Filomat 2023 Volume 37, Issue 9, Pages: 2717-2730
https://doi.org/10.2298/FIL2309717S
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The Mexican hat wavelet Stieltjes transform
Singh Abhishek (Department of Mathematics and Statistics Banasthali Vidyapith, Banasthali, India), mathdras@gmail.com
Rawat Aparna (Department of Mathematics and Statistics Banasthali Vidyapith, Banasthali, India), rawataparna18@gmail.com
In the present article, we define the Mexican hat wavelet Stieltjes transform
(MHWST) by applying the concept of Mexican hat wavelet transform [9]. The
proposed transform serves as a centralized method to analyze both discrete
and continuous time-frequency localization. Besides the formulation of all
the fundamental results, a reconstruction formula is also obtained for
MHWST. Further, a unified approach is applied to obtain the necessary and
sufficient conditions for the same. Moreover, simplified construction for
the jump operator is also presented for the Mexican hat wavelet Stieltjes
transform.
Keywords: Stieltjes and Lebesgue integral, Wavelet transform and Mexican hat wavelet transform
Show references
Bielecki, T., Chen, J., Lin, E. B., and Yau, S. (1999). Wavelet representations of general signals. Nonlinear Analysis: Theory, Methods and Applications, 35(1), 125-141.
Chui, C. K. (2016). An introduction to wavelets. Elsevier.
Lin, E. B. (1994). Wavelet transforms and wavelet approximations. In Approximation, Probability, and Related Fields. Springer, Boston, MA, 357-365.
Widder, D. V., and Hirschman, I. I. (2015). Convolution Transform. Princeton University Press.
Pandey, J. N. (2011). Wavelet transforms of Schwartz distributions. J. Comput. Anal. Appl., 13(1):47-83.
Pathak, R. S. (2017). Integral transforms of generalized functions and their applications. Amsterdam:Gordon and Breach Science Publishers.
Pathak, R. S. (2009). The wavelet transform. Amsterdam, Paris: Atlantis Press, World Scientific.
Pathak, R. S., and Singh, A. (2019). Paley-Wiener-Schwartz type theorem for the wavelet transform. Applicable Analysis, 98(7), 1324-1332.
Pathak, R. S., and Singh, A. (2016). Mexican hat wavelet transform of distributions. Integral Transforms and Special Functions, 27(6), 468-483.
Pathak, R. S., and Singh, A. (2017). Wavelet transform of Beurling-Bj¨orck type ultradistributions. Rend Sem Mat Univ Padova, 137(1), 211-222.
Rawat, A., and Singh, A. (2021). Mexican hat wavelet transforms on generalized functions in G′-space, Proc. Math. Sci., 131, 1-13. https://doi.org/10.1007/s12044-021-00627-6
Singh, A., Raghuthaman, N., Rawat, A., and Singh, J. (2020). Representation theorems for the Mexican hat wavelet transform. Mathematical Methods in the Applied Sciences, 43(7), 3914-3924.
Srivastava, H. M. (2022). Some general families of integral transformations and related results. Appl. Math. Comput. Sci, 6, 27-41.
Srivastava, H. M., and Y¨ urekli, O. (1995). A theorem on a Stieltjes-type integral transform and its applications. Complex Variables and Elliptic Equations, 28(2), 159-168.
Srivastava, H. M., and Tuan, V. K. (1995). New convolution theorem for the Stieltjes transform and its application to a class of singular integral equations. Arch. Math., 64, 144-149.
Srivastava, H. M. (1976). Some remarks on a generalization of the Stieltjes transform. Publ Math Debrecen, 23, 119-122.
Srivastava, H. M., Shah, F. A., and Teali, A. A. (2022). On Quantum Representation of the Linear Canonical Wavelet Transform. Universe, 8(9), 477.
Srivastava, H. M., Shukla, P., and Upadhyay, S. K. (2022). The localization operator and wavelet multipliers involving theWatson transform. Journal of Pseudo-Differential Operators and Applications, 13(4), 1-21.
Srivastava, H. M., Mishra, K. K., and Upadhyay, S. K. (2022). Characterizations of Continuous Fractional Bessel Wavelet Transforms. Mathematics, 10(17), 3084.
Srivastava, H. M., Shah, F. A., and Nayied, N. A. (2022). Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter-Saxton Equation. Applied Sciences, 12(15), 7738.
Srivastava, H. M., Irfan, M., and Shah, F. A. (2021). A Fibonacci wavelet method for solving dual-phase-lag heat transfer model in multi-layer skin tissue during hyperthermia treatment. Energies, 14(8), 2254.
Srivastava, H. M., Singh, A., Rawat, A., and Singh S. (2021) A family of Mexican hat wavelet transforms on greens function, Mathematical Methods in the Applied Sciences, 44 (14), 11340-11349.
Srivastava, H.M., Shah, F.A., Garg, T.K., Lone, W.Z., and Qadri, H.L. (2021) Non-separable linear canonical wavelet transform. Symmetry, 13(11), 2182.
Srivastava, H. M., Khatterwani, K., and Upadhyay, S. K. (2019). A certain family of fractional wavelet transformations. Mathematical Methods in the Applied Sciences, 42(9), 3103-3122.
Widder, D. V. (1966). The Laplace Transform: Princeton.