Herz spaces and restricted summability of Fourier transforms and Fourier series

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy-Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1,1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that over a cone-like set a.e. for all f∈L₁(Rd). Moreover, converges to f(x) over a cone-like set at each Lebesgue point of f∈L₁(Rd) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.     

Several Dimensional θ-Summability and Hardy Spaces

The d-dimensional Hardy spaces H_p ( T × … × T ) (d = d₁ + … + d_kand a general summability method of Fourier series and Fourier transforms are introduced with the help of integrable functions θ_j having integrable Fourier transforms. Under some conditions on θ_j we show that the maximal operator of the θ-means of a distribution is bounded from H_p ( T × … × T ) to L_p ( T ^d) where p₀ < p < ∞ and p₀ < 1 is depending only on the functions θ_j. By an interpolation theorem we get that the maximal operator is also of weak type ( L₁) (i = 1, …, k) where the Hardy space is defined by a hybrid maximal function and if k = 1. As a consequence we obtain that the θ-means of a function (log L)^k–1 converge a.e. to the function in question. If k = 1 then we get this convergence result for all f ∈ L₁. Moreover, we prove that the θ-means are uniformly bounded on the spaces H_p ( T × … × T ) whenever p₀ <p < ∞, thus the θ-means converge to f in ( T × … × T ) norm. The same results are proved for the conjugate θ-means and for d-dimensional Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations.     

Restricted summability of Fourier transforms and local Hardy spaces

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.     

Marcinkiewicz-\theta-Summability of Fourier Transforms

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function . Under some conditions on we show that the maximal operator of the Marcinkiewicz- -means of a tempered distribution is bounded from to for all and, consequently, is of weak type , where depends only on . As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz- -means of a function converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz- -means are uniformly bounded on the spaces and so they converge in norm . Some special cases of the Marcinkievicz- -summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski and Riesz summations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Local Hardy spaces and summability of Fourier transforms

New Wiener amalgam spaces are introduced for local Hardy spaces. A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from the amalgam space to W(L_p,ℓ_∞). This implies the almost everywhere convergence of the θ-means for all fW(L₁,ℓ_∞)L₁.     

Norm inequalities for certain classes of functions and their Fourier transforms

We apply tools of interpolation theory and a commutative property of the Hilbert transform to prove necessary and sufficient conditions related to trigonometric series. These results extend and improve related theorems proven by several authors, summarized by Boas. In addition, we explore inequalities and operators, both connected to Hardy's inequalities, on certain classes of functions, including quasimonotone functions.     

窗口Fourier变换反演公式的级数表示——献给陆善镇教授75华诞

本文研究连续窗口Fourier变换的反演公式.与经典的积分重构公式不同,本文证明当窗函数满足合适的条件时,窗口Fourier变换的反演公式可以表示为一个离散级数.此外,本文还研究这一重构级数的逐点收敛及其在Lebesgue空间的收敛性.对于L^2空间,本文给出重构级数收敛的充分必要条件.     

Herz spaces and summability of Fourier transforms

A general summability method is considered for functions from Herz spaces K^α_p,r (?^d ). The boundedness of the Hardy–Littlewood maximal operator on Herz spaces is proved in some critical cases. This implies that the maximal operator of the θ ‐means σ^θ _T f is also bounded on the corresponding Herz spaces and σ^θ _T f → f a.e. for all f ∈ K^{–d /p} _p,∞ (?^d ). Moreover, σ^θ _T f (x) converges to f (x) at each p ‐Lebesgue point of f ∈ K^{–d /p} _p,∞ (?^d ) if and only if the Fourier transform of θ is in the Herz space K^{d /p} _{p ′,1} (?^d ). Norm convergence of the θ ‐means is also investigated in Herz spaces. As special cases some results are obtained for weighted L_p spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Randomised circular means of Fourier transforms of measures

We explore decay estimates for circular means of the Fourier transform of a measure on in terms of its -dimensional energy. We find new upper bounds for the decay exponent. We also prove sharp estimates for a certain family of randomised versions of this problem.

     

A note on rearrangement of Fourier series

We give an example of a permutation of integers which preserves all convergent Fourier series and makes some divergent Fourier series converge.     

Characterizations of the Gelfand-Shilov spaces via Fourier transforms

We give symmetric characterizations, with respect to the Fourier transformation, of the Gelfand-Shilov spaces of (generalized) type and type . These results explain more clearly the invariance of these spaces under the Fourier transformations.

     

The interpolation of the Titchmarsh-Weyl function

By using the idea of interpolation of functions in Hardy spaces, we obtain a new representation of the Titchmarsh-Weyl function, known also as the m-function, in the case of short range potentials. The key idea is to use the Gelfand-Levitan-Marchenko theory which allows us to represent Jost solutions by the Fourier transform.     

Asymptotical behavior of one class of p-adic singular Fourier integrals

We study the asymptotical behavior of the p-adic singular Fourier integrals

     

Pitt and Boas inequalities for Fourier and Hankel transforms

We prove Pitt and Boas inequalities for products of radial functions and spherical harmonics in Rⁿ${\mathbb{R}}^n$. In the process, we obtain upper and lower estimates of the operator norm of the Hankel transform with power weights. Our inequalities are sharp in some specific cases.     

Periodic hyperfunctions and Fourier series

Every periodic hyperfunction is a bounded hyperfunction and can be represented as an infinite sum of derivatives of bounded continuous periodic functions. Also, Fourier coefficients of periodic hyperfunctions are of infra-exponential growth in , i.e., for every and every . This is a natural generalization of the polynomial growth of the Fourier coefficients of distributions.

To show these we introduce the space of hyperfunctions of growth which generalizes the space of distributions of growth and represent generalized functions as the initial values of smooth solutions of the heat equation.

     

含参变量富里叶级数的Laplace变换求和法

本文建立了含参变量富里叶级数的Laplace变换求和定理.利用Laplace变换表可以求得许多在力学上有重要应用的新的含参变量富里叶级数的和式.     

Almost everywhere convergence of inverse Fourier transforms

We show that if , then the inverse Fourier transform of converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when and . We also consider sequential convergence for general elements of .

     

Dirichlet characters,Gauss sums and arithmetic Fourier transforms

In this paper, a general algorithm for the computation of the Fourier coefficients of 2π-periodic(continuous) functions is developed based on Dirichlet characters, Gauss sums and the generalized M¨obius transform. It permits the direct extraction of the Fourier cosine and sine coefficients. Three special cases of our algorithm are presented. A VLSI architecture is presented and the error estimates are given.