time Fourier transform of f{hook} converges to f{hook} in Wiener amalgam norms, hence also in the L_{p} sense for L_{p} functions, and pointwise almost everywhere.

Original language | English |
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Pages (from-to) | 507-521 |

Number of pages | 15 |

Journal | Journal of Geometric Analysis |

Volume | 16 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 2006 |

### Fingerprint

### Keywords

- θ-summability
- Herz spaces
- Math Subject Classifications: 42B08, 42C15, 42C40, 42A38, 46B15
- short-time Fourier transform
- time-frequency analysis
- Wiener amalgam spaces

### ASJC Scopus subject areas

- Mathematics(all)
- Geometry and Topology

### Cite this

*Journal of Geometric Analysis*,

*16*(3), 507-521. https://doi.org/10.1007/BF02922064

**Inversion formulas for the short-time Fourier transform.** / Feichtinger, Hans G.; Weisz, F.

Research output: Contribution to journal › Article

*Journal of Geometric Analysis*, vol. 16, no. 3, pp. 507-521. https://doi.org/10.1007/BF02922064

}

TY - JOUR

T1 - Inversion formulas for the short-time Fourier transform

AU - Feichtinger, Hans G.

AU - Weisz, F.

PY - 2006/9

Y1 - 2006/9

N2 - The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L2. In the present article the so-called θ-summability (with a function parameter θ) is considered which induces norm convergence for a large class of function spaces. Under some conditions on θ we prove that the summation of the short-time Fourier transform of f{hook} converges to f{hook} in Wiener amalgam norms, hence also in the Lp sense for Lp functions, and pointwise almost everywhere.

AB - The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L2. In the present article the so-called θ-summability (with a function parameter θ) is considered which induces norm convergence for a large class of function spaces. Under some conditions on θ we prove that the summation of the short-time Fourier transform of f{hook} converges to f{hook} in Wiener amalgam norms, hence also in the Lp sense for Lp functions, and pointwise almost everywhere.

KW - θ-summability

KW - Herz spaces

KW - Math Subject Classifications: 42B08, 42C15, 42C40, 42A38, 46B15

KW - short-time Fourier transform

KW - time-frequency analysis

KW - Wiener amalgam spaces

UR - http://www.scopus.com/inward/record.url?scp=33748297786&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748297786&partnerID=8YFLogxK

U2 - 10.1007/BF02922064

DO - 10.1007/BF02922064

M3 - Article

VL - 16

SP - 507

EP - 521

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 3

ER -