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1
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Statistical extensions of some classical Tauberian theorems in nondiscrete setting

100%
Móricz F.
Colloquium Mathematicum
|
2007
|
tom 107
|
nr 1
45-56
EN
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 analogues of the famous Vijayaraghavan lemma, which seem to be new and may be useful in other contexts.
2
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The harmonic Cesáro and Copson operators on the spaces $L^{p}(ℝ)$, 1 ≤ p ≤ 2

100%
Móricz F.
Studia Mathematica
|
2002
|
tom 149
|
nr 3
267-279
EN
The harmonic Cesàro operator 𝓒 is defined for a function f in $L^{p}(ℝ)$ for some 1 ≤ p < ∞ by setting $𝓒(f)(x): = ∫^{∞}_{x} (f(u)/u)du$ for x > 0 and $𝓒(f)(x): = -∫_{-∞}^{x} (f(u)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L¹_{loc}(ℝ)$ by setting $𝓒*(f)(x): = (1/x) ∫^{x₀} f(u)du$ 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 $f ∈ L^{p}(ℝ)$ for some 1 ≤ p ≤ 2, then $(𝓒(f))^{∧}(t) = 𝓒*(f̂)(t)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f ∈ L^{p}(ℝ)$ for some 1 < p ≤ 2, then $(𝓒*(f))^{∧}(t) = 𝓒(f̂)(t)$ a.e. As a by-product of our proofs, we obtain representations of $(𝓒(f))^{∧}(t)$ and $(𝓒*(f))^{∧}(t)$ in terms of Lebesgue integrals in case f belongs to $L^{p}(ℝ)$ for some 1 < p ≤ 2. These representations are valid for almost every t and may be useful in other contexts.
3
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Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences

100%
Móricz F.
Studia Mathematica
|
2013
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tom 219
|
nr 2
109-121
EN
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 $lim_{t→∞} τ(t) = A$, where $τ(t):= 1/(log t) ∫_{1}^{t} s(u)/u du$. (*) 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 oscillating, by which we mean that for every ε > 0 there exist t₀ = t₀(ε) > 1 and λ = λ(ε) > 1 such that |s(u) - s(t)| ≤ ε whenever $t₀ ≤ t < u ≤ t^{λ}$, then the converse implication holds true: the ordinary convergence $lim_{t→∞} s(t) = A$ follows from (*). We also present necessary and sufficient Tauberian conditions under which the ordinary convergence of a numerical sequence $(s_{k})$ follows from its logarithmic summability. Furthermore, we give a more transparent proof of an earlier Tauberian theorem due to Kwee.
4
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Multiple conjugate functions and multiplicative Lipschitz classes

100%
Móricz F.
Colloquium Mathematicum
|
2009
|
tom 115
|
nr 1
21-32
EN
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 $Lip(α₁,..., α_N)$ for some $0 < α₁,...,α_N < 1$ and its marginal functions satisfy $f(·,x₂,...,x_N) ∈ Lip β₁,...,f(x₁,...,x_{N-1},·) ∈ Lip β_N$ for some $0 < β₁,...,β_N < 1$ uniformly in the indicated variables $x_{l}$, 1 ≤ l ≤ N, then $f̃^{(η₁, ..., η_N)} ∈ Lip(α₁, ..., α_N)$ for each choice of $(η₁,...,η_N)$ with $η_{l} = 0$ or 1 for 1 ≤ l ≤ N.
5
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On the uniform convergence and L¹-convergence of double Walsh-Fourier series

100%
Móricz F.
Studia Mathematica
|
1992
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tom 102
|
nr 3
225-237
EN
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 supremum norm. As special cases, we obtain the extensions of the Dini-Lipschitz test and the Dirichlet-Jordan test for double Walsh-Fourier series.
6
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Tauberian theorems for Cesàro summable double integrals over $ℝ^{2}_{+}$

100%
Móricz F.
Studia Mathematica
|
2000
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tom 138
|
nr 1
41-52
EN
Given ⨍ ∈ $L^1_loc (ℝ^2_+)$, 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 to be replaced either by Schmidt type slow decrease (or increase) conditions, or by Landau type one-sided Tauberian conditions. Second, we consider complex-valued functions and give a two-sided Tauberian condition which is necessary and sufficient in order that (**) follow from (*). In particular, this condition is satisfied if s(u,v) is slowly oscillating, or if f(x,y) obeys Landau type two-sided Tauberian conditions. At the end, we extend these results to the mixed case, where the (C, 1, 0) mean, that is, the average value of s(w,v) with respect to the first variable over the interval [0,u], is considered instead of $σ_11 (u,v) := σ(u,v)$
7
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Absolutely convergent Fourier series and generalized Lipschitz classes of functions

100%
Móricz F.
Colloquium Mathematicum
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2008
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tom 113
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nr 1
105-117
EN
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 $|f(x+h) - f(x)| ≤ Ch^{α}L(1/h)$ for all x ∈ 𝕋, h > 0, where the constant C does not depend on x and h. The above sufficient conditions are also necessary in the case of a certain subclass of Fourier coefficients. As a corollary, we deduce that if a function f with Fourier coefficients in this subclass belongs to one of these generalized Lipschitz classes, then the conjugate function f̃ also belongs to the same generalized Lipschitz class.
8
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Tauberian theorems for Cesàro summable double sequences

100%
Móricz F.
Studia Mathematica
|
1994
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tom 110
|
nr 1
83-96
EN
$(s_{jk}: j,k = 0,1,...)$ 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 $(s_{jk})$ converges in Pringsheim's sense. These conditions are satisfied if $(s_{jk})$ 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 $(s_{jk})$ is summable (C,1,1) to a finite limit and there exist constants $n_1 > 0$ and H such that $jk(s_{jk} - s_{j-1,k} - s_{j-1,k} + s_{j-1,k-1}) ≥ -H$, $j(s_{jk} - s_{j-1, k}) ≥ -H$ and $k(s_{jk} - s_{j,k-1}) ≥ -H$ whenever $j,k > n_1$, then $(s_{jk})$ converges. We always mean convergence in Pringsheim's sense. Our method is suitable to obtain analogous Tauberian results for double sequences of complex numbers or for those in an ordered linear space over the real numbers.
9
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Ordinary convergence follows from statistical summability (C,1) in the case of slowly decreasing or oscillating sequences

100%
Móricz F.
Colloquium Mathematicum
|
2004
|
tom 99
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nr 2
207-219
EN
Schmidt's Tauberian theorem says that if a sequence (x_k) of real numbers is slowly decreasing and $lim_{n→ ∞} (1/n) ∑^{n}_{k=1} x_k = L$, then $lim_{k→ ∞} x_k = L$. 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 lemma under less restrictive conditions, which may be useful in other contexts.
10
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On the maximal Fejér operator for double Fourier series of functions in Hardy spaces

100%
Móricz F.
Studia Mathematica
|
1995
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tom 116
|
nr 1
89-100
EN
We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1,0)}(𝕋^2)$, $H^{(0,1)}(𝕋^2)$, or $H^{(1,1)}(𝕋^2)$. We prove that the maximal Fejér operator is bounded from $H^{(1,0)}(𝕋^2)$ or $H^{(0,1)}(𝕋^2)$ into weak-$L^1(𝕋^2)$, and also bounded from $H^{(1,1)}(𝕋^2)$ into $L^1(𝕋^2)$. These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1} log^{+} L(𝕋^2)$, $L^1(log^{+}L)^2(𝕋^2)$, and $L^μ(𝕋^2)$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures.
11
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Best possible sufficient conditions for the Fourier transform to satisfy the Lipschitz or Zygmund condition

100%
Móricz F.
Studia Mathematica
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2010
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tom 199
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nr 2
199-205
EN
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.
12
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Regular statistical convergence of double sequences

100%
Móricz F.
Colloquium Mathematicum
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2005
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tom 102
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nr 2
217-227
EN
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 ${x_{jk}: (j,k) ∈ ℕ²}$ 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 ${x_{jk} : k ∈ ℕ}$ is statistically convergent to some $ξ_j ∈ ℂ$ for each fixed j ∈ ℕ ∖ 𝓢₁, (iii) the single sequence ${x_{jk}: j ∈ ℕ}$ is statistically convergent to some $η_k ∈ ℂ$ for each fixed $k ∈ ℕ ∖ 𝓢₂$, where 𝓢₁ and 𝓢₂ are subsets of ℕ whose natural density is zero. We prove that under conditions (i)-(iii), both ${ξ_j}$ and ${η_k}$ are statistically convergent to ξ. As an application, we prove that if f ∈ L log⁺L(𝕋²), then the rectangular partial sums of its double Fourier series are regularly statistically convergent to f(u,v) at almost every point (u,v) ∈ 𝕋². Furthermore, if f ∈ C(𝕋²), then the regular statistical convergence of the rectangular partial sums of its double Fourier series holds uniformly on 𝕋².
13
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Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊

64%
Kórus P., Móricz F.
Studia Mathematica
|
2010
|
tom 201
|
nr 3
287-304
EN
We investigate the convergence behavior of the family of double sine integrals of the form $∫_{0}^{∞} ∫_{0}^{∞} f(x,y) sin ux sin vy dxdy$, 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 $∫^{b₁}_{a₁} ∫^{b₂}_{a₂}$ to zero in (u,v) ∈ ℝ²₊ as max{a₁,a₂} → ∞ and $b_{j} > a_{j} ≥ 0$, j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial integrals $∫_{0}^{b₁} ∫_{0}^{b₂}$ in (u,v) ∈ ℝ²₊ as min{b₁,b₂} → ∞ (called uniform convergence in Pringsheim's sense). These sufficient conditions are the best possible in the special case when f(x,y) ≥ 0.
14
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On the uniform convergence of double sine series

64%
Kórus P., Móricz F.
Studia Mathematica
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2009
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tom 193
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nr 1
79-97
EN
Let a single sine series (*) $∑^{∞}_{k=1} a_{k} sin kx$ 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 $ka_{k} → 0$ 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 (**) $∑^{∞}_{k=1} ∑^{∞}_{l=1} c_{kl} sin kx sin ly$, even with complex coefficients ${c_{kl}}$. We also give a uniform boundedness test for the rectangular partial sums of series (**), and slightly improve the results on single sine series.
15
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On the bundle convergence of double orthogonal series in noncommutative $L_2$-spaces

64%
Móricz F., Le Gac B.
Studia Mathematica
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2000
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tom 140
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nr 2
177-190
EN
The notion of bundle convergence in von Neumann algebras and their $L_2$-spaces for single (ordinary) sequences was introduced by Hensz, Jajte, and Paszkiewicz in 1996. Bundle convergence is stronger than almost sure convergence in von Neumann algebras. Our main result is the extension of the two-parameter Rademacher-Men'shov theorem from the classical commutative case to the noncommutative case. To our best knowledge, this is the first attempt to adopt the notion of bundle convergence to multiple series. Our method of proof is different from the classical one, because of the lack of the triangle inequality in a noncommutative von Neumann algebra. In this context, bundle convergence resembles the regular convergence introduced by Hardy in the classical case. The noncommutative counterpart of convergence in Pringsheim's sense remains to be found.
16
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Bundle Convergence in a von Neumann Algebra and in a von Neumann Subalgebra

64%
Le Gac B., Móricz F.
Bulletin of the Polish Academy of Sciences. Mathematics
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2004
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tom 52
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nr 3
283-295
EN
Let H be a separable complex Hilbert space, 𝓐 a von Neumann algebra in 𝓛(H), ϕ a faithful, normal state on 𝓐, and 𝓑 a commutative von Neumann subalgebra of 𝓐. Given a sequence (Xₙ: n ≥ 1) of operators in 𝓑, we examine the relations between bundle convergence in 𝓑 and bundle convergence in 𝓐.
17
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Addendum to the paper "Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅ ²₊" (Studia Math. 201 (2010), 287-304)

64%
Kórus P., Móricz F.
Studia Mathematica
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2012
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tom 212
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nr 3
285-286
18
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On the integrability and L¹-convergence of double trigonometric series

38%
Móricz F.
Studia Mathematica
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1991
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tom 98
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nr 3
203-225
19
Content available remote

On the integrability and L¹-convergence of sine series

32%
Móricz F.
Studia Mathematica
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1989
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tom 92
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nr 2
187-200
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