Extension of Integral Transforms with Wavelet Kernels to Generalized Functions

Abstract

The classical concept of point function is insufficient for analyzing many physical phenomena. Another way of representing a physical variable is to define them as functional, that act on a set of functions called test functions . Such a particular type of functional is called a generalized function or distribution. Laurent Schwartz extended the classical concept of point functions to generalized functions or distributions so that every point function regarded as a functional has a defined derivative in the new theory of generalized functions. He extended Fourier and Laplace transform to distributions. This extension allowed Fourier transform to solve many problems in partial differential equations, which classical methods cannot do. Motivated by this result, several integral transforms like the Radon transform, Hilbert Transform, Hankel transform, Mellin transform, curvelet transform, ridgelet transform, wavelet transform, and fractional Fourier transform have been extended to distributions. But there are numerous integral transforms which are idle in the theory of distributions. Among them, integral transforms with wavelet kernels have enormous applications in the digital world and still need to be extended to generalized functions. Recently, many novel transforms have been developed with wavelet kernels. Transforms such as wave packet transform, Bessel wavelet transform, fractional Hankel wavelet transform, and linear canonical wavelet transform are extended to distributions in this thesis. For this extension, it is requisite to build a suitable test function space depending on the kernel of the particular transform. (abstract attached). newline