This paper studies several problems,which are potentially relevant for the construction of adaptive numerical schemes,First,Biorthogonal spline wavelets on [0,1] are chosen as a starting point for characteri zations of functions in Besov spaces Br.r^σ(0,1) with 0<σ<∞ and (1+σ)^-1<r<∞,Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered.Besides characterization results Jackshon type estimates for various tree-type and tresholding algorithms are investigated.Finally knowm approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.
In this paper we constructh bivariate polynomials attached to a bivariate function,that approoximate with Jacksan-type rate involving a bivariate Ditzian-Totik ω2-modulus of smoothness and presrves some natural kinds of bivariate monotonicity and convexity of function.The result extends that in univariate case-of D.Leviatan in [5-6],improves that in bivariate case of the author in  and in some special cases,that in bivariate case of G.Anastassiou in .
It is well known that the Walsh-Fourier expansion of a function from the block space Kq([0,1)),1<q≤∞,converges pointwise a.e.We prove that the same result is true for the expansion of a function from Kq in certain periodized smooth periodic non-stationary wavelet packets bases based on the Haar filters.We also consider wavelet packets based on the Shannon filters and show that the expansion of L^p-functions,1<q<∞,converges in norm and pointwise almost everywhere.
The authors esstablish the boundedness on homogeneous weighted Herz spaces for a large class of rough operators and their commutators with BMO functions.In particular ,the Cadderon-Zygmund singular integrals and the rough R.Fefferman singular integral operators and the rough Ricci-Stein oscillatory singular integrals and the corresponding commutators are considered.
In this paper ,the dimension of the spaces of bivariate spline with degree less that 2r and smoothness order r on the Morgan-Scott triangulation is consdered.The concept of the instability degree in the dimension of spaces of bivariate spline is presented.The resulats in the paper make us conjecture the instability degree in the dimension of spaces of bivariate spline is infinity.
A degree elevation formula for multivariate simplex splines was given by Micchellis and extended to hold for multivariate Dirichlet splines in .We report similar formulae for multivariate cone splines and box splines.To this and ,we utilize a relation due to Dahmen and Micchelli that connects box splines and cone splines and a degree reduction formula given by Cohen,Lyche,and Riesenfeld in .
Let C( R+^2) be a class of continuous functions f on R+^2.A bivariate extension Ln(f,x,y)of Bleiman-Butzer-Hahn operator is defined and its standard convergence properties are given.Moreover,a local analogue of Voronovskaja theorem is also give for a subclass of C(R+^2).