Absolute Convergence Of Fourier Series Research Articles

Green’s Function Estimates for Elliptic Differential Operators with Singular Coefficients and Absolute Convergence of Fourier Series

Green’s Function Estimates for Elliptic Differential Operators with Singular Coefficients and Absolute Convergence of Fourier Series

Absolute convergence factors of Lipschitz class functions for general Fourier series

Abstract The main aim of this paper is to investigate the sequences of positive numbers, for which multiplication with Fourier coefficients of functions f ∈ Lip ⁡ 1 {f\in\operatorname{Lip}1} class provides absolute convergence of Fourier series. In particular, we found special conditions for the functions of orthonormal system (ONS), for which the above sequences are absolute convergence factors of Fourier series of functions of Lip ⁡ 1 {\operatorname{Lip}1} class. It is established that the resulting conditions are best possible in certain sense.

General orthonormal systems and absolute convergence

We study problems of the absolute convergence of Fourier series of differentiable functions with respect to general orthonormal systems (ONS). We obtain results which show that under certain conditions on functions in the ONS, the Fourier series of an arbitrary function in a certain differentiability class is absolutely convergent. We finally note that our results cannot be improved.

On Changes of Variable Preserving the Convergence and Absolute Convergence of Fourier Series in the Haar Wavelet System

It is established that, among all continuously differentiable homeomorphic changes of variable, the absolute convergence of Fourier series in the Haar wavelet system is preserved by only those for which $$\varphi^{-1}(0)$$ is binary-rational and $$\varphi'(x)=\pm 2^m$$ , where $$m$$ is an integer and $$x\in\mathbb R$$ . It is also established that this condition is necessary for a continuously differentiable homeomorphic change of variable to preserve the convergence of Fourier series in the Haar wavelet system.

Absolutely Convergent Fourier Series of Functions over Homogeneous Spaces of Compact Groups

This paper presents a systematic study for classical aspects of functions with absolutely convergent Fourier series over homogeneous spaces of compact groups. Let G be a compact group, H be a closed subgroup of G, and μ be the normalized G-invariant measure over the left coset space G/H associated with Weil’s formula with respect to the probability measures of G and H. We introduce the abstract notion of functions with absolutely convergent Fourier series in the Banach function space L1(G/H,μ). We then present some analytic characterizations for the linear space consisting of functions with absolutely convergent Fourier series over the compact homogeneous space G/H.

Some properties of an odd-dimensional space

Abstract In this paper, we investigate the absolute convergence of Fourier series of functions in several variables for an odd-dimensional space when these functions have continuous partial derivatives. It should be noted that similar properties for an even-dimensional space were given in [L. D. Gogoladze and V. S. Tsagareishvili, On absolute convergence of multiple Fourier series (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 2015, 9, 12–21; translation in Russian Math. (Iz. VUZ) 59 (2015), no. 9, 9–17]. The obtained results are the best possible in a certain sense.

On the generalized absolute convergence of Fourier series

Abstract In the present paper the sufficient conditions are obtained for the generalized r-absolute convergence ( 0 < r < 2 {0<r<2} ) of the single Fourier trigonometric series in terms of the modulus of δ-variation of a function. It is proved that these conditions are unimprovable in a certain sense. The classical results of Berstein, Szasz, Zygmund and others, related to the absolute convergence of single trigonometric Fourier series, were previously generalized by [L. Gogoladze and R. Meskhia, On the absolute convergence of trigonometric Fourier series, Proc. A. Razmadze Math. Inst. 141 2006, 29–40].

On the absolute convergence of Fourier series with respect to general orthonormal systems

Abstract The problems of absolute convergence of multiple Fourier series with respect to both general orthonormal systems and multiple Haar series are studied.

On the absolute convergence of Fourier series and generalisation of Lipschitz spaces, defined by differences of fractional order

We obtain the necessary and sufficient conditions in terms of Fourier coefficients of $2\pi$-periodic functions $f$ with absolutely convergent Fourier series, for $f$ to belong to the generalized Lipschitz classes $H^{\omega, \alpha}_{\mathbb{C}}$, and to have the fractional derivative of order $\alpha$ ($0 < \alpha < 1$).

Absolute convergence of Fourier series in bounded complete double orthonormal systems

Absolute convergence of Fourier series in bounded complete double orthonormal systems

On β-absolute convergence of Vilenkin-Fourier series with small gaps

The study of absolute convergence of Fourier series is one of the most important problems of Fourier Analysis and the problem has been studied intensively by many researchers in the setting of circle group in particular and classical groups in general. Recently, in [Math. Inequal. Appl. 17 (2) (2014), 749-760], we have studied the β-absolute convergence (0 < β ≤ 2) of Vilenkin-Fourier series for the functions of various classes of functions of generalized bounded fluctuation and given sufficient conditions in terms of modulus of continuity. In this paper, we prove that this is a matter only of local fluctuation for functions with the Vilenkin-Fourier series lacunary with small gaps. Our results, as in the case of trigonometric Fourier series, illustrate the interconnection between 'localness' of the hypothesis and type of lacunarity and allow us to interpolate the results.

Absolute convergence of Fourier series in bounded complete double orthonormal systems

Absolute convergence of Fourier series in bounded complete double orthonormal systems

Wiener–Ingham type inequality for Vilenkin groups and its application to harmonic analysis

An important trigonometric inequality essentially due to Wiener but later on made precise by Ingham concerning the lacunary trigonometric sums \(f(x)=\sum A_ke^{in_kx}\), where \(A_k\)’s are complex numbers, \(n_{-k}=-n_k\) and \(\{n_k\}\) satisfies the small gap condition \((n_{k+1}-n_k)\ge q\ge 1\) for \(k=0,1,2,\ldots \), says that if I is any subinterval of \([-\pi ,\pi ]\) of length \(|I|=2\pi (1+\delta )/q>2\pi /q\) then \(\sum |A_k|^2\le A_{\delta }|I|^{-1}\int _I|f|^2\), \(|A_k|\le A_{\delta }|I|^{-1}\int _I|f|\), wherein \(A_{\delta }\) depends only on \(\delta \). Such an inequality is proved here in the setting of the Vilenkin groups G. The inequality is then applied to generalize the Bernstěin, Szasz and Stechkin type results concerning the absolute convergence of Fourier series on G.

Absolute convergence of fourier series of almost-periodic functions

We present necessary and sufficient conditions for the absolute convergence of the Fourier series of almost-periodic (in the sense of Besicovitch) functions when the Fourier exponents have limit points at infinity or at zero. The structural properties of the functions are described by the modulus of continuity or the modulus of averaging of high order, depending on the behavior of the Fourier exponents. The case of uniform almost-periodic functions of bounded variation is considered.

On the divergence of Fourier series of functions in several variables

The paper deals with the problems of divergence of the series from absolute values of the Fourier coefficients of functions in several variables. It is proved that as the dimension of the space increases, the absolute convergence of Fourier series with respect to any complete orthnormal system (ONS) of functions with continuous partial derivatives becomes worse. For instance, for any ɛ ∈ (0, 2) there exists a function in variables \(k > \frac{{2(2 - \varepsilon )}} {\varepsilon }\) having all the continuous partial derivatives, however the series of absolute values of its coefficients with respect to any complete orthnormal system diverges in power 2 − ɛ.

Absolute convergence of Fourier series of functions of class and of functions of bounded variation

We study problems of the absolute convergence of Fourier series with respect to general orthonormal systems for functions in certain classes.

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space $A(\mathbb T)$ of all continuous functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z\}$ belongs to $l^1(\mathbb Z)$. The norm on $A(\mathbb T)$ is defined by $\|f\|_{A(\mathbb T)}=\|\hat{f}\|_{l^1(\mathbb Z)}$. According to the known Beurling--Helson theorem, if $\phi : \mathbb T\rightarrow\mathbb T$ is a continuous mapping such that $\|e^{in\phi}\|_{A(\mathbb T)}=O(1), ~n\in\mathbb Z,$ then $\phi$ is linear. It was conjectured by Kahane that the same conclusion about $\phi$ is true under the assumption that $\|e^{in\phi}\|_{A(\mathbb T)}=o(\log |n|)$. We show that if $\|e^{in\phi}\|_{A(\mathbb T)}=o((\log\log |n|/\log\log\log |n|)^{1/12})$ then $\phi$ is linear.

Operator ideals and the absolute convergence of Fourier series

In this paper we give some relationships between the absolutely convergent Fourier series of functions belonging to Besov spaces and their connection with the theory of operator ideals. In this context, we give results in operator ideals associated with generalized approximation numbers, Weyl numbers and entropy numbers.

Sufficient conditions for the Lebesgue integrability of Fourier transforms

We give sufficient conditions for the Lebesgue integrability of the Fourier transform of a function f ∈ Lp(ℝ) for some 1 < p ≤ 2. These sufficient conditions are in terms of the Lp 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, Szasz, Zygmund and Salem, and recent theorems of Gogoladze and Meskhia on the absolute convergence of Fourier series.

Absolute convergence of fourier series of superpositions of functions

In this paper we establish conditions for the absolute convergence of series of Fourier coefficients with respect to a generalized Haar system of a superposition of two functions.