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On the approximation by convolution operators in homogeneous Banach spaces of periodic functions. (English) Zbl 1237.41004

Summary: The paper is concerned with establishing direct estimates for convolution operators on homogeneous Banach spaces of periodic functions by means of an appropriately defined \(K\)-functional. The differential operator in the \(K\)-functional is defined by means of a strong limit and described explicitly in terms of its Fourier coefficients. The description is simple and independent of the homogeneous Banach space. Saturation of such operators is also considered.

MSC:

41A25 Rate of convergence, degree of approximation
41A27 Inverse theorems in approximation theory
41A35 Approximation by operators (in particular, by integral operators)
41A36 Approximation by positive operators
41A40 Saturation in approximation theory
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42A85 Convolution, factorization for one variable harmonic analysis
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