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- Author or Editor: Ferenc Móricz x

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## Abstract

The well-known characterization indicated in the title involves the moving maximal dyadic averages of the sequence (*X*
_{
k
}: *k* = 1, 2, …) of random variables in Probability Theory. In the present paper, we offer another characterization of the SLLN
which does not require to form any maximum. Instead, it involves only a specially selected sequence of moving averages. The
results are also extended for random fields (*X*
_{
kℓ}: *k*, ℓ = 1, 2, …).

<a name="abs1"/>Abstract??We give sufficient conditions for the convergence of the double Fourier integral of a complex-valued function*f*?*L*
^{1}(?^{2}) with bounded support at a given point (*x*
_{0},*y*
_{0}) ? ?^{2}. It turns out that this convergence essentially depends on the convergence of the single Fourier integrals of the marginal functions*f*(*x*,*y*
_{0}),*x*? ?, and*f*(*x*
_{0},*y*),*y*? ?, at the points*x*:=*x*
_{0}and*y*:=*y*
_{0}, respectively. Our theorem applies to functions in the multiplicative Zygmund classes of functions in two variables.

## Abstract

We prove sufficient conditions for the convergence of the integrals conjugate to the double Fourier integral of a complex-valued
function *f* ∈ *L*
^{1} (ℝ^{2}) with bounded support at a given point (*x*
_{0}, *g*
_{0}) ∈ ℝ^{2}. It turns out that this convergence essentially depends on the convergence of the integral conjugate to the single Fourier
integral of the marginal functions *f*(*x, y*
_{0}), *x* ∈ ℝ, and *f*(*x*
_{0}, *y*), *y* ∈ ℝ, at *x*:= *x*
_{0} and *y*:= *y*
_{0}, respectively. Our theorems apply to functions in the multiplicative Lipschitz and Zygmund classes introduced in this paper.

## Abstract

This is a survey paper on the recent progress in the study of the continuity and smoothness properties of a function *f* with absolutely convergent Fourier series. We give best possible sufficient conditions in terms of the Fourier coefficients
of *f* which ensure the belonging of *f* either 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. We also discuss the termwise differentiation of Fourier series. Our theorems generalize those by R. P. Boas Jr., J.
Németh and R. E. A. C. Paley, and a number of them are first published in this paper or proved in a simpler way.

## Abstract.

We consider complex-valued functions *f*∊*L*
^{1}(ℝ_{+}), where ℝ_{+}:=[0,∞), and prove sufficient conditions under which the sine Fourier transform and the cosine Fourier transform belong 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 if *f*(*x*)≧0 almost everywhere.

## Abstract

We introduce the higher order Lipschitz classes Λ_{
r
}(*α*) and *λ*
_{
r
}(*α*) of periodic functions by means of the *r*th order difference operator, where *r* = 1, 2, ..., and 0 < *α* ≦ *r*. We study the smoothness property of a function *f* with absolutely convergent Fourier series and give best possible sufficient conditions in terms of its Fourier coefficients
in order that *f* belongs to one of the above classes.

## Abstract

*N*-multiple trigonometric series whose complex coefficients

*c*

_{ j1},...,

*j*

_{ N }, (

*j*

_{1},...,

*j*

_{ N }) ∈ ℤ

^{ N }, form an absolutely convergent series. Then the series

*f*, which is continuous on the

*N*-dimensional torus

^{ N },

*f*belong to one of the multiplicative Lipschitz classes Lip (α

_{1},..., α

_{ N }) and lip (α

_{1},..., α

_{ N }) for some α

_{1},..., α

_{ N }> 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related

*N*-multiple numerical series may be useful in other investigations, too.

## Abstract

We give sufficient conditions for the Lebesgue integrability of the Fourier transform of a function *f* ∈ *L*
^{
p
}(ℝ) for some 1 < *p* ≤ 2. These sufficient conditions are in terms of the *L*
^{
p
} integral modulus of continuity of *f*; in particular, they apply for functions in the integral Lipschitz class Lip(*α, p*) and for functions of bounded *s*-variation for some 0 < *s* < *p*. Our theorems are nonperiodic versions of the classical theorems of Bernstein, Szász, Zygmund and Salem, and recent theorems
of Gogoladze and Meskhia on the absolute convergence of Fourier series.