The directional short-time fractional Fourier transform of distributions. (English) Zbl 07914978
Summary: We introduce the directional short-time fractional Fourier transform (DSTFRFT) and prove an extended Parseval’s identity and a reconstruction formula for it. We also investigate the continuity of both the directional short-time fractional Fourier transform and its synthesis operator on the appropriate space of test functions. Using the obtained continuity results, we develop a distributional framework for the DSTFRFT on the space of tempered distributions \(\mathcal{S}' (\mathbb{R}^n)\). We end the article with a desingularization formula.
MSC:
| 46F12 | Integral transforms in distribution spaces |
| 44Axx | Integral transforms, operational calculus |
| 42Axx | Harmonic analysis in one variable |
Keywords:
fractional Fourier transform; short-time fractional Fourier transform; directional short-time fractional Fourier transform; distributions; desingularization formulaReferences:
| [1] | Gröchenig, K., Foundation of Time-Frequency Analysis, 2001, Boston: Birkhäuser, Boston · Zbl 0966.42020 · doi:10.1007/978-1-4612-0003-1 |
| [2] | Bultheel, A.; Sulbaran, HM, An introduction to the fractional Fourier transform and friends, Cubo. Mat. Educ., 7, 201-221, 2005 · Zbl 1099.42004 |
| [3] | Kober, H., Wurzeln aus der Hankel- und Fourier und anderen stetigen Transformationen, Quart. J. Math. Oxford Ser., 10, 45-49, 1939 · JFM 65.0489.01 · doi:10.1093/qmath/os-10.1.45 |
| [4] | Namias, V., The fractional order Fourier transform and its application to quantum mechanics, J. Inst. Maths. Appl., 25, 241-265, 1980 · Zbl 0434.42014 · doi:10.1093/imamat/25.3.241 |
| [5] | Almeida, LB, The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal Process., 42, 11, 3084-3091, 1994 · doi:10.1109/78.330368 |
| [6] | Zayed, AI, Fractional Fourier transform of generalized functions, Integral Transf. Spec. Funct., 7, 3-4, 229-312, 1998 · Zbl 0941.46021 · doi:10.1080/10652469808819206 |
| [7] | Pathak, RS; Prasad, A.; Kumar, M., Fractional Fourier transform of temepered distributions and generalized pseudo-differential operator, J. Pseudo-Differ. Oper. Appl., 3, 239-254, 2012 · Zbl 1266.46031 · doi:10.1007/s11868-012-0047-8 |
| [8] | Ozaktas, H.; Zalevsky, Z.; Kutay, M., The Fractional Fourier Transform: With Application in Optics and Signal Processing, 2001, New York: Wiley, New York · doi:10.23919/ECC.2001.7076127 |
| [9] | Kamalakkannan, R.; Roopkumar, R., Multidimensional fractional Fourier transform and generalized fractional convolution, Integral Transforms Spec. Funct., 31, 2, 152-165, 2019 · Zbl 1431.42010 · doi:10.1080/10652469.2019.1684486 |
| [10] | Toft, J.; Bhimani, DG; Manna, R., Fractional Fourier transforms, harmonic oscillator propagators and Strichartz estimates on Pilipović and modulation spaces, Appl. Comput. Harmon. Anal., 67, 2023 · Zbl 1535.46049 · doi:10.1016/j.acha.2023.101580 |
| [11] | Zayed, A., Two-dimensional fractional Fourier transform and some of its properties, Integral Transforms Spec. Funct., 29, 7, 553-570, 2018 · Zbl 1393.42011 · doi:10.1080/10652469.2018.1471689 |
| [12] | Tao, R.; Li, YL; Wang, Y., Short-time fractional Fourier transform and its applications, IEEE Trans. Signal Process., 58, 5, 2568-2580, 2010 · Zbl 1392.94484 · doi:10.1109/TSP.2009.2028095 |
| [13] | Gao, W.; Li, B., Convolution theorem involving n-dimensional windowed fractional Fourier transform, Sci. China Inf. Sci., 64, 2021 · doi:10.1007/s11432-020-2909-5 |
| [14] | Atanasova, S.; Maksimović, S.; Pilipović, S., Abelian and Tauberian results for the fractional Fourier and short-time Fourier transforms of distributions, Integral Transforms Spec. Funct., 2023 · Zbl 1533.46038 · doi:10.1080/10652469.2023.2255367 |
| [15] | Candès, EJ, Harmonic Analysis of Neural Networks, Appl. Comput. Harmon. Anal., 6, 2, 197-218, 1999 · Zbl 0931.68104 · doi:10.1006/acha.1998.0248 |
| [16] | Grafakos, L.; Sansing, C., Gabor frames and directional time-frequency analysis, Appl. Comput. Harmon. Anal., 25, 1, 47-67, 2008 · Zbl 1258.42032 · doi:10.1016/j.acha.2007.09.004 |
| [17] | Giv, HH, Directional short-time Fourier transform, J. Math. Anal. Appl., 399, 1, 100-107, 2013 · Zbl 1256.42050 · doi:10.1016/j.jmaa.2012.09.053 |
| [18] | Saneva, K.; Atanasova, S., Directional short-time Fourier transform of distributions, J. Inequal. Appl., 124, 1-10, 2016 · Zbl 1335.42043 · doi:10.1186/s13660-016-1065-5 |
| [19] | Hörmander, L., The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 1983, Berlin: Springer, Berlin · Zbl 0521.35001 |
| [20] | Schwartz, L., Théorie des Distributions, 1966, Paris: Hermann, Paris · Zbl 0149.09501 |
| [21] | Treves, F., Topological Vector Spaces, Distributions and Kernels, 1967, New York-London: Academic press, New York-London · Zbl 0171.10402 |
| [22] | Hertle, A., Continuity of the radon transform and its inverse on Euclidean space, Math. Z., 184, 165-192, 1983 · doi:10.1007/BF01252856 |
| [23] | Helgason, S., The Radon Transform, 1999, Boston: Birkhäuser Boston Inc, Boston · Zbl 0932.43011 · doi:10.1007/978-1-4757-1463-0 |
| [24] | Atanasova, S., Pilipović, S., Saneva, K.: Directional time-frequency analysis and directional regularity. Bull. Malays. Math. Sci. Soc. 42, 2075-2090 (2019). doi:10.1007/s40840-017-0594-5 · Zbl 1439.46031 |
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