Determination Of Jumps For Functions Via Derivatives Of Convolution Operators

The calculation of jumps for functions is an important problem in many applications.People have used reveral different methods to do this.such as Fourier coeffcient method,classic concentration factor method,Gabor derivatives series method and Hilbert transform method etcIn this paper we discussed how to calculate generalized jumps via deriva-tives of some convolution operators.set De(f):=limh�'O+1/h∫Oh(f(ξ+x)-f(ξ-x))dx, and P(x)is one of the following three kernels: Poisson ker-nel1/1+x2、Gauss kernel e-x2and the kernel1/1+x4.We proved that if f(x)is a bounded integratiable function, and Tn(f):=Pn*f(x),then λTn’(f)(ξ)/n�'Dξ(f)(n�'∞),where λ=∫R P(x)dxFurthermore we give some estimations of convergence rate for piecewise Lip α functions f(x)with a simple discontinuity ξ. Set dξ(f):=limt�'O+[f(ξ+t)-f(ξ-t)],if the kernel P(x)is one of above three kernels,and Pn(x)=nP(nx),Tn(f)(x):=Pn*f(x),we proved that the approximation degree of λ·Tn’(f)(ξ)/n to dξ(f) is O(1/nα).