### Extension of a Theorem of Fejér to Double Fourier-Stieltjes Series.

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We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces ${H}^{(1,0)}{(}^{2})$, ${H}^{(0,1)}{(}^{2})$, or ${H}^{(1,1)}{(}^{2})$. We prove that the maximal Fejér operator is bounded from ${H}^{(1,0)}{(}^{2})$ or ${H}^{(0,1)}{(}^{2})$ into weak-${L}^{1}{(}^{2})$, and also bounded from ${H}^{(1,1)}{(}^{2})$ into ${L}^{1}{(}^{2})$. These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces ${L}^{1}lo{g}^{+}L{(}^{2})$, ${L}^{1}{\left(lo{g}^{+}L\right)}^{2}{(}^{2})$, and ${L}^{\mu}{(}^{2})$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures....

$({s}_{jk}:j,k=0,1,...)$ be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which $\left({s}_{jk}\right)$ converges in Pringsheim’s sense. These conditions are satisfied if $\left({s}_{jk}\right)$ is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If $\left({s}_{jk}\right)$ is summable (C,1,1) to a finite limit and there exist constants ${n}_{1}>0$ and H such that $jk({s}_{jk}-{s}_{j-1,k}-{s}_{j-1,k}+{s}_{j-1,k-1})\ge -H$, $j({s}_{jk}-{s}_{j-1,k})\ge -H$ and $k({s}_{jk}-{s}_{j,k-1})\ge -H$ whenever $j,k>{n}_{1}$, then $\left({s}_{jk}\right)$ converges....

In 1970 C. W. Onneweer formulated a sufficient condition for a periodic W-continuous function to have a Walsh-Fourier series which converges uniformly to the function. In this paper we extend his results from single to double Walsh-Fourier series in a more general setting. We study the convergence of rectangular partial sums in ${L}^{p}$-norm for some 1 ≤ p ≤ ∞ over the unit square [0,1) × [0,1). In case p = ∞, by ${L}^{p}$ we mean ${C}_{W}$, the collection of uniformly W-continuous functions f(x, y), endowed with the...

Given ⨍ ∈ ${L}_{l}^{1}oc\left({\mathbb{R}}_{+}^{2}\right)$, denote by s(w,z) its integral over the rectangle [0,w]× [0,z] and by σ(u,v) its (C,1,1) mean, that is, the average value of s(w,z) over [0,u] × [0,v], where u,v,w,z>0. Our permanent assumption is that (*) σ(u,v) → A as u,v → ∞, where A is a finite number. First, we consider real-valued functions ⨍ and give one-sided Tauberian conditions which are necessary and sufficient in order that the convergence (**) s(u,v) → A as u,v → ∞ follow from (*). Corollaries allow these Tauberian conditions...

The concepts of statistical convergence of single and double sequences of complex numbers were introduced in [1] and [7], respectively. In this paper, we introduce the concept indicated in the title. A double sequence ${x}_{jk}:(j,k)\in \mathbb{N}\xb2$ is said to be regularly statistically convergent if (i) the double sequence ${x}_{jk}$ is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence ${x}_{jk}:k\in \mathbb{N}$ is statistically convergent to some ${\xi}_{j}\in \u2102$ for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence ${x}_{jk}:j\in \mathbb{N}$ is statistically convergent to some ${\eta}_{k}\in \u2102$ for...

Schmidt’s classical Tauberian theorem says that if a sequence $({s}_{k}:k=0,1,...)$ of real numbers is summable (C,1) to a finite limit and slowly decreasing, then it converges to the same limit. In this paper, we prove a nondiscrete version of Schmidt’s theorem in the setting of statistical summability (C,1) of real-valued functions that are slowly decreasing on ℝ ₊. We prove another Tauberian theorem in the case of complex-valued functions that are slowly oscillating on ℝ ₊. In the proofs we make use of two nondiscrete...

We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class $Lip(\alpha \u2081,...,{\alpha}_{N})$ for some $0<\alpha \u2081,...,{\alpha}_{N}<1$ and its marginal functions satisfy $f(\xb7,x\u2082,...,{x}_{N})\in Lip\beta \u2081,...,f(x\u2081,...,{x}_{N-1},\xb7)\in Lip{\beta}_{N}$ for some $0<\beta \u2081,...,{\beta}_{N}<1$ uniformly in the indicated variables ${x}_{l}$, 1 ≤ l ≤ N, then $f{\u0303}^{(\eta \u2081,...,{\eta}_{N})}\in Lip(\alpha \u2081,...,{\alpha}_{N})$ for each choice of $(\eta \u2081,...,{\eta}_{N})$ with ${\eta}_{l}=0$ or 1 for 1 ≤ l ≤ N.

Schmidt’s Tauberian theorem says that if a sequence (xk) of real numbers is slowly decreasing and $li{m}_{n\to \infty}(1/n){\sum}_{k=1}^{n}{x}_{k}=L$, then $li{m}_{k\to \infty}{x}_{k}=L$. The notion of slow decrease includes Hardy’s two-sided as well as Landau’s one-sided Tauberian conditions as special cases. We show that ordinary summability (C,1) can be replaced by the weaker assumption of statistical summability (C,1) in Schmidt’s theorem. Two recent theorems of Fridy and Khan are also corollaries of our Theorems 1 and 2. In the Appendix, we present a new proof of Vijayaraghavan’s...

The harmonic Cesàro operator is defined for a function f in ${L}^{p}\left(\mathbb{R}\right)$ for some 1 ≤ p < ∞ by setting $\left(f\right)\left(x\right):={\int}_{x}^{\infty}(f\left(u\right)/u)du$ for x > 0 and $\left(f\right)\left(x\right):=-{\int}_{-\infty}^{x}(f\left(u\right)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L{\xb9}_{loc}\left(\mathbb{R}\right)$ by setting $*\left(f\right)\left(x\right):=(1/x){\int}^{x\u2080}f\left(u\right)du$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f\in {L}^{p}\left(\mathbb{R}\right)$ for some 1 ≤ p ≤ 2, then ${\left(\left(f\right)\right)}^{\wedge}\left(t\right)=*\left(f\u0302\right)\left(t\right)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f\in {L}^{p}\left(\mathbb{R}\right)$ for some 1 < p ≤ 2, then ${(*\left(f\right))}^{\wedge}\left(t\right)=\left(f\u0302\right)\left(t\right)$ a.e. As...

We consider complex-valued functions f ∈ L¹(ℝ), and prove sufficient conditions in terms of f to ensure that the Fourier transform f̂ belongs to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes zyg(α) and zyg(α) for some 0 < α ≤ 2. These sufficient conditions are best possible in the sense that they are also necessary in the case of real-valued functions f for which either xf(x) ≥ 0 or f(x) ≥ 0 almost everywhere.

Let s: [1,∞) → ℂ be a locally Lebesgue integrable function. We say that s is summable (L,1) if there exists some A ∈ ℂ such that $li{m}_{t\to \infty}\tau \left(t\right)=A$, where $\tau \left(t\right):=1/\left(logt\right){\int}_{1}^{t}s\left(u\right)/udu$. (*) It is clear that if the ordinary limit s(t) → A exists, then also τ(t) → A as t → ∞. We present sufficient conditions, which are also necessary, in order that the converse implication hold true. As corollaries, we obtain so-called Tauberian theorems which are analogous to those known in the case of summability (C,1). For example, if the function s is slowly...

We investigate the order of magnitude of the modulus of continuity of a function f with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that f belong to one of the generalized Lipschitz classes Lip(α,L) and Lip(α,1/L), where 0 ≤ α ≤ 1 and L = L(x) is a positive, nondecreasing, slowly varying function such that L(x) → ∞ as x → ∞. For example, a 2π-periodic function f is said to belong to the class Lip(α,L) if $|f(x+h)-f\left(x\right)|\le C{h}^{\alpha}L(1/h)$ for all x ∈ , h >...

We investigate the convergence behavior of the family of double sine integrals of the form ${\int}_{0}^{\infty}{\int}_{0}^{\infty}f(x,y)sinuxsinvydxdy$, where (u,v) ∈ ℝ²₊:= ℝ₊ × ℝ₊, ℝ₊:= (0,∞), and f: ℝ²₊ → ℂ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals ${\int}_{a\u2081}^{b\u2081}{\int}_{a\u2082}^{b\u2082}$ to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and ${b}_{j}>{a}_{j}\ge 0$, j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial integrals...

Let a single sine series (*) ${\sum}_{k=1}^{\infty}{a}_{k}sinkx$ be given with nonnegative coefficients ${a}_{k}$. If ${a}_{k}$ is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $k{a}_{k}\to 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) ${\sum}_{k=1}^{\infty}{\sum}_{l=1}^{\infty}{c}_{kl}sinkxsinly$, even with complex coefficients ${c}_{kl}$. We also give a uniform...

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