Wiener-Plancherel Formulas Related To Dunkl Transform And Hankel Transform

The Wiener-Plancherel formula is an analogue of the Plancherel formula on the Fourier transform,with respect to a class of non-square-integrable functions.It plays an important role in the quantitative analysis of signals with non-square-integrable noise or random components.The aim of the thesis is to study the Wiener-Plancherel formulas associated with the Dunkl transform on the line and the Hankel transform on the half-line respectively.The Dunkl transform is a generalization of the Fourier transform.In the thesis,the Wiener transform associated with the Dunkl transform on the line is defined,and a matching generalized symmetric difference operator is introduced.As a result,the Wiener-Plancherel formula associated with the Dunkl transform is proved for a class of non-weighted-square-integrable functions.Associated with the Hankel transform,the thesis defines a sequence of the Wiener transforms on the half-line.It is found that the generalized difference operators of various orders introduced by Li and Peng in 2007 match them.For all functions with polynomial growth of any order at infinity on the half-line,the Wiener-Plancherel formula of corresponding order related to the Hankel transform is established.