Conditional bounds and best L∞-approximations in probability spaces

• Published in: Journal of approximation theory Vol. 56; no. 1; pp. 1 - 12
• Main Authors: Cuesta, Juan A., Matrán, Carlos
• Format: Journal Article
• Language: English
• Published: Brugge Elsevier Inc 1989; Academic Press
• Subjects: Exact sciences and technology; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Sciences and techniques of general use; Characterization; Best approximation; Approximation; Algorithm analysis; Convergence; Probability space
• ISSN: 0021-9045; 1096-0430
• DOI: 10.1016/0021-9045(89)90128-7

The paper deals with L ∞-approximations in a probability space ( Ω, σ, P) by means of α-measurable random variables for α ⊂ σ, a σ-lattice. Attention is paid to the characterization of the set of all best L ∞-approximations in terms of the notion of “conditional bounds,” developed in the paper. On the other hand we study in the framework above the Pólya algorithm, showing that, if ƒ r denotes a best L r -approximation and r( n) → ∞, then lim inf ƒ r(n) and lim sup ƒ r(n) are best L ∞-approximations. We also point out an error in an article on this subject by Darst and discuss the validity of subsequent articles by Darst, Al-Rashed, and others.