Determination Of Jumps For Functions In Terms Of Convolution Operators

Determination of jumps for functions is an important part of detection of singularities of a function. People have used many different methods to do this, such as Fourier coefficient method, concentration factor method, MCM conjugate series method and Gabor derivatives series method etc.In this paper we discussed how to calculate generalized jumps via deriva-tives of convolution operators and its Hilbert Transforms.This paper includes three chapters. The first chapter is an introduction, mainly discussing the background and significance of the research, and the main conclusions of this paper. The second chapter is to prove the approxima-tion theorems, including how to calculate jumps via any odd order derivatives of convolution operators and the Hilbert Transforms of convolution operators’ any even order derivatives. In the third chapter we give the estimations of approximation rate via Poisson kernel and Gaussian kernel.