Elijah Liflyand

• Prof. PhD
• Professor (Associate) at Bar Ilan University

Pólya-type functions are of special importance in probability and harmonic analysis. We introduce and study their multidimensional extensions.

The aim of this work is to derive a symbol calculus on \(L^2(\mathbb {R}^n)\) for multidimensional Hausdorff operators. Two aspects of this activity result in two almost independent parts. While throughout the perturbation matrices are supposed to be self-adjoint and form a commuting family, in the second part they are additionally assumed to be po...

The aim of this work is to derive a symbol calculus on $L^2(\mathbb{R})$ for one-dimensional Hausdorff operators in apparently the most general form.

The aim of this work is to derive a symbol calculus on $L^2(\mathbb{R}^n)$ for multidimensional Hausdorff operators. Two aspects of this activity result in two almost independent parts. While throughout the perturbation matrices are supposed to be self-adjoint and form a commuting family, in the second part they are additionally assumed to be posit...

The aim of this work is to derive a symbol calculus on L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\mathbb {R})$$\end{document} for one-dimensional Hausdor...

A very general asymptotic type formula is presented for the Lebesgue constants of both Fourier partial sums and linear methods of summability of Fourier series in the multivariate Euclidean spaces. More precisely, the obtained formula represents the norm of a trigonometric polynomial with coefficients being the values of a multiplier as the norm of...

The author’s recent attempt to generalize the problem of L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} convergence of trigonometric series to th...

Let \(W_0(\mathbb {R})\) be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proved in the paper that, in particular, a trigonometric series \(\sum \nolimits _{k=-\infty }^\infty c_k e^{ikt}\) is the Fourier series of an integrable function if and only if there exists a \(\phi \in...

We give an asymptotic formula for the Lebesgue constants of the Riesz means of negative order in dimension one. The obtained result allows us to extend the asymptotic relations to those for the Lebesgue constants of more general means.

This is the first attempt to generalize the problem of \(L^1\) convergence of trigonometric series to the non-periodic case. We extend one of the most general results and then show the way how to derive its prototype from the obtained extension.

New sufficient conditions for the almost everywhere convergence (at the Lebesgue points) of general summability means of the conjugate integral to the Hilbert transform are obtained in the paper. Consequences and applications are also discussed.

Salem type conditions for trigonometric series are extended to functions from the Wiener algebra. While in the earlier one-dimensional generalization the conditions are given in terms of the Hilbert transform, for the multivariate setting all reasonable singular integrals are equally involved.

A new class of functions is introduced closely related to that of functions with bounded Tonelli variation and to the real Hardy space. For this class, conditions for integrability of the Fourier transform are established.

Let $W_0(\mathbb R)$ be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proven in the paper that, in particular, a trigonometric series $\sum\limits_{k=-\infty}^\infty c_k e^{ikt}$ is the Fourier series of an integrable function if and only if there exists a $\phi\in W_0(\mathbb...

In this paper we obtain new sufficient conditions for representation of a function as an absolutely convergent Fourier integral. Unlike those known earlier, these conditions are given in terms of belonging to weighted spaces. Adding weights allows one to extend the range of application of such results to Fourier multipliers with unbounded derivativ...

Salem’s necessary conditions for a trigonometric series to be the Fourier series of an integrable function are generalized to the nonperiodic case for functions in the Wiener algebra. Applications of the obtained result are given.

New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. The main result of the paper is an asymptotic formula for the cosine Fourier transform. Such relations have previously been known only for the sine Fourier transform. For this, not only a different space is consid...

In this paper we obtain new sufficient conditions for representation of a function as an absolutely convergent Fourier integral. Unlike those known earlier, these conditions are given in terms of belonging to weighted spaces. Adding weights allows one to extend the range of application of such results to Fourier multipliers with unbounded derivativ...

Найденное новое доказательство асимптотической формулы для синус-преобразования Фурье функции ограниченной вариации проведено полностью\linebreak в рамках теории пространств Харди, в первую очередь, с помощью неравенства Харди. Показано, что все стороны поведения преобразования Фурье функции ограниченной вариации с производной в пространстве Харди...

For the asymptotic formula for the Fourier sine transform of a function of bounded variation, we find a new proof entirely within the framework of the theory of Hardy spaces, primarily with the use of the Hardy inequality. We show that, for a function of bounded variation whose derivative lies in the Hardy space, every aspect of the behavior of its...

We transform Leray’s formula on the Fourier transform of a radial function in such a way that it preserves the form of a one-dimensional Fourier transform and the transformed function is close to the initial function as much as possible.

The Paley-Wiener theorem states that the Hilbert transform of an integrable odd function, which is monotone on \(\mathbb{R}_{+}\), is integrable. In this paper we prove weighted analogs of this theorem for sequences and their discrete Hilbert transforms under the assumption of general monotonicity for an even/odd sequence.

A one-dimensional version of the Poisson summation formula for functions of bounded variation due to R. M. Trigub is extended to the multivariate case under minimal assumptions on functions.

A known Hardy-Littlewood theorem asserts that if both the function and its conjugate are of bounded variation, then their Fourier series are absolutely convergent. It is proved in the paper that the same result holds true for functions on the whole axis and their Fourier transforms with certain adjustments. The proof of the original Hardy-Littlewoo...

We compare, in the multidimensional case, the Fourier integral of a function lambda of bounded variation and the corresponding trigonometric series with the coefficients lambda(k). Posing additional smoothness conditions on the function, we infer that the difference between the two mentioned values is controlled not only by the total variation as i...

A condition of proved worth guarantees almost everywhere convergence of Fourier integrals of functions from an essentially wider class than known earlier.

New sucient conditions for representation of a function via the absolutely convergent Fourier integral are obtained in the paper. In the main result, Theorem 1.1, this is controlled by the behavior near innity of both the function and its derivative. This result is extended to any dimension d 2:

Certain relations between the Fourier transform of a function and the Hilbert transform of its derivative are revealed. They concern the integrability/non-integrability of both transforms. Certain applications are discussed.

Fourier transform estimates for parallel to(f) over cap parallel to L-q,L-(w) over tilde via parallel to f parallel to L-p,L- w from above and from below are studied. For p = q, equivalence results, i.e., C-1 parallel to f parallel to L-p,L- w <= parallel to(f) over cap parallel to L-p,L- (w) over tilde <= C-2 parallel to f parallel to L-p,L- w, (w...

We introduce an amalgam type space, a subspace of $L^1(\mathbb R_+).$ Integrability results for the Fourier transform of a function with the derivative from such an amalgam space are proved. As an application we obtain estimates for the integrability of trigonometric series.

In this survey, results on the representation of a function as an absolutely convergent Fourier integral are collected, classified and discussed. Certain applications are also given.

New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. After various preceding works of the last 25 years where the behavior of the Fourier transform has been considered on specific subspaces of the space of functions of bounded variation, in this paper such problems...

We study an extension to Fourier transforms of the old problem on absolute convergence of the re-expansion in the sine (cosine) Fourier series of an absolutely convergent cosine (sine) Fourier series. The results are obtained by revealing certain relations between the Fourier transforms and their Hilbert transforms.

Weighted L p (ℝn ) → L q (ℝn ) Fourier inequalities are studied. We prove Pitt-Boas type results on integrability with power weights of the Fourier transform of a radial function. Extensions to general weights are also given.

Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained in the paper. The results are given in terms of $L^p$ integrability of the function and its partial derivatives, each with the corresponding $p$. These $p$ are subject to certain relations known earlier o...

In this paper we investigate properties of the general monotone sequences and functions, a generalization of monotone sequences and functions as well as of those of bounded variation. Some applications to various problems of analysis are given. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Analyzing several classical tests for convergence/divergence of number series, we relax the monotonicity assumption for the sequence of terms of the series. We verify the sharpness of the obtained results on corresponding classes of sequences and functions.

We compare the Fourier integral of a function of bounded variation and the corresponding trigonometric series, generated by that function, in the multidimensional case. Several known notions of bounded variation are used and a new one is introduced. The obtained results are applied to integrability of multidimensional trigonometric series.

New sufficient conditions for the representation of a function via an absolutely convergent Fourier integral are obtained in the paper. In the main result, this is controlled by the behavior near infinity of both the function and its derivative. This result is extended to any dimension d≥2d≥2.

New sufficient conditions for representation of a function as an absolutely convergent Fourier integral are obtained in the paper. Mathematical Reviews subject classification: Primary: 42A38; Secondary: 42A25 Key words: Fourier integral, absolute convergence, Hardy inequality

New sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained in the paper.

Firstly, we study the uniform convergence of cosine and sine Fourier transforms. Secondly, we obtain Pitt–Boas type results on Lp-integrability of Fourier transforms with the power weights. The solutions of both problems are written as criteria in terms of general monotone functions.

We prove weighted analogues of the Paley–Wiener theorem on integrability of the Hilbert transform of an integrable odd function which is monotone on R+. This extends Hardy–Littlewood's and Flett's results to the case p=1 under the assumption of (general) monotonicity for an even/odd function.

We obtain new sufficient conditions for the representability of a function by an absolutely convergent Fourier integral in ℝ d . These conditions are given in terms of the simultaneous behavior of a function and its derivatives at ∞. We test the sharpness of the conditions using well-known examples.

We study the multiplicative, tensor, Sobolev and convolution inequalities in certain Banach spaces, the so-called Bilateral Grand Lebesgue Spaces. We also give examples to show the sharpness of these inequalities when possible.

We give an example of a function which belongs only to the largest space in the chain of embedded spaces, important in various problems of analysis.

Extending the notion of the general monotonicity for sequences to functions, we exploit it to investigate integrability problems for Fourier transforms. The problem of controlling integrability properties of the Fourier transform separately near the origin and near infinity is examined. We then apply the obtained results to the problems of integrab...

We prove the necessary conditions for the integrability of the Fourier transform. The result is a generalization, on one hand, of the well known necessary condition for absolutely convergent Fourier series and, on the other hand, of an earlier multidimensional result of Trigub.

We give criteria for a function to be in the Hardy space on a bounded complete Reinhardt domain. Using these and known one-dimensional results, we obtain boundedness conditions for Hausdorff operators on Hardy spaces in Reinhardt domains. The only known earlier result for the polydisk is a paticular case of the obtained results.

Weighted Lp→Lq Fourier inequalities are studied. We prove Boas' conjecture on integrability with power weights of the Fourier transform. One-dimensional as well as multidimensional versions (for radial functions) are obtained for general monotone functions. To cite this article: E. Liflyand, S. Tikhonov, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

In this paper we study the multiplicative, tensor, Sobolev's and convolution inequalities in certain Banach spaces, the so-called Bide - Side Grand Lebesque Spaces, and give examples to show their sharpness.

Tests for the integrability of the Fourier transform of a function are given in terms of belonging of the function simultaneously to two spaces of smooth functions. These are, in a sense, generalizations of Zygmund´s test for the absolute convergence of Fourier series.

For a wide family of multivariate Hausdorff operators, the boundedness of an operator from this family is proved on the real Hardy space. By this we extend and strengthen previous results due to K.F. Andersen and F. Móricz.

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R 2, and acting on the real Hardy space H 1(R), or the product Hardy space H 11(RR), or one of the hybrid Hardy spaces H 10(R 2) and H 01(R 2), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator g...

Weak and strong estimates in weighted Lp spaces are obtained for linear means of Fourier integrals defined by a single function with support in a specially organized set.

Two types of spaces of sequences as well as their analogs for func-tions are compared. One of them was inspired by results of A. Beurling in spectral synthesis. The other has appeared in the work of R. P. Boas in trigonometric series. It turns out that natural additional assumptions provide the equivalence of these two types of spaces. Applications...

Following R. Fefferman, the product Hardy space H 11 (ℝ×ℝ) of functions f∈L 1 (ℝ 2 ) is defined by the requirement that the Hilbert transforms f ˜ 10 , f ˜ 01 and f ˜ 11 also belong to L 1 (ℝ 2 ). The proof of the statement claimed in the title relies on the closed graph theorem and on the fact that if a function f∈L 1 (ℝ 2 ) is such that its Fouri...

Complementary spaces for Fourier series were introduced by G. Goes and generalized by M. Tynnov. In this paper we investigate a notion of complementary space for double Fourier series of functions of bounded variation. Various applications are given.

We prove that the Hausdor operator generated by a function ' 2 L 1 (R) is bounded on the real Hardy space H 1 (R). The proof is based on the closed graph theorem and on the fact that if a function f in L 1 (R )i s such that its Fourier transform b f(t )e quals 0f or t< 0( or for t> 0), then f2 H 1 (R). 1. Preliminaries We recall that the Fourier tr...

A mean-value characterization of holomorphic and pluriharmonic functions in n>1, was obtained by the authors in their first paper on this problem under additional assumption of sufficient smoothness of a given function. Here not only this restriction is removed but a mean-value characterization is obtained for distributions of finite order.

Some problems of summability and localization are considered for square linear moans of Fourier series defined by hyperbolically symmetric multipliers. © 2014, Oldenbourg Wissenschaftsverlag GmbH, Rosenheimer Str. 145, 81671 München. All rights reserved.

A connection is established between the multidimensional Fourier transform of a radial function f from a given class and the one-dimensional Fourier transform of a related function. This is applied to give an asymptotic formula for the Fourier transform of f. The function class in question is compared with related classes already considered in the...

Asymptotics is obtained for the Lebesgue constants of the Cesàro means of spherical harmonic expansions. Precise constant in the main term is found and the order of growth of the remainder term is given.

Beurling's algebra A * = {f : ∞ P k=0 sup k≤|m| f (m)| < ∞} is considered. A * arises quite naturally in problems of summability of the Fourier series at Lebesgue points, whereas Wiener's algebra A of functions with absolutely convergent Fourier series arises when studying the norm convergence of linear means. Certainly, both algebras are used in s...

We show that holomorphic functions in are characterized by a special mean-value condition, and also indicate a new mean-value characteristic for pluriharmonic functions, that is real parts of holomorphic functions.

Estimates from below for the norms of linear means of multiple Fourier series are obtained. These means are given by some function λ and generalize the well-known Bochner-Riesz means. Sharpness of these estimates is established. The assumptions on λ are rather weak and of local character. Our results contain as particular cases a number of earlier...

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