On the Uniform Convergence of Walsh-Fourier Series

We study the uniform convergence of Walsh-Fourier series of functions on the generalized Wiener class BV (p(n)↑∞) This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Strong Summability of Orthogonal Series

It is well known that not every summability method implies the strong summability with any positive exponent. We give easygoing additional conditions on the terms of a positive regular Toeplitz-matrix implying the strong summability for any positive exponent. The classical (C, α > 0)- and Abel-summabilities satisfy our conditions plainly. We treat the generalized Abel, the Euler, the Riesz and the generalized de la Vallée Poussin methods, as well.     

On the Very Strong Summability of Orthogonal Series

We prove two theorems pertaining to the very strong summability of orthogonal series for general summability methods and present four consequences of them for classical summability methods.     

On the Absolute Matrix Summability Factors of Fourier Series

In this paper, two known theorems on |N?, p _n | _k summability methods of Fourier series have been generalized for |A, p _n | _k summability factors of Fourier series by using different matrix transformations. New results have been obtained dealing with some other summability methods.

     

Summability of Hermite Series

Let f∈L ^p (R), 1≤p≤t8, and c _j be the inner product of f and the Hermite function h _j . Assume that c _j 's satisfy $\left| {c_j } \right| \cdot f = o\left( 1 \right)\;\quad as\;j \to \infty $ If r=5/4, then the Hermite series Σc _j h _j conerges to f almost everywhere. If r=9/4-1/p, the Σ c _j h _j converges to f in L ^p (R).     

Summability of Hermite Series

Let fL ^p (R), 1pt8, and c _j be the inner product of f and the Hermite function h _j . Assume that c _j 's satisfy If r=5/4, then the Hermite series c _j h _j conerges to f almost everywhere. If r=9/4-1/p, the c _j h _j converges to f in L ^p (R).     

On Strong Summability of Fourier Series of Summable Functions

In the metric of L, we obtain estimates for the generalized means of deviations of partial Fourier sums from an arbitrary summable function in terms of the corresponding means of its best approximations by trigonometric polynomials.     

On the Absolute Summability of Series with Respect to Block-Orthonormal Systems

Theorems determining Weyl's multipliers for the summability almost everywhere by the |c, 1| method of the series with respect to block-orthonormal systems are proved. In particular, it is stated that if the sequence {(n)} is the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all orthogonal series, then there exists a sequence {N _k} such that {(n)} will be the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all series with respect to the _k -orthonormal systems.     

傅里叶级数的求和理论与方法

通过Dirichlet积分算子构造了一个新的积分算子Hn(f; k,x). 对于f(x)∈Cj2π, 0jk (其中k为任意自然数),Hn(f; k,x)的逼近阶达到了最佳.     

Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series

It follows from results of A. Yudin, V. Yudin, E. Belinskii, and I. Liflyand that if $m \geqslant 2$ and a $2\pi $ -periodic (in each variable) function $f(x) \in C(T^m )$ belongs to the Nikol'skii class $h_\infty ^{(m - 1)/2} (T^m )$ , then its multiple Fourier series is uniformly convergent over hyperbolic crosses. In this paper, we establish the finality of this result. More precisely, there exists a function in the class $h_\infty ^{(m - 1)/2} (T^m )$ whose Fourier series is divergent over hyperbolic crosses at some point.     

Convergence of Double Walsh-Fourier Series and Hardy Spaces

It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H _p to L _p (2/3<p<∞) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejér means of a function f∈L ₁ converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on H _p whenever 2/3<p<∞. Thus, in case f∈H _p , the Marczinkiewicz-Fejér means conv f in H _p norm. The same results are proved for the conjugate means, too.     

Convergence of Double Walsh-Fourier Series and Hardy Spaces

It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H _p to L _p (2/3<p<) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejér means of a function fL ₁ converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on H _p whenever 2/3<p<. Thus, in case fH _p , the Marczinkiewicz-Fejér means conv f in H _p norm. The same results are proved for the conjugate means, too.     

The Abel Summability of Conjugate Laplace Series of Measures

This paper deals with the summability of conjugate Laplace series. In particular, the Abel summability is proved and an integral representation of the relevant sum is given.     

On the Summability of Trigonometric Interpolation Processes

The aim of this paper is to show that several processes studied in trigonometric interpolation theory can be obtained by -sums of discrete Fourier series. We shall investigate the uniform convergence of the sequences of thesepolynomials. We show that the convergence of several processes can be seen immediately from suitable explicit forms of the corresponding polynomials. Error estimates for the approximation can be also obtained by certain general results.     

Divergence of the (C, 1) Means of d-dimensional Walsh-Fourier Series

In 1992, Móricz, Schipp and Wade [MSW] proved for functions in L log⁺ L(I ²) (I ² is the unit square) the a.e. convergence of the double (C, 1) means of the Walsh-Fourier series _n f f as min(n ₁, n ₂) , n = (n ₁, n ₂ N ²). In the same paper, they also proved the restricted convergence of the (C, 1) means of functions in L(I ²): (2 ⁿ 1,2 ⁿ 2)f f a.e. as min (n ₁, n ₂) provided |n ₁ – n ₂| < C. The aim of this paper is to demonstrate the sharpness of these results of Móricz, Schipp and Wade with respect to both the space L log⁺ L(I ²) and the restrictedness |n ₁ – n ₂| < C.