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A recent theorem of S. Alesker, S. Artstein-Avidan and V. Milman characterises the Fourier transform on R-n as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. In this note we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group.

In this paper for functions on locally compact Vilenkin groups, we prove an analogue of one classical Titchmarsh theorem on the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L2.

We study convergence almost everywhere of multiple trigonometric Fourier series over cubes defined on the d d -dimensional torus T d \mathbb {T}^d . We provide a new approach which allows us to prove the novel interpolation estimates for the Carleson maximal operators generated by the partial sums of the multiple Fourier series and all its conjugate series. Combining these estimates we show that these operators are bounded from a variant of the Arias-de-Reyna space Q A d Q\!A^d to the weak L 1 L^1 -space on T d \mathbb {T}^d . This implies that the multiple Fourier series of every function f ∈ Q A d f\in Q\!A^d and all its conjugate series converge over cubes almost everywhere. By a close analysis of the space Q A d Q\!A^d we prove that it contains a Lorentz space that strictly contains the Orlicz space L ( log L ) d log ⁡ log ⁡ log L ( T d ) L(\log \,L)^{d} \log \log \log \,L(\mathbb {T}^d) . This yields a significant improvement of a deep theorem proved by Antonov which was the best known result on the convergence of multiple Fourier series over cubes.

We give the solution of the Turan, Fejer, Delsarte, Logan, and Bohman extremal problems for the Fourier transform on the hyperboloid ℍ d or Lobachevsky space. We apply the averaging function method over the sphere and the solution of these problems for the Jacobi transform on the half-line.

In this paper, for functions on the group Q p , we prove an analogue of the classical Titchmarsh theorem on description of the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L 2.

We prove analogues of the Bombieri–Vinogradov Theorem and the Barban–Davenport–Halberstam Theorem on primes in arithmetic progressions for Fourier coefficients of cusp forms.

We study solutions to convolution equations for functions with discrete support in ℝ n , a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holomorphic function in some domains in ℂ n , and we determine possible domains in terms of the properties of the convolution operator.

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 considered but also a new way of proving such theorems is applied. Interrelations of various function spaces are studied in this context. The obtained results are used for obtaining completely new results on the integrability of trigonometric series.

A brief introduction to the fractional Fourier cosine transform as well as fractional Fourier sine transform and their basic properties are given. Fractional Fourier cosine (fractional Fourier sine) transform of tempered distributions is studied. Pseudo-differential operators involving these transformations are investigated and discussed the continuity on certain spaces $\mathcal{S}_e $ and $\mathcal{S}_o$ .

The analytic functions in tubes which obtain the distributional finite Fourier transform as boundary value are shown to have a strong boundedness property and to be recoverable as a Fourier-Laplace transform, a distributional finite Fourier transform, and as a Cauchy integral of a distribution associated with the boundary value.

We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups $$G= \hbox {exp}({\mathfrak {g}})$$ by showing that operator fields defined on suitable sub-manifolds of $${\mathfrak {g}}^*$$ are images of Schwartz functions under the Fourier transform. As an application of this result, we provide a complete characterisation of a large class of invariant prime closed two-sided ideals of $$L^1(G)$$ as kernels of sets of irreducible representations of G.

We investigate the pointwise and uniform convergence of the symmetric rectangular partial (also called Dirichlet) integrals of the double Fourier integral of a function that is Lebesgue integrable and of bounded variation over ℝ2. Our theorem is a two-dimensional extension of a theorem of Móricz (see Theorem 3 in [10]) concerning the single Fourier integrals, which is more general than the two-dimensional extension given by Móricz himself (see Theorem 3 in [11]).

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 Fourier transform can somehow be expressed in terms of the Hilbert transform of the derivative.

We provide a variant of Hytonen’s embedding theorem, which allows us to extend and unify several sufficient conditions for a function to be a Fourier multiplier on the real Hardy spaces.

It was proved earlier that, for multiple Fourier series whose coefficients are monotone in each index, the classicalHardy–Littlewood theorem is not valid for p ≤ 2m/(m+1), where m is the dimension of the space. We establish how the theorem must be modified in this case.

We establish a general sufficient condition for the possibility of representation of functions $$ f\left( \max \left\{\left|{x}_1\right|,\left|{x}_2\right|\right\}\right) $$ in the form of absolutely convergent double Fourier integrals and study the possibility of its application to various problems of summability of double Fourier series, in particular, by using the Marcinkiewicz–Riesz method.

The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LlogL and more general Lorentz-Zygmund spaces., Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s00041-015-9455-5

For a wide class of Kato-smoothing estimates with radial weights, the Funk–Hecke theorem is used to generate a new expression for the optimal constant in terms of the Fourier transform of the weight, from which several applications are given. For example, we are able to easily establish a unified theorem, assuming natural power-like asymptotic estimates for the Fourier transform of the weight, from which many well-studied smoothing estimates immediately follow, as well as sharpness of the decay and smoothness exponents. Furthermore, observing that the weight has an everywhere positive Fourier transform in many well-studied cases, our approach allows sharper information regarding the optimal constant and extremisers, substantially extending earlier work of Simon. These observations are very closely related to the Mizohata–Takeuchi conjecture regarding the equivalence of weighted L 2 bounds for the Fourier extension operator on the sphere and the uniform boundedness of the X-ray transform of the weight. For radial weights, this has been independently established by Barcelo–Ruiz–Vega and Carbery–Soria; we provide a short alternative proof in three and higher dimensions of this equivalence when the Fourier transform of the weight is positive, with the optimal relationship between constants. Finally, our approach works for the closely connected trace theorems on the sphere where analogous results are given, including the optimal constant and characterisation of extremisers for the inhomogeneous H s ( R d ) → L 2 ( S d − 1 ) trace theorem.

T. Qian As it will turn out in this paper, the recent hype about most of the Clifford–Fourier transforms is not thoroughly worth the pain. Almost everyone that has a real application is separable, and these transforms can be decomposed into a sum of real valued transforms with constant multivecor factors. This fact makes their interpretation, their analysis, and their implementation almost trivial. Copyright © 2015 John Wiley & Sons, Ltd.

For multiplicative Fourier transforms, analogs of results obtained by R. P. Boas, F. Moricz, M. I. D’yachenko, I. P. Liflyand, and S. Yu. Tikhonov and dealing with conditions for the uniform convergence and weighted integrability with power weight of classical Fourier transforms as well as conditions for these transforms to belong to Lipschitz classes are proved. Certain results of C. W. Onneweer concerning conditions for multiplicative Fourier transforms to belong to Lipschitz—Besov and Herz spaces are also generalized.

We give a simple proof of the fact that there exist measures on the real line of discrete support, whose Fourier Transform is also a measure of discrete support, yet this Fourier pair cannot be constructed by repeatedly applying the Poisson Summation Formula finitely many times. More specifically the support of both the measure and its Fourier Tranform are not contained in a finite union of arithmetic progressions., The bibliography was missing from the previous version

In this paper, we estimate the order of magnitude of the double Fourier coefficients of a function of the class ( Λ 1 , Λ 2 ) BV ( p ( n ) ↑ ∞ , ϕ ) ${(\Lambda ^1,\Lambda ^2)\operatorname{BV}(p(n)\uparrow \infty ,\varphi )}$ over [ 0 , 2 π ] 2 ${[0,2\pi ]^2}$ .

In this paper, we prove the Hardy’s and Cowling–Price’s uncertainty principles for the generalized Fourier transform associated to a Cherednik type operator on the real line.

This paper continues the study of Fourier transforms on finite inverse semigroups, with a focus on Fourier inversion theorems and FFTs for new classes of inverse semigroups. We begin by introducing four inverse semigroup generalizations of the Fourier inversion theorem for finite groups. Next, we describe a general approach to the construction of fast inverse Fourier transforms for finite inverse semigroups complementary to an approach to FFTs given in previous work. Finally, we give fast inverse Fourier transforms for the symmetric inverse monoid and its wreath product by arbitrary finite groups, as well as fast Fourier and inverse Fourier transforms for the planar rook monoid, the partial cyclic shift monoid, and the partial rotation monoid., Comment: v2: Streamlined presentation and added results for the planar rook monoid, partial cyclic shift monoid, and partial rotation monoid. 24 pages

One Szokefalvi-Nagy theorem is utterly strengthened, and a new relationship between Fourier series and integrals is established.

A recent extension due to Trigub of one of the classical Sz.-Nagy’s results on the integrability of the Fourier transform is generalized to several dimensions for functions with bounded Hardy variation.

We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in S 1 , 0 m ( n , 2 ) and non-degenerate phase functions, from L p × L q → L r under the assumptions that m ⩽ − ( n − 1 ) ( | 1 p − 1 2 | + | 1 q − 1 2 | ) and 1 p + 1 q = 1 r . This is a bilinear version of the classical theorem of Seeger–Sogge–Stein concerning the L p boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of quasi-Banach target spaces.

We consider a nontrivial example of distributional eigenfunction of the planar Fourier transform. This eigenfunction is not a tensor product of univariate eigenfunctions. As a consequence, we obtain a formula for multi-dimensional eigenfunctions in dimension 2N.

WOS: 000416484100008 We generalize a main theorem dealing with absolute weighted mean summability of Fourier series to the |A, p(n)|(k) summability factors of Fourier series under weaker conditions. Also some new and known results are obtained.

International audience; We propose a generalization of a classical result on random Fourier series, namely the Billard Theorem, for random Fourier series over the d-dimensional torus. We provide an investigation of the independence with respect to a choice of a sequence of partial sums (or method of summation). We also study some probabilistic properties of the resulting sum field such as stationarity and characteristics of the marginal distribution.

A fundamental theme in classical Fourier analysis relates smoothness properties of functions to the growth and/or integrability of their Fourier transform. By using a suitable class of $$L^{p}$$ -multipliers, a rather general inequality controlling the size of Fourier transforms for large and small argument is obtained. As consequences, quantitative Riemann–Lebesgue estimates are obtained and an integrability result for the Fourier transform is developed extending ideas used by Titchmarsh in the one dimensional setting.

Associated with the Dirac operator and partial derivatives, this paper establishes some real Paley-Wiener type theorems to characterize the Clifford-valued functions whose Clifford Fourier transform (CFT) has compact support. Based on the Riemann-Lebesgue theorem for the CFT, the Boas theorem is provided to describe the CFT of Clifford-valued functions that vanish on a neighborhood of the origin.

Analytic reconstruction of a function defined in an affine space from data of its integrals along lines or rays is in focus of the paper. Basic tools are the Fourier transform of homogeneous distributions and a self-duality equation in integral geometry. Three dimensional case is of special interest.

Using the Fragmen-Lindelof principle, we prove that, in the Paley-Wiener-Schwartz theorem, the condition imposed on the function can be replaced by two conditions whose validity is easier to verify in a number of cases.

We study the inversion formula for the continuous windowed Fourier transform. Different from the classical ones where a single or double integral is involved, we show that for a large class of window functions, a function can be reconstructed from its continuous windowed Fourier transform with a discrete series. Moreover, we show that the series is convergent almost everywhere on R as well as in L p (R) if the function to be reconstructed is. In particular, for the case of p = 2, we give a necessary and sufficient condition for the series to be convergent to the original function.

A Hardy-Littlewood-Paley inequality is obtained for Bellman type means of Fourier coefficients of functions from anisotropic Lorentz spaces.

We establish new sufficient conditions for the absolute |C, α|-summability of the Fourier series of functions almost periodic in a sense of Besicovitch whose spectrum has limit points at infinity and at the origin for $$ \alpha \ge \frac{1}{2} $$ .

This work is dedicated to the investigation of strong summability of Fourier series in the context of periodic Morrey spaces. First, we study the Hilbert transform in the periodic vector-valued context. Boundedness of the Hilbert transform implies uniform estimates of the operator norms of the partial sums of the Fourier series. Afterwards, we study the Lizorkin-Triebel-Morrey and Nikol’skij-Besov-Morrey spaces. Here we concentrate on Lizorkin representations and embeddings into the scale of Holder-Zygmund spaces. In the final section, we study consequences for strong summability of Fourier series.

For a Lebesgue integrable complex-valued function f defined over the m -dimensional torus Tm := [0,2π)m , let f (n) denote the Fourier coefficient of f , where n = (n(1), . . . ,n(m)) ∈ Zm . Recently, in one of our papers [to appear in Mathematical Inequalities & Applications], we have defined the notion of bounded φ -variation for a complex-valued function on a rectangle [a1,b1 ]× . . .× [am,bm ] and studied the order of magnitude of Fourier coefficients of such functions on [0,2π]m . In this paper, the order of magnitude of Fourier coefficients of a function of bounded φ -variation from [0,2π]m to C and having lacunary Fourier series with certain gaps is studied and a generalization of our earlier result (Theorem in [Acta Sci. Math. (Szeged), 78, (2012), 97–109]) is proved. Interestingly, the Jensen’s inequality for integrals is used to prove the main result. Mathematics subject classification (2010): 42B05, 26B30, 26D15.

We characterize the entire functions of exponential type, whose restriction to the real line is in L p , in different ways: by the usual classical Paley–Wiener growth estimates in the complex plane, by Bernstein inequalities using derivatives or differences, by L p -growth properties of iterated derivatives or differences, and by support properties of the Fourier image. We also establish Paley–Wiener theorems for the Fourier series of a function on the circle.

We present necessary and sufficient conditions for the absolute convergence of the Fourier series of almost-periodic (in the sense of Besicovitch) functions when the Fourier exponents have limit points at infinity or at zero. The structural properties of the functions are described by the modulus of continuity or the modulus of averaging of high order, depending on the behavior of the Fourier exponents. The case of uniform almost-periodic functions of bounded variation is considered.

This is a survey paper on recent results indicated in the title. In contrast to the famous examples of Kolmogorov and Fejer on the pointwise divergence of Fourier series, the statistical convergence of the Fourier series of any integrable function takes place at almost every point; and the statistical convergencr of the Fourier series of any continuous function is uniform. Furthermore, Tauberian conditions are also presented, under which ordinary convergence of any sequence of real or complex numbers follows from its statistical summability.

Ramanujan’s Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the “Master Theorem”. In this paper we prove an analogue of Ramanujan’s Master theorem for the hypergeometric Fourier transform associated with root systems. This theorem generalizes to arbitrary positive multiplicity functions the results previously proven by the same authors for the spherical Fourier transform on semisimple Riemannian symmetric spaces.

We consider Fourier coefficients for functions of bounded variation with respect to general orthonormal systems (GONS). We show that in general case the sequence of coefficients may have an arbitrarily prescribed order of vanishing. In this connection we seek for a class of GONS such that Fourier coefficients of a function from V(0, 1) satisfy the same inequality as in the case of classical systems (trigonometric, Walsh, and Haar ones). In the present paper we study this problem and related issues.

In this paper, we extended some classical results of Fourier series to rational Fourier series. We give an estimate of convergence rate of the rational Fourier series of functions of bounded variation and an analogous one for the conjugate rational Fourier series. As its applications, we deduce the Dirichlet-Jordans theorem and W. H. Youngs theorem for rational Fourier series of functions of bounded variation. Finally, we extend these two theorems to harmonic bounded variation.

We show that if f f is in an anisotropic Hardy space H A p H_A^p , 0 > p ≤ 1 0 > p \leq 1 , with respect to a dilation matrix A A , then its Fourier transform f ^ \hat {f} satisfies the pointwise estimate \[ | f ^ ( ξ ) | ≤ C | | f | | H A p ρ ∗ ( ξ ) 1 p − 1 . |\hat f(\xi )| \le C ||f||_{H^p_A} \rho _*(\xi )^{\frac {1}{p}-1}. \] Here, ρ ∗ \rho _* is a quasi-norm associated with the transposed matrix A ∗ A^* . This leads to necessary conditions for functions m m to be multipliers on H A p H_A^p , as well as further pointwise characterizations on f ^ \hat {f} and a generalization of the Hardy-Littlewood inequality on the integrability of f ^ \hat {f} . This last result is strengthened through the use of rearrangement functions.