General Orthonormal Systems Research Articles

Optimal convergence factors for general Fourier coefficients

Abstract Stefan Banach proved that even the Fourier series of the function f ⁢ ( x ) = 1 {f(x)=1} ( x ∈ [ 0 , 1 ] ) {(x\in[0,1])} might not be convergent for some orthonormal systems (ONS). Thus we can conclude that the Fourier series of functions belonging to a certain differential class cannot be convergent in general. On the other hand, for classical ONS (trigonometric, Haar, Walsh systems, …), the convergence of Fourier series of differentiable class functions is a simple subject. In this paper, we investigate the sequence of positive numbers such that, multiplying the Fourier coefficients of Lipschitz class functions by these numbers, one obtains a convergent series of the special form. From the convergence of these special form series, we derive the convergence of the special Fourier series of Lipschitz class functions with respect to general ONS. The obtained results are best possible.

Some Problems of Convergence of General Fourier Series

Banach [1] proved that good differential properties of function do not guarantee the a.e. convergence of the Fourier series of this function with respect to general orthonormal systems (ONS). On the other hand it is very well known that a sufficient condition for the a.e. convergence of an orthonormal series is given by the Menshov–Rademacher Theorem. The paper deals with sequence of positive numbers $$(d_{n})$$ such that multiplying the Fourier coefficients $$(C_{n}(f))$$ of functions with bounded variation by these numbers one obtains a.e. convergent series of the form $$\sum_{n=1}^{\infty}d_{n}C_{n}(f)\varphi_{n}(x).$$ It is established that the resulting conditions are best possible.

Lipschitz class functions and their general Fourier coefficients

Abstract In the present paper we consider some specific properties of special series of Fourier coefficients of the Lipschitz class functions with respect to general orthonormal systems (ONS). The obtained results demonstrate that the properties of the general ONS and of the subsequence of this system are essentially different. It is shown that the problems for the general ONS are trivial for classical ONS (trigonometric, Haar and Walsh systems).

Smooth functions and general Fourier coefficients

As is known, the Fourier series of differentiable functions for classical orthonormal systems (trigonometric, Haar, Walsh, . . . ) are absolutely convergent. However, for general orthonormal systems (ONS) this fact does not hold. In the present paper, we consider some specific properties of special series of Fourier coefficients of differentiable functions with respect to the general ONS. The obtained results demonstrate that the properties of the general ONS and of the subsequence of this system are essentially different. Here we have shown that the received results are best possible.

Differentiable functions and their Fourier coefficients

Abstract In the paper we consider the properties of Fourier coefficients of functions that possess derivatives of bounded variation. We investigate the convergence of the special series of Fourier coefficients with respect to general orthonormal systems (ONS). The obtained results are the best possible. We also describe the behavior of subsequences of general ONS.

Some Properties of General Fourier Coefficients of Lipschitz Functions

In this paper, the properties of Lipschitz functions are considered and the convergence of the special type of series of Fourier coefficients with respect to general orthonormal systems (ONS) is investigated. For the functions of ONS we find a condition under which the special series of Fourier coefficients of Lipschitz functions with respect to general orthonormal systems are convergent. The obtained results indicate that the character of classical ONS (trigonometric, Haar, Walsh systems) and of the general ONS are different in the general case. In addition, a result on subsequences of general ONS is given.

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.

Some particular properties of general orthonormal systems

The present paper deals with some particular properties of special series of Fourier coefficients of the class of functions with bounded variation with respect to general orthonormal systems (ONS). The obtained results demonstrate that the properties of the general ONS and of the classical ONSs (trigonometric, Haar, Walsh system) are different in some cases.

General Fourier coefficients for Lip α class functions

We investigate the behavior of Fourier coefficients of Lip α class functions with respect to general orthonormal systems (ONS). The generalizations in a certain sense of some theorems by S. Szasz, B. I. Golubov and S. V. Bochkarev are obtained.

General orthonormal systems and absolute convergence

В работе рассмотрены вопросы абсолютной сходимости рядов Фурье дифференцируемых функций относительно общих ортонормированных систем (ОНС). Получены результаты, которые указывают, что если функции ОНС удовлетворяют некоторым условиям, то ряд Фурье любой функции из заданного дифференциального класса сходится абсолютно. Здесь же отмечается, что полученные результаты не могут быть усилены. Библиография: 9 наименований.

Some properties of general orthonormal systems

Some properties of general orthonormal systems

Differentiable Functions and General Orthonormal Systems

Differentiable Functions and General Orthonormal Systems

General Fourier coefficients

It is well known that if $${f \in L_{2}(0,1)}$$ is an arbitrary function ( $${{f(x) \nsim 0}, x \in [0,1]}$$ ) then its Fourier coefficients with respect to general orthonormal systems (ONS) may belong only to $${\ell_2}$$ . Thus in the general case it is impossible to estimate these coefficients by moduli of continuity or moduli of smoothness of the given functions. In the present paper conditions are found which should be satisfied by ONS so that the coefficients of some classes of functions can be estimated by modulus of continuity or modulus of smoothness of these functions.

Peak-to-Average Power Control via Tone Reservation in General Orthonormal Transmission Systems

In this paper, we study the tone reservation method for reducing the peak-to-average power ratio (PAPR) in general orthonormal transmission systems. We prove that strong solvability, where the peak value of the transmit signal has to be bounded by a constant times the energy of the information symbols, is equivalent to weak solvability, where the peak value of the transmit signal has to be only bounded. Further, we show that in the case where the PAPR problem is not weakly solvable, almost all information sequences lead to an unbounded transmit signal.

Unconditional Convergence of Fourier Series for Functions of Bounded Variation

This article concerns the unconditional convergence a.e. of Fourier series with respect to general orthonormal systems. We find certain conditions to be satisfied by the functions in the orthonormal system so that the Fourier series of each function of finite variation unconditionally converge a.e. The results are best possible.

Functions of the Lipschitz class and summability of Fourier series

In this paper we investigate summability of the general orthonormal systems (GOS) of the functions f(x) ∈ Lip 1.

Convergence of the Fourier Series of Lip 1 Functions with Respect to General Orthonormal Systems

We establish sufficient conditions that should be satisfied by functions of a general orthonormal system (ONS) {φ n (x)} in order that the Fourier series in this system be convergent almost everywhere on [0, 1] for any function from the class Lip 1. It is shown that the obtained conditions are best possible in a certain sense.

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 Fourier coefficients of functions with respect to general orthonormal systems

We present results describing some properties of the Fourier coefficients of functions with respect to general orthonormal systems (ONS). We note that good differential properties of the functions do not ensure the `good' behaviour of the Fourier coefficients (in the sense of convergence to zero) of these functions with respect to general ONS. We find conditions on the functions forming an ONS (, , for which the series of Fourier coefficients of the functions , where , are absolutely convergent. We consider relationships between ONS, that is, problems of absolute independence for orthonormal systems.

The Fourier coefficients of continuous functions with respect to certain orthonormal systems

The properties of the Fourier coefficients for some classes of functions with respect to both the Haar system and general orthonormal systems are studied. It is established that the results obtained in the paper cannot be strengthened.