Spaces of Test Functions via the STFT

By Karlheinz Gr, G. Zimmermann. 2004

176 citations recorded by the source corpus.

We characterize several classes of test functions, among them Bjorck's ultra-rapidly decaying test functions and the Gelfand-Shilov spaces of type S, in terms of the decay of their short-time Fourier transform and in terms of their Gabor coefficients.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Signal Processing

View Paper: https://www.semanticscholar.org/paper/9a3ece78f4aa0c5fdd2ca167afaeca9fda4053a3

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In Applied Mathematics

1. Karlheinz Gröchenig
2. L. Rodino
3. Nenad Teofanov
4. Michael Ruzhansky
5. Stevan Pilipovic
6. Todor Gramchev
7. Aparajita Dasgupta
8. J. Toft

In Computer Vision and Pattern Recognition

1. Nenad Teofanov
2. Stevan Pilipovic
3. L. Rodino
4. Aparajita Dasgupta
5. Michael Ruzhansky
6. Karlheinz Gröchenig
7. Qiyu Sun
8. J. Toft

In Signal Processing

1. L. Rodino
2. Aparajita Dasgupta
3. Michael Ruzhansky
4. Karlheinz Gröchenig
5. Stevan Pilipovic
6. Nenad Teofanov
7. Luís Daniel Abreu
8. Qiyu Sun

Citations in the corpus

• Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces (2012, methodology): “We also mention characterisations of the Gelfand–Shilov spaces and their duals on Rn in terms of the short-time Fourier transform, see Gröchenig and Zimmerman [7] and Toft [18].”
• Classes of Degenerate Elliptic Operators in Gelfand-Shilov Spaces (2008, background): “It is well known (see Pilipovi´ c and Teofanov [23], Gr¨ ochenig and Zimmermann [ 12 ], Cordero et al [8], Cappiello et al [7], and the references therein) that the Wigner and Fourier transform continuously map S σ (R n ) × S σ σ (R n )i ntoS σ σ (R 2n ). Moreover, we have:”
• Ultradistributions and Time-Frequency Analysis (2006, methodology): “We combine the ideas of the proof of similar assertions given in [23] and [7].”
• Weight Functions in Time-Frequency Analysis (2006, background): “oups [9]. In time-frequency analysis we may ask an analogous question about the modulation spaces M1 v because these are the preferred spaces of test functions. Indeed we have the following statement [24]. Theorem 8.2. Let v be a submultiplicative weight on R2d and assume that v(x,ξ) ≤ Cv(ξ,−x). The”
• Weight Functions in Time-Frequency Analysis (2006, unknown): “Indeed we have the following statement [24].”
• Modulation Spaces: Looking Back and Ahead (2006, unknown): “[126] K.”