The functional spaces formed by the Riesz potentials, generated by Bessel shift and their characterization with the aid of wavelet type translation

The differantial operator, named Laplace differential operator, Δ=((∂²)/(∂x₁²))+((∂²)/(∂x₂²))+⋅⋅⋅+((∂²)/(∂x_{n}²)) is one of the most famous operators in mathematics. For example, classical wave equation and heat equation in n-dimensional space ℝⁿ, are expressed with the aid of this differential operator. The subject, studied in this thesis, is related to Harmonic Analysis which is named Fourier-Bessel Harmonic Analysis, arised as the sum of Laplace differential operator and singular Bessel differential operator and generated by a hibrit differential operator named Laplace-Bessel differential operator. One of the most important means of Fourier-Bessel Harmonic Analysis, starting to develop after the 1950s, is the Riesz potentials which is associated with Bessel shift. The role of Riesz potentials in Classical Fourier Harmonic Analiysis, plays generalized Riesz potentials in Fourier-Bessel Harmonic Analysis. The real aim of this thesis study which deals with the generalized Riesz potentials, one of the important technical tools of Fourier-Bessel Harmonic Analysis, is to give a new presentation for this potentials and its inverse and a new characterization for the spaces generated by these potentials. Firstly for this, a new semi group was identified which is generalized both of Abel-Poisson and Gauss-Weierstrass semi groups and a new one dimensional integral repesentation is given with the aid of this semi group. A wavelet type transform is identified, the inverse formula is obtained and the opposite of this potentials are found. Moreover, a new representation is given to generalized Riesz potentials, by applying generalized shift to a few of the variables, not only to one variable, and finally, the characterization of the space which is generated by these potentials, is presented.