Best mean-square approximations by entire functions of exponential type and mean ν-widths of classes of functions on the line.

For the classes L(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, where μ ∈ ℕ, 1/μ ≤ p ≤ 2, and χ(1, t), where 0 < t ≤ π/(2 σ), 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. [ABSTRACT FROM AUTHOR]