Estimates of approximations by interpolation trigonometric polynomials on the classes of convolutions of high smoothness

2023/08/23 by Serdyuk, A. S., Stepaniuk, T. A.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.12184

Abstract

We establish interpolation analogues of Lebesgue type inequalities on the sets of C^ψ_βL₁ 2π-periodic functions f, which are representable as convolutions of generating kernel Ψ_β(t) = ∑_k=1^∞ψ(k)cos (kt-(βπ)/(2)), ψ(k)≥ 0, ∑_k=1^∞ψ(k)<∞, β∈ℝ, with functions φ from L₁ . In obtained inequalities for each x∈ℝ the modules of deviations |f(x)- S_n-1(f;x)| of interpolation Lagrange polynomials S_n-1(f;⋅) are estimated via best approximations E_n(φ)_L₁ of functions φ by trigonometric polynomials in L₁-metrics. When the sequences ψ(k) decrease to zero faster than any power function, the obtained inequalities in many important cases are asymptotically exact. In such cases we also establish the asymptotic equalities for exact upper boundaries of pointwise approximations by interpolation trigonometric polynomials on the classes of convolutions of generating kernel Ψ_β with functions φ, which belong to the unit ball from the space L₁.

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• Approximation by interpolation trigonometric polynomials in metrics of the spaces L_p on the classes of periodic entire functions
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• Uniform approximations by Fourier sums on the sets of convolutions of periodic functions of high smoothness
• Asymptotic behavior of best approximations of classes of infinitely differentiable functions defined by moduli of continuity
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