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It is proved that if a multiple series in the Franklin system converges in the sense of Pringsheim everywhere, except, perhaps, on a set that is a Cartesian product of sets of measure zero, to an everywhere finite integrable function, then it is the Fourier–Franklin series of this function. A uniqueness theorem is also proved for multiple Franklin series whose rectangular partial sums at each point have a sequential limit.

В работе для простых тригонометрических рядов, в частности, доказано, что если тригонометрический ряд методом Римана по мере суммируется к интегрируемой функции $f$ и мажоранта Римана всюду, кроме, быть может, некоторого счетного множества, конечна, то этот ряд является рядом Фурье функции $f$. С применением этой теоремы получены теоремы единственности для кратных тригонометрических рядов. Библиография: 14 названий.

В работе доказано, что если кратный ряд по системе Франклина всюду, кроме быть может, некоторого множества, являющегося декартовым произведением множеств меры нуль, сходится по Прингсхейму к всюду конечной интегрируемой функции, то он является рядом Фурье-Франклина этой функции. Доказана также теорема единственности для кратных рядов Франклина, прямоугольные частичные суммы которого в каждой точке имеют последовательный предел. Библиография: 25 названий.

It is proven that, if the Riemann sums of a trigonometric series are bounded everywhere, perhaps, except for some countable set and converge in measure to a bounded function $$f$$ , then this series is the Fourier series of the function $$f$$ . This implies the generalizations of the Cantor and Young theorems.

The paper contains two main results. First we describe one-dimensional Franklin series converging everywhere except possibly on a finite set to an everywhere-finite integrable function. Second we establish a class of subsets of with the following property. If a double Franklin series converges everywhere except on this set to an everywhere-finite integrable function, then it is the Fourier–Franklin series of this function. In particular, all countable sets are in this class.

We generalize an important property of trigonometric series to the case of series by orthonormal spline systems corresponding to the dyadic sequence of grid points. We prove that Ciesielski series cannot diverge to infinity on a set of positive measure.

We prove that, for every 0 1 − ϵ such that, for all f ∈ L1(T) and 0 < η < 1, we can find $$\tilde f \in {L^1}(T)$$ with $$\int\!\!\!\int_T {{\rm{|}}f(x,y) - \tilde f(x,y){\rm{|}}dxdy \le \eta } $$ coinciding with f(x, y) on E whose double Fourier-Franklin series converges absolutely to f almost everywhere on T.

In this paper we prove that the quadratic partial sums of a multiple Franklin series with indices $$2^{\nu}$$, $$\nu=1,2,...$$, cannot converge to $$+\infty$$ on a set of positive measure.

Статья содержит два основных результата. В первом описаны однократные ряды Франклина, сходящиеся всюду, кроме, быть может, некоторого конечного множества, к всюду конечной интегрируемой функции. Во втором результате устанавливается некоторый класс подмножеств множества $[0, 1]^2$ со свойством: если двойной ряд Франклина всюду, кроме, быть может, этого множества сходится к всюду конечной интегрируемой функции, то он является рядом Фурье-Франклина этой функции. В этот класс входит, в частности, любое счетное множество. Библиография: 27 наименований.

A criterion for almost everywhere convergence of Franklin series on a set is proved. The criterion is similar to the ones obtained by Y.S. Chow for martingales and F.G. Arutyunyan for Haar series.

It is proved that the partial sums of a series in the Franklin system with numbers 2μ, μ ∈ ℕ, cannot approach +∞ on a set of positive measure. In particular, a Franklin series cannot converge to +∞ on a set of positive measure.

Доказывается, что частичные суммы ряда по системе Франклина с номерами $2^{\mu}$, $\mu\in \mathbb{N}$, не могут стремиться к $+\infty$ на множестве положительной меры. В частности, ряд Франклина на множестве положительной меры не может сходиться к $+\infty$. Библиография: 15 названий.

Simple Franklin series are investigated that converge to zero everywhere except for one point (or several points). It is also proved that the one-point set (or a finite set) is a uniqueness set for Pringsheim convergent double series.

In this paper, we prove that a set E is an M*-set or an AM*-set for the Franklin system if and only if E contains a nonempty perfect set.

The paper considers the general Franklin system corresponding to a regular by couples partition of the segment [0; 1]. For series by this system, we prove uniqueness theorems and obtain restoration formulas for coefficients, provided that the series converge in measure and satisfy some necessary condition.

A uniqueness theorem and a recovery theorem for the coefficients of series in generalized Haar systems are proved under the assumption that the series converge in measure and satisfy a certain necessary condition on the distribution function of the majorant of partial sums.

In this paper we obtain representation formulas for coefficients of Haar series that converge to zero everywhere except possibly some finite set.

In this paper, we prove uniqueness theorems and restoration formulas for coefficients of series by Vilenkin system. The series is assumed to be convergent in measure and the distribution function of the majorant of partial sums satisfies some necessary condition.

In this paper, we obtain recovery formulas for coefficients of multiple Franklin series by means of its sum, if the series satisfies the following conditions: 1) the square partial sums with indices 2 ν converge almost everywhere, 2) the majorant of partial sums with indices 2 ν satisfies some necessary condition.

In this paper we obtain a necessary and sufficient condition on the sequence of natural numbers {q n } such that the almost everywhere convergence of the cubic partial sums S qn (x) of the multiple Haar series Σn a nχn(x) and the condition lim inf $$\lambda \cdot mes\left\{ {x:\begin{array}{*{20}{c}} {\sup } \\ n \end{array}\left| {S{}_{qn}\left( x \right)} \right| \succ \lambda } \right\} = 0$$ , imply that the coefficients a n can be uniquely determined by the sum of the series. Also, we have obtained a necessary and sufficient condition for the series $$\sum\limits_{n = 1}^\infty {{\varepsilon _n}{a_n}} {\chi _n}\left( x \right)$$ with an arbitrary bounded sequence {e n} to be a Fourier-Haar series of an A-integrable function.

In this paper we prove that the majorant of partial sums and the Paley function of Franklin series have equivalent norms in the space L p (I), p > 0, provided that the “peak” intervals of Franklin functions with non-vanishing coefficients lie in I. Examples of series emphasizing that this condition is essential are also given.

The paper presents the proof of the uniqueness theorem formultiple series in the Franklin systemthat converge inmeasure and whosemajorant of cubic partial sums with numbers 2 ν satisfies a certain necessary condition. This result is new in the one-dimensional case as well.

For an admissible sequence T we define an orthonormal system consisting of piecewise linear functions with vanishing integrals on R. Necessary and sufficient conditions on T are found for the corresponding system to be a basis in H 1(R).

For a general Franklin system generated by an admissible sequence , we obtain necessary and sufficient conditions on under which the corresponding system is a basis or an unconditional basis in .

A general Franklin system on R, generated by an admissible sequence T is defined. We show that such defined system forms an unconditional basis in the space Lp(R), 1 < p < ∞, and prove theorems on locally uniform convergence of Fourier-Franklin series by this system.

The paper considers general Franklin systems generated by strong regular partitions of the segment [0; 1]. For such systems we prove the following assertions: 1) the absolute convergence of a Fourier-Franklin series at a point is a local property; 2) the Fourier-Franklin series of a function of bounded variation absolute converges almost everywhere; 3) any almost everywhere finite measurable function can be represented by an almost everywhere absolutely convergent series by a general Franklin system.

We prove that if the admissible sequence \(\mathcal{T}\) is strongly regular or is quasi-dyadic and weakly regular, then the Paley operator by the corresponding general Franklin system is of weak type (1, 1). Examples of general Franklin system with some properties are also discussed.

For a general Franklin system generated by a quasidyadic and strongly regular sequence, it is proved, in particular, that any subseries of a Fourier series in the general Franklin system is a Fourier series of an A-integrable function in the sense of A-integration.

We give some sufficient condition for the uniqueness of series with respect to general Franklin systems and discuss analogous results for orthonormal spline systems of higher order and some wavelet systems.

We prove that if and then the Walsh series has the following property. For any measurable function which is finite almost everywhere, there are numbers such that the series converges to almost everywhere. This assertion complements and strengthens previously known results about universal Walsh series and Walsh null-series.

New non-traditional methods are given that ensure uniform convergence for summation of trigonometric Fourier series of continuous functions.

It is proved that if a series in the Franklin system converges almost everywhere to a function f(t) and the distribution function of the majorant of partial sums satisfies the condition mes{t∈[0,1]∶s(t)>λ}=o(1/λ) as λ→∞, then this series is a Fourier series for Lebesgue integrable functions f(t). In the general case the coefficients of the series are reconstructed by means of anA-integral.

Trigonometric series summable by the Riemann method almost everywhere are considered. In particular, it is proved that if a multiple trigonometric series sums almost everywhere by the Riemann method to an integrable function f(x), and the Riemann majorant of this series satisfies a certain necessary condition, then the series is the Fourier series of the function f(x).Bibliography: 11 titles.

Formulas are established for determining the coefficients of a trigonometric series summable almost everywhere by Riemann's method whose Riemann majorant satisfies a certain necessary condition that guarantees uniqueness. Examples are also presented of trigonometric series that show that the results obtained are definitive.

It is proved that (1)is the Fourier series of an integrable function if and only if1) almost everywhere, and2) \lambda\,\}=0$ SRC=http://ej.iop.org/images/0025-5734/68/2/A02/tex_sm_2107_img4.gif/>,where 0}\vert S(x,\,h)\vert.$ SRC=http://ej.iop.org/images/0025-5734/68/2/A02/tex_sm_2107_img5.gif/>Bibliography: 6 titles.