The Polya algorithm in L∞ approximation

Best monotone approximation in 𝐿₁[0,1]

“…The Polya property fails for an arbitrary / in Lx [2,1] but, as is shown in this note, the Polya-one property obtains.…”

Asymptotic behavior of 𝑝-predictions for vector valued random variables

“…Notice that the study of the convergence as p -* oo of p-predictions on uniformly convex spaces was carried out in [7], but the proof of the existence of the limit is not satisfactory as pointed out in [4]. (Also in connection with the Pólya algorithm see [5].)…”

Tubular sets and multivariate Polya algorithm

“…In particular, it generalizes the simultaneous-Polya-algorithm-convergence result of [12], and it establishes the convergence of the Polya algorithm in a general context which includes the approximation of multivariate by univariate functions [1]. [2] Tubular sets and multivariate Polya algorithm 233 Ifw = (w u ...,w n )eW let ' l < p < c x ) and llwlloo = max \x t \.…”

“…Current usage of the Polya algorithm is primarily theoretical. In [2] it was used to show that best uniform approximations exist, in [6] its limit was shown to be a Lipschitz uniform selection operator, and in [3] and [8] its convergence was studied in new contexts. The theme underlying these studies is the evolution of L p to L x as p -> oo.…”

“…Again we assume without loss of generality that a = 0. Let X' be the hyperplane (x 1 ,..., x"" 2 , v) and let S 1 and S 2 be the two open half spaces defined by X'. For i = 1,2 choose y' 6 S' D K if possible and, if this is not possible, let y' = 0.…”