A uniqueness theorem for Stromberg series

For simple trigonometric series it is shown, in particular, that if the trigonometric series is Riemann summable in measure to an integrable function and if the Riemann majorant is finite everywhere except possibly on a countable set, then this series is the Fourier series of the function . Uniqueness theorems for multiple trigonometric series are obtained on the basis of this result. Bibliography: 14 titles.

A survey of uniqueness questions in multiple trigonometric series

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.

On the unique factorization theorem for formal power series

On the unique factorization theorem for formal power series II

A Uniqueness Theorem for Exponential Series

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.

In this paper our primary interest is in developing further insight into convergence properties of multiple trigonometric series, with emphasis on the problem of uniqueness of trigonometric series. Let $E$ be a subset of positive (Lebesgue) measure of the $k$ dimensional torus. The principal result is that the convergence of a trigonometric series on $E$ forces the boundedness of the partial sums almost everywhere on $E$ where the system of partial sums is the one associated with the system of all rectangles situated symmetrically about the origin in the lattice plane with sides parallel to the axes. If $E$ has a countable complement, then the partial sums are bounded at every point of $E$. This result implies a uniqueness theorem for double trigonometric series, namely, that if a double trigonometric series converges unrestrictedly rectangularly to zero everywhere, then all the coefficients are zero. Although uniqueness is still conjectural for dimensions greater than two, we obtain partial results and indicate possible lines of attack for this problem. We carry out an extensive comparison of various modes of convergence (e.g., square, triangular, spherical, etc.). A number of examples of pathological double trigonometric series are displayed, both to accomplish this comparison and to indicate the “best possible” nature of some of the results on the growth of partial sums. We obtain some compatibility relationships for summability methods and finally we present a result involving the $(C,\alpha ,0)$ summability of multiple Fourier series.

On generalized uniqueness theorems for Walsh series

In 2011, Li [A uniqueness theorem for Dirichlet series satisfying a Riemann type functional equation. Adv Math. 2011;226(5):4198–4211] proved that if two L-functions and satisfy the same functional equation with and for two distinct finite complex numbers and then We prove that if two L-functions and have positive degree and satisfy the same functional equation with and for a finite set where and are three distinct finite complex values, then The main results of this paper are concerning some questions posed by Gross [Factorization of meromorphic functions and some open problems. Complex Analysis, Kentucky 1976 (Proc.Conf.); Berlin: Springer-Verlag; 1977. p. 51–69. (Lecture Notes in Mathematics; 599)].

A Uniqueness Theorem for Multiple Orthonormal Spline Series

A Uniqueness Theorem for Haar and Walsh Series

Dirichlet sets and some uniqueness theorems for Walsh series

A uniqueness theorem for Dirichlet series satisfying a Riemann type functional equation

It is shown that the main uniqueness properties of a multiple trigonometric series are equivalent to similar properties of the corresponding series with respect to a multiple Haar system with variable coefficients. Uniqueness theorems for multiple trigonometric series are proved under different conditions on the coefficients and on the derivative with respect to random binary nets of the sums of the series resulting from their single integration.