ON A CLASS OF ABELIAN AND TAUBERIAN-TYPE RESULTS FOR QUASIASYMPTOTIC BEHAVIOR OF DISTRIBUTIONS VIA CERTAIN INTEGRAL TRANSFORMS
Keywords:
shearlet transform, fractional Stockwell transform, short-time linear canonical transform, quasiasymptotic behavior, distributions, Lizorkin distributionsAbstract
We review a particular class of Abelian- and Tauberian-type results relating the quasiasymptotic behavior of distributions to the asymptotic behavior of several integral transforms. The discussion covers the shearlet transform, the fractional Stockwell transform, and the short-time linear canonical transform. A unified presentation highlights the common structure of the corresponding Abelian results and the specific features of each transform. Particular attention is given to the interpretation of the associated Tauberian-type statements, with emphasis on the role of their normalization and limiting behavior.
References
Atanasova, S., Gajić, S., Jakšić, S. and Maksimović, S., Survey of quasiasymptotic behavior of distributions in relation to the properties of their fractional transforms, AppliedMath, vol. 6, no. 1, pp. 1-23, 2026.
Bartolucci, F., Pilipović, S. and Teofanov, N., Continuity properties of the shearlet transform and the shearlet synthesis operator on the Lizorkin type spaces, Math. Nachr., vol. 295, no. 12, pp. 2318-2337, 2022.
Buralieva, J. V., Abelian- and Tauberian-type results for the generalized Stockwell and wavelet transforms, Asian-Eur. J. Math., vol. 18, no. 1, article 2450088, 2025.
Buralieva, J. V., Hadzi-Velkova Saneva, K. and Atanasova, S., Directional short-time Fourier transform and quasiasymptotics of distributions, Funct. Anal. Its Appl., vol. 53, pp. 3-10, 2019.
Estrada, R. and Kanwal, R. P., A Distributional Approach to Asymptotics. Theory and Applications, 2nd ed., Birkhäuser, Boston, 2002.
Ferizi, A. and Hadzi-Velkova Saneva, K., Directional short-time fractional Fourier analysis of quasiasymptotic properties of distributions, Anal. Math., vol. 52, pp. 475-492, 2026.
Ferizi, A. and Hadzi-Velkova Saneva, K., Tauberian theorems for the fractional Stockwell transform of Lizorkin distributions, J. Pseudo-Differ. Oper. Appl., vol. 17, 2026.
Ferizi, A. and Hadzi-Velkova Saneva, K., The shearlet transform and asymptotic behavior of Lizorkin distributions, Appl. Anal., vol. 103, no. 13, pp. 2465-2476, 2024.
Ferizi, A., Maksimović, S. and Hadzi-Velkova Saneva, K., Abelian- and Tauberian-type results for the directional short-time fractional Fourier transform with fixed direction, Filomat, vol. 39, no. 13, pp. 4285-4298, 2025.
Guo, K., Kutyniok, G. and Labate, D., Sparse multidimensional representations using anisotropic dilation and shear operators, in: Chen, G. and Lai, M. (eds.), Wavelets and splines: Athens 2005, Nashboro Press, Brentwood, TN, pp. 189-201, 2006.
Hadzi-Velkova Saneva, K. and Ferizi, A., Short-time coupled fractional Fourier transform and asymptotic behaviour of distributions, Integral Transforms Spec. Funct., 2026.
Hadzi-Velkova Saneva, K. and Ferizi, A., The short-time linear canonical transform and quasiasymptotic behavior of distribution, Open Math., vol. 24, no. 1, article 20250275, 2026.
Healy, J. J., Kutay, M. A., Ozaktas, H. M. and Sheridan, J. T. (eds.), Linear Canonical Transforms: Theory and Applications, Springer, New York, 2016.
Holschneider, M., Wavelets: An Analysis Tool, Clarendon/Oxford University Press, New York, 1995.
Kostadinova, S., Pilipović, S., Saneva, K. and Vindas, J., The short-time Fourier transform of distributions of exponential type and Tauberian theorems for shift-asymptotics, Filomat, vol. 30, no. 11, pp. 3047-3061, 2016.
Kutyniok, G. and Labate, D. (eds.), Shearlets: Multiscale Analysis for Multivariate Data, Birkhäuser/Springer, New York, 2012.
Maksimović, S., Asymptotic results for the distributional fractional Stockwell and fractional wavelet transforms, J. Anal., vol. 34, pp. 1311-1325, 2026.
Moshinsky, M. and Quesne, C., Linear canonical transformations and their unitary representation, J. Math. Phys., vol. 12, no. 8, pp. 1772-1780, 1971.
Pilipović, S., Stanković, B. and Vindas, J., Asymptotic Behavior of Generalized Functions, Series on Analysis, Applications and Computation, vol. 5, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
Saneva, K., Asymptotic behaviour of wavelet coefficients, Integral Transforms Spec. Funct., vol. 20, no. 3-4, pp. 333-339, 2009.
Saneva, K., Aceska, R. and Kostadinova, S., Some Abelian and Tauberian results for the short-time Fourier transform, Novi Sad J. Math., vol. 43, no. 2, pp. 81-89, 2013.
Saneva, K., Atanasova, S. and Buralieva, J. V., Tauberian theorems for the Stockwell transform of Lizorkin distributions, Appl. Anal., vol. 99, no. 4, pp. 596-610, 2020.
Saneva, K. and Buchkovska, A., Tauberian theorems for distributional wavelet transform, Integral Transforms Spec. Funct., vol. 18, no. 5, pp. 359-368, 2007.
Saneva, K. and Buchkovska, A., Asymptotic behaviour of the distributional wavelet transform at 0, Math. Balkanica (NS), vol. 18, pp. 437-441, 2004.
Schwartz, L., Théorie des distributions, Hermann, Paris, 1966.
Seneta, E., Regularly Varying Functions, Springer-Verlag, Berlin-Heidelberg-New York, 1976.
Trèves, F., Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.
Vladimirov, V. S., Drozhzhinov, Yu. N. and Zav'yalov, B. I., Tauberian Theorems for Generalized Functions, Kluwer Academic Publishers, Dordrecht, 1988.
Vindas, J., Pilipović, S. and Rakić, D., Tauberian theorems for the wavelet transform, J. Fourier Anal. Appl., vol. 17, pp. 65-95, 2011.
Wei, D. and Zhang, Y., Fractional Stockwell Transform: Theory and Applications, Digit. Signal Process., vol. 115, article 103090, 2021.
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