Abstract
For the classes L r2 (ℝ), r ∈ ℤ+, we establish the upper and lower bounds for the quantities
where µ, r ∈ ℤ+, µ ≤ r, k ∈ ℕ, 0 < p ≤ 2, 0 < σ < ∞, 0 < t ≤ π/σ, and ψ is a nonnegative, measurable function summable on the closed interval [0, t] and not equivalent to zero. In the cases χ σ, k, r, μ, p , where μ ∈ ℕ, 1/μ ≤ p ≤ 2, and χ σ, k, r, μ, 2/k (1, t), where 0 < t ≤ π/(2σ), we obtain the exact values of these quantities. We also obtain the exact values of the average ν-widths of classes of functions defined in terms of the modulus of continuity ω* and the majorant ψ.
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Original Russian Text © S. B. Vakarchuk, 2014, published in Matematicheskie Zametki, 2014, Vol. 96, No. 6, pp. 827–848.
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Vakarchuk, S.B. Best mean-square approximations by entire functions of exponential type and mean ν-widths of classes of functions on the line. Math Notes 96, 878–896 (2014). https://doi.org/10.1134/S000143461411025X
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DOI: https://doi.org/10.1134/S000143461411025X