Moduli of smoothness and best approximation of functions with singularities
Let \(\varphi_{\lambda,\alpha,0}(x)=|x|^\lambda\exp(iA|x|^{-\alpha}),\) where \(\lambda>0\), \(\alpha>0\), \(A\in R\backslash\{0\}\) and \(\varphi_{\lambda,\alpha,1}(x)= \operatorname {Re}\varphi_{\lambda,\alpha,0}(x),\) \(\varphi_{\lambda,\alpha,2}(x)= \operatorname {Im}\varphi_{\lambda,\alpha,0}(x).\) The question of the best approximation of functions \(\varphi_{\lambda,\alpha,i}\), \(i=0,1,2,\) by algebraic plynomials (resp., by entire functions of exponential type) with respect to the norm of the space \(L_p[-a,a]\), \(p\in[1,\infty]\) (resp., \(L_p(-\infty,\infty)\)) are studied. For example it is proved that the exact order of the best approximation by polynomials of order \(n\) is \(n^{-(\lambda+1/p)/(1+\alpha)}.\) Besides error estimates for the best approximation of functions \(\psi_{\lambda,\alpha,i,s}(\tau)=\varphi_{\lambda, \alpha,i}(\sin\tau)\cos^s \tau\), \(i=0,1,2,\) \(s\in N\cup\{0\}\) by means of trigonometric polynomials and periodic splines are given.
scientific article; zbMATH DE number 1471518
scientific article