A simple observation about compactness and fast decay of Fourier coefficients.
A simple proof of a weighted inequality for the Hardy-Littlewood maximal operator in .
A simple proof of the atomic decomposition for , 0 < p ≤ 1
A Singular Convolution Equation In The Space Of Distributions
A singular convolution equation in the space of distributions.
A special Gaussian rule for trigonometric polynomials.
A stability result on Muckenhoupt's weights.
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 stable method for the inversion of the Fourier transform in
A general method is given for recovering a function , , knowing only an approximation of its Fourier transform.
A strengthened form of a theorem of Wiener.
A strengthening of a theorem of Marcinkiewicz
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.
A strong convergence theorem for H¹(𝕋ⁿ)
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, .
A strong version of Poisson summation.
A study of the real Hardy inequality.
A study on degree of approximation by summability means of the Fourier-Laguerre expansion.
A subelliptic Bourgain–Brezis inequality
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 .
A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform
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.
A survey of certain results on strong approximation by orthogonal series
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.
A survey of weighted polynomial approximation with exponential weights.
A survey on D-semiclassical orthogonal polynomials.
A survey on uncertainty principles related to quadratic forms.