Abstract
We show that D. Lépingle’s L_1(ell _2)-inequality ∑nE[fn|Fn-1]21/21≤2·∑nfn21/21,fn∈Fn,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\| \\left( \\sum _n {\\mathbb {E}}[f_n | {\\mathscr {F}}_{n-1}]^2 \\right) ^{1/2}\\right\\| _1 \\le 2\\cdot \\left\\| \\left( \\sum _n f_n^2 \\right) ^{1/2} \\right\\| _1, \\quad f_n\\in {\\mathscr {F}}_n, \\end{aligned}$$\\end{document}extends to the case where we substitute the conditional expectation operators with orthogonal projection operators onto spline spaces and where we can allow that f_n is contained in a suitable spline space {mathscr {S}}({mathscr {F}}_n). This is done provided the filtration ({mathscr {F}}_n) satisfies a certain regularity condition depending on the degree of smoothness of the functions contained in {mathscr {S}}({mathscr {F}}_n). As a by-product, we also obtain a spline version of H_1-{{,mathrm{BMO},}} duality under this assumption.
Highlights
This article is part of a series of papers that extend martingale results to polynomial spline sequences of arbitrary order
In order to explain those martingale type results, we have to introduce a little bit of terminology: Let k be a positive integer, (Fn) an increasing sequence of σ -algebras of sets in [0, 1] where each Fn is generated by a finite partition of [0, 1] into intervals of positive length
To the definition of martingales, we introduce the following notion: letn≥0 be a sequence of integrable functions
Summary
This article is part of a series of papers that extend martingale results to polynomial spline sequences of arbitrary order (see e.g. [11,14,16,17,18,19,22]). Properties (ii) and (iii) are proved in [19] and properties (iv) and (v) in [14], but see [18], where it is shown that, in analogy to the martingale case, the validity of (iv) and (v) for all k-martingale spline sequences with values in a Banach space X characterize the Radon–Nikodým property of X (for background information on that material, we refer to the monographs [6,20]) We continue this line of transferring martingale results to k-martingale spline sequences and extend Lépingle’s L1( 2)-inequality [12], which reads. Garsia’s book [8] on Martingale Inequalities
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