We investigate the approximation properties of the trigonometric system in L p. / 2 : We consider the moduli of smoothness of fractional order and obtain direct and inverse approximation theorems together with a constructive characterization of a Lipschitz-type class.

Abstract. We use the approximation properties of the Faber polynomials to obtain some direct theorems of the polynomial approximation in Smirnov-Orlicz classes.

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This work deals with basic approximation problems such as direct, inverse and simultaneous theorems of trigonometric approximation of functions of weighted Lebesgue spaces with a variable exponent on weights satisfying a variable Muckenhoupt A p(·) type condition. Several applications of these results help us transfer the approximation results for weighted variable Smirnov spaces of functions defined on sufficiently smooth finite domains of complex plane ℂ.

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