Optimal recovery of operators and multidimensional Carlson type inequalities

1. General setting Let T be a nonempty set, Σ be the σ -algebra of subsets of T , and μ be a nonnegative σ -additive measure on Σ . We denote by Lp(T , Σ, μ) (or simply Lp(T , μ)) the set of all Σ-measurable functions with values in R or in C for which ∥x(·)∥Lp(T ,μ) =  T |x(t)|p dμ(t) 1/p < ∞, 1 ≤ p < ∞, ∥x(·)∥L∞(T ,μ) = ess sup t∈T |x(t)| < ∞, p = ∞. Put W = {x(·) ∈ Lp(T , μ) : ∥φ(·)x(·)∥Lr (T ,μ) < ∞}, W = {x(·) ∈ W : ∥φ(·)x(·)∥Lr (T ,μ) ≤ 1}, where 1 ≤ p, r ≤ ∞, and φ(·) is a measurable function on T . Consider the problem of recovery of operator Λ : W → Lq(T , μ), 1 ≤ q ≤ ∞, defined by equality Λx(·) = ψ(·)x(·), where ψ(·) E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jco.2015.07.004 0885-064X/© 2015 Elsevier Inc. All rights reserved. 54 K.Yu. Osipenko / Journal of Complexity 32 (2016) 53–73 is a measurable function on T , on the class W by the information about functions x(·) ∈ W given inaccurately. More precisely, we assume that for any function x(·) ∈ W we know y(·) ∈ Lp(T0, μ), where T0 is not empty μ-measurable subset of T , such that ∥x(·) − y(·)∥Lp(T0,μ) ≤ δ, δ ≥ 0. We want to approximate the value Λx(·) knowing y(·). As recovery methods we consider all possible mappings m : Lp(T0, μ) → Lq(T , μ). The error of a methodm is defined as e(p, q, r,m) = sup x(·)∈W , y(·)∈Lp(T0,μ) Lp(T0,μ) ≤δ ∥Λx(·) − m(y)(·)∥Lq(T ,μ). The quantity E(p, q, r) = inf m : Lp(T0,μ)→Lq(T ,μ) e(p, q, r,m) (1) is known as the optimal recovery error, and a method on which this infimum is attained is called optimal. Various settings of optimal recovery theory and examples of such problems may be found in [11,12,17,18,15,13]. Much of them are devoted to optimal recovery of linear functionals. There are not so many results about optimal recovery of linear operators when non-Euclidean metrics is used [12, Theorem 12 on p. 45], [6,14]. In [14] we considered problem (1) when any two of p, q, and r coincide. Here we analyze the case when all metrics can be different and 1 ≤ q < p, r < ∞. We construct optimal method of recovery, find its error, and apply this result to obtain exact constants in Carlson type inequalities. The case p = ∞ and/or r = ∞ requires a slightly different approach. Some particular results of such kind may be found in [8] (T = Z) and [9] (T = R). 2. Main results Let χ0(·) be the characteristic function of the set T0: χ0(t) =  1, t ∈ T0, 0, t ∉ T0. Theorem 1. Let 1 ≤ q < p, r < ∞, λ1, λ2 ≥ 0, λ1 + λ2 > 0, φ(t) ≠ 0 for almost all t ∈ T \ T0, x(t) =x(t, λ1, λ2) ≥ 0 be a solution of equation − q|ψ(t)|q + pλ1x(t)χ0(t) + rλ2|φ(t)|x(t) = 0, (2) and λ1, λ2 such that  T0 xp(t) dμ(t) ≤ δp,  T |φ(t)|rx r(t) dμ(t) ≤ 1,