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Let G ⊂ C be a bounded Jordan domain with a rectifiable Dini-smooth boundary Γ and let G− := ext Γ. In terms of the higher order modulus of smoothness the direct and inverse problems of approximation theory in the variable exponent Smirnov classes Ep(·)(G) and Ep(·)(G−) are investigated. Moreover, the Marcinkiewicz and Littlewood–Paley type theorems are proved. As a corollary some results on the constructive characterization problems in the generalized Lipschitz classes are presented.

Multiplier and approximation theorems in Smirnov classes with variable exponent