In this paper, we discuss representation theorems for Mexican hat wavelet transform with determining function α2false(yfalse) belonging to different class of functions. The classes H, Afalse[p,qfalse], Bfalse[p,qfalse], and Cfalse[p,qfalse] are characterized by the Mexican hat wavelet transform. Also, we investigate the convergence of the transform with its boundedness property. Furthermore, some representation theorems for locally bounded variation and locally Lebesgue integrable function are established.

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In this paper, Green's function of the heat equation is used to obtain one of the forms in a family of Mexican hat wavelet transforms. By implementing the theory of reproducing kernels, an isometry in the heat equation is also investigated. Further, a natural inversion is established for the Mexican hat wavelet transform. Several related recent investigations are also considered.

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