×

Directional short-time Fourier transform. (English) Zbl 1256.42050

Summary: A directionally sensitive variant of the short-time Fourier transform is introduced which sends functions on \(\mathbb R^{n}\) to those on the parameter space \(S^{n - 1}\times \mathbb R\times \mathbb R^{n}\). This transform, which is named directional short-time Fourier transform (DSTFT), uses functions in \(L^{\infty }(\mathbb R)\) as window and is related to the celebrated Radon transform. We establish an orthogonality relation for the DSTFT and explore some operator-theoretic aspects of the transform, mostly in terms of proving a variant of the Hausdorff-Young inequality. The paper is concluded by some reconstruction formulas.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Candés, E. J., Harmonic analysis of neural networks, Appl. Comput. Harmon. Anal., 6, 2, 197-218 (1999) · Zbl 0931.68104
[3] Candés, E. J.; Donoho, D. L., New tight frames of curvelets and optimal representations of objects with piecewise-\(C^2\) singularities, Comm. Pure Appl. Math., 57, 2, 219-266 (2004) · Zbl 1038.94502
[4] Gabor, D., Theory of communications, J. Inst. Electr. Eng. (London), 93, 429-457 (1946)
[5] Gröchenig, K., Foundations of Time-Frequency Analysis (2001), Birkhäuser: Birkhäuser Boston · Zbl 0966.42020
[6] Grafakos, L.; Sansing, C., Gabor frames and directional time-frequency analysis, Appl. Comput. Harmon. Anal., 25, 1, 47-67 (2008) · Zbl 1258.42032
[7] Deans, S. R., The Radon Transform and Some of its Applications (1993), Robert E. Krieger Publishing Co. Inc.: Robert E. Krieger Publishing Co. Inc. Malabar, FL, (revised reprint of the 1983 original) · Zbl 0868.44001
[8] Helgason, S., The Radon Transform (1999), Birkhäuser: Birkhäuser Boston · Zbl 0932.43011
[9] Radon, J., Über die Bestimmung von Funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten, Akad. Wiss., 69, 262-277 (1917) · JFM 46.0436.02
[10] Folland, G. B., Real Analysis, Modern Techniques and their Applications (1999), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York · Zbl 0924.28001
[11] Rudin, W., Real and Complex Analysis (1987), McGraw-Hill, Inc. · Zbl 0925.00005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.