On sharp estimates of the convergence of double Fourier–Bessel series

The problem of approximation of a differentiable function of two variables by partial sums of a double Fourier–Bessel series is considered. Sharp estimates of the rate of convergence of the double Fourier–Bessel series on the class of differentiable functions of two variables characterized by a generalized modulus of continuity are obtained. The proofs of four theorems on this issue, which can be directly applied to solving particular problems of mathematical physics, approximation theory, etc., are presented.     

On Fourier–Jacobi expansions of real analytic Eisenstein series of degree 2

We discuss the Fourier–Jacobi expansion of certain vector valued Eisenstein series of degree $2$ , which is also real analytic. We show that its coefficients of index $\pm 1$ can be described by using a generating series of real analytic Jacobi forms. We also describe all the coefficients of general indices in suitable manners. Our method can be applied to study another Fourier series of Saito-Kurokawa type that is associated with a cusp form of one variable and half-integral weight. Then, following the arguments in the holomorphic case, we find that the Fourier series indeed defines a real analytic Siegel modular form of degree 2.     

On absolute convergence of Haar—Fourier series of superpositions of functions

В работе решены некот орые задачи, поставле нные П. Л. Ульяновым [5]. ПустьA _χ обозначает мн ожество тех функций?∈L(0, l), ряды Фурье которых вс юду абсолютно сходятся. Если? — вещественная функция на интервале [a, b], то для того, чтобы для каж дой?∈A _χ удовлетворяющей ус ловиюa≦?(t)≦b, 0≦t≦1, выполнялось включен ие?○?∈A _χ , необходимо и достато чно, чтобы? была лине йной функцией, т.е.?(x)=cx+d,x∈[a,b], гдеc иd — вещественны е постоянные. То же самое утвержден ие остается в силе, есл и заменить классA _? наA _χ ?C[0, 1]. Пусть далее? непреры вно дифференцируема я функция на [a,b]. Если некоторый мо дуль непрерывностиω и мод уль непрерывностиω(δ;?′) удовлетворяют услов ию \(\sum\limits_{n = 0}^\infty {\omega (\omega (2^{ - n} } ;\varphi '))\omega (2^{ - n} )< \infty ,\) , то?○?∈A _χ для каждой функци и?∈A _χ ∩H ^ω , гдеa≦?(t)≦b, 0≦t≦1.     

On Λ2-strong convergence of numerical sequences and Fourier series

We introduce the Λ²-strong convergence of numerical sequences and with it we generalize the concept of Λ-strong convergence of the results published by F. Móricz [2].     

On some estimates for best approximations of bivariate functions by Fourier–Jacobi sums in the mean

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.     

On the divergence of double Fourier–Walsh and Fourier–Walsh–Kaczmarz series of continuous functions

Acta Mathematica Hungarica - We prove that there exists a continuous function on $$[0,1]^2$$ , with a certain smoothness, whose double Fourier–Walsh series diverges on a set of positive...     

On the L1-convergence and behavior of coefficients of Fourier–Vilenkin series

In this paper, we prove the following statement that is true for both bounded and some type of unbounded Vilenkin systems: for any \( \varepsilon \in (0,1)\), there exists a measurable set \(E\subset [0,1)\) of measure bigger than \(1-\varepsilon \) such that for any function \(f \in L^{1}[0,1)\), it is possible to find a function \(g\in L^{1}[0,1)\) coinciding with f on E, Fourier series of g with respect to Vilenkin system are convergent in \(L^{1}\)-norm and the absolute values of non zero Fourier coefficients of g are monotonically decreasing.     

Uniform convergence of basic Fourier–Bessel series on a q-linear grid

We study Fourier–Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier–Bessel series.

     

On the Fourier coefficients of Siegel–Eisenstein series of degree 2 for odd level

The Ramanujan Journal - We calculate the Fourier coefficients of the Siegel–Eisenstein series of degree 2, level $$p^n$$ with trivial character. The main result is to compute the Siegel...     

On the local convergence of the Douglas–Rachford algorithm

We discuss the Douglas–Rachford algorithm to solve the feasibility problem for two closed sets A,B in \({\mathbb{R}^d}\) . We prove its local convergence to a fixed point when A,B are finite unions of convex sets. We also show that for more general nonconvex sets the scheme may fail to converge and start to cycle, and may then even fail to solve the feasibility problem.     

On the convergence and divergence of multiple Fourier series with respect to the Walsh—Paley system in the spacesC andL

Работа касается вопр осов сходимости и рас ходимости кратных рядов Фурье п о системе Уолша-Пэли в метрикахС и L. Из доказанных тео рем следует, в частности, ч то еишf∈Е (Е=С или E=L) и существует н атуральноеi ₀ (1≦i ₀ ≦N) так ое, что и $$\begin{gathered} \omega (\delta _{i_0 } ;f)_E = o\left( {\frac{1}{{1n^N \frac{1}{{\delta _{i_0 } }}}}} \right) (\delta _{i_0 } \to 0) \hfill \\ \omega (\delta _k ;f)_E = o\left( {\frac{1}{{1n^N \frac{1}{{\delta _k }}}}} \right), k \ne i_0 (\delta _k \to 0), k = 1,2, ...,N, \hfill \\ \end{gathered}$$ тоN-кратный ряд Фурье функцииf по системе У олша-Пэли сходится по Прингсхе йму в смысле метрики пространств аЕ. Доказано также, что вы шеотмеченное утверж дение неусиляемо в метрикеL не только для системы Уолша, но и для некоторого класс а ОНС, ограниченных в совок упности.     

On everywhere divergence of the strong Φ-means of Walsh–Fourier series

Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems was established by Rodin in 1990. We prove, that if the growth of a function Φ(t):[0,∞)→[0,∞)$\Phi (t):[0,\infty )\to [0,\infty )$ is bigger than the exponent, then the strong Φ-summability of a Walsh–Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the authors of this paper.     

On the Abel–Jacobi maps of Fermat Jacobians

We study the Abel–Jacobi image of the Ceresa cycle ${W_k-W_k^-}$ , where W _k is the image of the kth symmetric product of a curve X on its Jacobian variety. For the Fermat curve of degree N, we express it in terms of special values of generalized hypergeometric functions and give a criterion for the non-vanishing of ${W_k-W_k^-}$ modulo algebraic equivalence, which is verified numerically for some N and k.     

Multiplicative convolutions of functions from Lorentz spaces and convergence of series of Fourier–Vilenkin coefficients

Let f and g be functions from different Lorentz spaces L ^{p, q} [0, 1), h be theirmultiplicative convolution and xxxx be Fourier coefficients of h with respect to a multiplicative system with bounded generating sequence. We estimate the remainder of the series of xxxx with multiplicators of type k ^b in terms of the best approximations of f and g in the corresponding Lorentz spaces. We establish sharpness of this result and of its corollaries for the Lebesgue spaces.