Abstract
In this chapter, we discuss a unique method to time-frequency analysis which gives a centralized way to represent discrete and continuous time-frequency. This serves as a straightforward way to include all possible (countable) discrete and continuous time scales in one model. We consider the Mexican hat wavelet which is one of the basic wavelet functions formulated by the second derivative of Gaussian function to define the Mexican hat wavelet transform (MHWT). Further, the theory of MHWT is implemented to obtain the Mexican hat wavelet Stieltjes transform (MHWST) of a bounded variation function. Some convenient properties of MHWST are also presented. Further, a standard method is introduced for representing functions of class B(m, n). Besides, an integral transform is constructed with the help of the Fourier summation kernel. This construction results in a flexible way to present some conditions that are necessary and sufficient for a function of class B(m, n) to be Mexican hat wavelet and MHWST.
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