A simple observation about compactness and fast decay of Fourier coefficients.
We prove that Muckenhoupt's A1-weights satisfy a reverse Hölder inequality with an explicit and asymptotically sharp estimate for the exponent. As a by-product we get a new characterization of A1-weights.
A general method is given for recovering a function , , knowing only an approximation of its Fourier transform.
We consider a problem of intervals raised by I. Ya. Novikov in [Israel Math. Conf. Proc. 5 (1992), 290], which refines the well-known theorem of J. Marcinkiewicz concerning structure of closed sets [A. Zygmund, Trigonometric Series, Vol. I, Ch. IV, Theorem 2.1]. A positive solution to the problem for some specific cases is obtained. As a result, we strengthen the theorem of Marcinkiewicz for generalized Cantor sets.
Let ⁿ denote the usual n-torus and let , u > 0, denote the Bochner-Riesz means of order δ > 0 of the Fourier expansion of f ∈ L¹(ⁿ). The main result of this paper states that for f ∈ H¹(ⁿ) and the critical index α: = (n-1)/2, .
We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space by functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for on the Heisenberg group .
Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space . We show that if W and its inverse both satisfy a matrix reverse Hölder property introduced by Christ and Goldberg, then the weighted Hilbert transform and also all weighted dyadic martingale transforms are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.
This is a survey of results in a particular direction of the theory of strong approximation by orthogonal series, related mostly with author's contributions to the subject.