Asymptotic Behaviour of the Error of Polynomial Approximation of Functions Like $$\vert x\vert ^{{\alpha }+i{\beta }}$$

Abstract

New asymptotic relations between the $$L_p$$ -errors of polynomials approximation of univariate functions by algebraic polynomials and entire functions of exponential type are obtained for $$p\in (0,{\infty }]$$ . General asymptotic relations are applied to functions $$\vert x\vert ^{{\alpha }+i{\beta }},\,\vert x\vert ^{{\alpha }}\cos ({\beta }\log \vert x\vert )$$ , and $$\vert x\vert ^{{\alpha }}\sin ({\beta }\log \vert x\vert )$$ .