The surprising almost everywhere convergence of Fourier-Neumann series

For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in L^p requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L² is the celebrated Carleson theorem, proved in 1966 (and extended to L^p by Hunt in 1967).In this paper, we take the system

     

有理Fourier级数在变差条件下的收敛性研究

本文把Fourier级数的一些经典结论推广到有理Fourier级数的情况下. 首先给出了有理Fourier级数和共轭有理Fourier级数在有界变差条件下的收敛速度估计. 利用此结论, 得到了类似于Fourier级数的Dirichlet-Jordan定理和W. H. Young定理. 最后, 证明了这两个定理在调和有界变差条件下也成立.     

The quartile operator and pointwise convergence of Walsh series

The bilinear Hilbert transform is given by

It satisfies estimates of the type

In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in , with converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.

     

On the absolute convergence of multiple fourier series

Let f: R ^N → C be a periodic function with period 2π in each variable. We prove suffcient conditions for the absolute convergence of the multiple Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to multiple Fourier series. This research was started while the first author was a visiting professor at the Department of Mathematics, Texas A&M University, College Station during the fall semester in 2005; and it was also supported by the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

Divergence almost everywhere of rectangular partial sums of multiple fourier series of bounded functions

In this paper we establish the following results, which are the multidimensional generalizations of well-known theorems:

Neumann series and Lommel functions of two variables

We obtain several new results for Neumann series of Bessel functions as well as for its various special cases. The generalization of some well-known results for these kind of series, such as the Graf's addition theorem, are also established.     

Multipliers for the absolute euler summability of fourier series

In this paper, the author has investigated necessary and sufficient conditions for the absolute Euler summability of the Fourier series with miltipliers. These conditions are weaker than those obtained earlier by some workers. It is further shown that the multipliers are best possible in certain sense.     

Convergence of multiple fourier series for functions of bounded variation

For functions of bounded variation in the sense of Hardy, we consider the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 583–595, April, 1997. Translated by S. A. Telyakovskii and V. N. Temlyakov     

Expansion of entire functions into Lagrange series

This paper is devoted to the problem of representing entire functions, in spaces described by the order and the type of these functions, by Lagrange series that converge in the natural topology in these spaces; this topology is stronger than the topology of compact convergence. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 119–124, January, 1997. Translated by M. A. Shishkova     

Mean convergence of orthogonal Fourier series of modified functions

We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the -metric for any 2. $"> We prove also that if is a uniformly bounded ONS which is complete in all the spaces , then there exists a rearrangement of the natural numbers such that the system has the strong -property for all 2$">; that is, for every and for every and 0 $">there exists a function which coincides with except on a set of measure less than and whose Fourier series with respect to the system converges in      

Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L ^p and weak L ^p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L _Φ having the property L ^∞ ? L _Φ L ^p , 1 ? p > ∞. The second contains spaces L _Φ that resemble L ^p spaces.     

Some estimates for the error in mixed Fourier-Bessel expansions of functions of two variables

Some issues concerning expansions of functions of two variables in mixed Fourier-Bessel series are considered. In particular, the rate of their convergence in the classes of functions characterized by generalized moduli of continuity are estimated, and estimates of the remainder terms are obtained.     

Divergent Fourier series of integrable functions with respect to orthonormal systems

There exist two orthonormal systems such that the Fourier series of each functionƒ ∈ L[0, 1],ƒ ≠ 0, with respect to at least one of these systems diverges on a set of positive measure. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 717–724, May, 1998.     

A complete orthonormal system of divergence

A complete orthonormal system of functions {Θ_n}_n=1^∞,Θ_n∈L_[0,1]^∞, defined on the closed interval [0,1] is constructed such that ∑_n=1^∞a_nΘ_n diverges almost everywhere for any .For the constructed system the following result is true:Corollary 1.Any nontrivial series in the system {Θ_n}_n=1^∞which converges in measure to zero diverges almost everywhere.     

k解析函数的Fourier级数

讨论了复平面上k解析函数的性质,并利用k解析函数的泰勒展开定理研究了k解析函数的Fourier级数,推广了经典的解析函数的Fourier级数理论.     

Mean convergence of Fourier-Dunkl series

In the context of the Dunkl transform a complete orthogonal system arises in a very natural way. This paper studies the weighted norm convergence of the Fourier series expansion associated to this system. We establish conditions on the weights, in terms of the Ap classes of Muckenhoupt, which ensure the convergence. Necessary conditions are also proved, which for a wide class of weights coincide with the sufficient conditions.     

Comparison of some tests for convergence of positive series

In this survey, aimed at professors and students of undergraduate analysis courses, two hierarchies of tests are considered. The first is originated from the ratio test and the second from the root test. The test construction techniques are exposed and relations between the tests in each family are discussed. The examples of convergent and divergent series clarify the range of application for each of the introduced tests and situations when they are not conclusive. The behaviour of the partial sums of these series is illustrated geometrically and the level of complexity of each series is evaluated in terms of the rate of its convergence or divergence.     

On the coefficients of Neumann series of Bessel functions

Recently Pogány and Süli (Proc. Amer. Math. Soc. 137 (7) (2009) 2363-2368) derived a closed-form integral expression for Neumann series of Bessel functions. In this note we precisely characterize the class of functions α that generate the integral representation of a Neumann series of Bessel functions in the sense that the restriction αN_|=(αn) of a function α to the set N of all positive integers is the sequence of coefficients of the initial Neumann series.     

On the Absolute Convergence of Fourier Series

The necessary and sufficient conditions of the absolute convergence of a trigonometric Fourier series are established for continuous 2-periodic functions which in [0, 2] have a finite number of intervals of convexity, and whose nth Fourier coefficients are O((l/n; f)/n), where ( f) is the continuity modulus of the function f.