Absolute Riesz summability of Fourier series
Absolutely convergent Fourier series and generalized Lipschitz classes of functions
We investigate the order of magnitude of the modulus of continuity of a function f with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that f belong to one of the generalized Lipschitz classes Lip(α,L) and Lip(α,1/L), where 0 ≤ α ≤ 1 and L = L(x) is a positive, nondecreasing, slowly varying function such that L(x) → ∞ as x → ∞. For example, a 2π-periodic function f is said to belong to the class Lip(α,L) if for all x ∈ , h >...
Accelerating the convergence of trigonometric series
A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed.
Accuracy of Several Multidimensional Refinable Distributions.
Adaptive convex optimization in Banach spaces: a multilevel approach
This is mainly a review paper, concerned with some applications of the concept of Nonlinear Approximation to adaptive convex minimization. At first, we recall the basic ideas and we compare linear to nonlinear approximation for three relevant families of bases used in practice: Fourier bases, finite element bases, wavelet bases. Next, we show how nonlinear approximation can be used to design rigorously justified and optimally efficient adaptive methods to solve abstract minimization problems in...
Adaptive density estimation under weak dependence
Assume that (Xt)t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue's measure. [Donoho et al. Ann. Stat.24 (1996) 508–539] propose near-minimax estimators based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are...
Adaptive Deterministic Dyadic Grids on Spaces of Homogeneous Type
In the context of spaces of homogeneous type, we develop a method to deterministically construct dyadic grids, specifically adapted to a given combinatorial situation. This method is used to estimate vector-valued operators rearranging martingale difference sequences such as the Haar system.
Adaptive wavelet methods for saddle point problems
Adaptive wavelet methods for saddle point problems
Recently, adaptive wavelet strategies for symmetric, positive definite operators have been introduced that were proven to converge. This paper is devoted to the generalization to saddle point problems which are also symmetric, but indefinite. Firstly, we investigate a posteriori error estimates and generalize the known adaptive wavelet strategy to saddle point problems. The convergence of this strategy for elliptic operators essentially relies on the positive definite character of the operator....
Addendum to a paper "On singular integrals"
Addition au mémoire sur les fonctions discontinues
Additions to the Telyakovskiĭ's class .
Adèles et séries trigonométriques spéciales
Affine frames, GMRA's, and the canonical dual
We show that if the canonical dual of an affine frame has the affine structure, with the same number of generators, then the core subspace V₀ is shift invariant. We demonstrate, however, that the converse is not true. We apply our results in the setting of oversampling affine frames, as well as in computing the period of a Riesz wavelet, answering in the affirmative a conjecture of Daubechies and Han. Additionally, we completely characterize when the canonical dual of a quasi-affine frame has the...
Affine Systems in L...(...) II: Dual Systems.
Algebrability of the set of non-convergent Fourier series
We show that, given a set E ⊂ 𝕋 of measure zero, the set of continuous functions whose Fourier series expansion is divergent at any point t ∈ E is dense-algebrable, i.e. there exists an infinite-dimensional, infinitely generated dense subalgebra of 𝓒(𝕋) every non-zero element of which has a Fourier series expansion divergent in E.
Algebras of Special Singular Integrals over Local Fields.
Algorithm 79. An algorithm for Fourier series expansion in a subinterval of the circumference
Allied integrals, functions, and series for the unit sphere.
Almost everywhere convergence of convolution powers on compact abelian groups
It is well-known that a probability measure on the circle satisfies for every , every (some) , if and only if for every non-zero ( is strictly aperiodic). In this paper we study the a.e. convergence of for every whenever . We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of , for the strong sweeping out property (existence of a Borel set with a.e. and a.e.). The results are extended to general compact Abelian groups with Haar...