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

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