Fourier-Haar Series Research Articles

Erratum to: “Coefficients of multiple Fourier-Haar series and variational modulus of continuity” [Bulletin of the Karaganda University. Mathematics series, No. 4(112), 2023, pp. 21–29

Prof. Tuncay Aktosun requests retraction of his co-authorship for the following reason. Prof. Tuncay Aktosun has not been involved in the publication process and has not participated in any discussions, therefore, he cannot hold responsibility for any results in the published paper.

Coefficients of multiple Fourier-Haar series and variational modulus of continuity

In this paper, we introduce the concept of a variational modulus of continuity for functions of several variables, give an estimate for the sum of the coefficients of a multiple Fourier-Haar series in terms of the variational modulus of continuity, and prove theorems of absolute convergence of series composed of the coefficients of multiple Fourier-Haar series. In this paper, we study the issue of the absolute convergence for multiple series composed of the Fourier-Haar coefficients of functions of several variables of bounded p-variation. We estimate the coefficients of a multiple Fourier-Haar series in terms of the variational modulus of continuity and prove the sufficiency theorem for the condition for the absolute convergence of series composed of the Fourier-Haar coefficients of the considered function class. This paper researches the question: under what conditions, imposed on the variational modulus of continuity of the fractional order of several variables functions, there is the absolute convergence for series composed of the coefficients of multiple Fourier-Haar series.

Approximation by Haar polynomials in variable exponent grand Lebesgue spaces

In this paper, we give direct theorems on approximation by Haar and Walsh polynomials in variable exponent grand Lebesgue space. Also the degree of approximation by Borel, Abel-Poisson, Riesz-Zygmund and Euler linear means of Haar-Fourier series are treated in the above-cited space.

On the Sets of Divergence of Multiple Fourier–Haar Series

On the Sets of Divergence of Multiple Fourier–Haar Series

Multipliers of double Fourier–Haar series

Multipliers of double Fourier–Haar series

On Changes of Variable that Preserve the Absolute Convergence of Fourier–Haar Series of Continuous Functions

On Changes of Variable that Preserve the Absolute Convergence of Fourier–Haar Series of Continuous Functions

On Changes of Variable that Preserve the Absolute Convergence of Fourier-Haar Series of Continuous Functions

Известно, что среди всех дифференцируемых гомеоморфных замен переменной только функции $\varphi_1(x)=x$ и $\varphi_2(x)=1-x$, $x\in[0,1]$, сохраняют абсолютную сходимость всюду рядов Фурье-Хаара. Устанавливается, что класс всех дифференцируемых гомеоморфных замен переменной, сохраняющих абсолютную сходимость всюду, не станет шире, если ограничиться непрерывными внешними функциями. Библиография: 8 названий.

On the Absolute Convergence of Fourier–Haar Series in the Metric of Lp(0, 1), 0 < p < 1

It is proved that for every ∈ > 0 there exists a measurable set E ⊂ [0, 1] with measure |E| > 1 − ∈ such that for every function f(x) ∈ L[0, 1] one can find a function g(x) ∈ L[0, 1] coinciding with f(x) on E such that its Fourier–Haar series absolutely converges in the metric of Lp(0, 1), 0 < p < 1.

On changes of variable that preserve convergence and absolute convergence of Fourier-Haar series

It is established that among all the differentiable homeomorphic changes of variable only the functions and for preserve convergence everywhere of the Fourier-Haar series. The same is true for absolute convergence everywhere. Bibliography: 8 titles.

On changes of variable that preserve convergence and absolute convergence of Fourier-Haar series

Устанавливается, что среди всех дифференцируемых гомеоморфных замен переменной только функции $\varphi_{1}(x)=x$ и $\varphi_{2}(x)=1-x$, $x\in [0,1]$, сохраняют сходимость всюду рядов Фурье-Хаара. Показывается, что аналогичный результат имеет место также для абсолютной сходимости. Библиография: 8 названий.

Fourier series and Fourier–Haar series for stochastic measures

Fourier series and Fourier–Haar series for stochastic measures

On Haar series of A–integrable functions

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.

AN EXAMPLE OF DOUBLE FOURIER–HAAR SERIES WITH A NONREGULAR SUBSERIES

We bring an example of integrable function on $[0;1]^2$, so that the double Fourier–Haar series has a subseries, whose majorant of partial sums does not belong to $ L^{1, \infty}$.

Integration of Banach-valued functions and Haar series with Banach-valued coefficients

It is proved that for any Banach space each everywhere convergent Haar series with coefficients from this space is the Fourier–Haar series in the sense of Henstock type integral with respect to a dyadic differential basis. At the same time, the almost everywhere convergence of a Fourier–Henstock–Haar series of a Banach-space-valued function essentially depends on properties of the space.

The convergence of double Fourier-Haar series over homothetic copies of sets

The paper is concerned with the convergence of double Fourier- Haar series with partial sums taken over homothetic copies of a given bounded set containing the intersection of some neighbourhood of the origin with . It is proved that for a set from a fairly broad class (in particular, for convex ) there are two alternatives: either the Fourier-Haar series of an arbitrary function converges almost everywhere or is the best integral class in which the double Fourier-Haar series converges almost everywhere. Furthermore, a characteristic property is obtained, which distinguishes which of the two alternatives is realized for a given . Bibliography: 12 titles.

Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by finite Fourier-Haar series

The paper deals with the space consisting of classes of real measurable functions on with finite integral . If , then the space can be made into a Banach space with the norm . The inequality , which is an analogue of the first Jackson theorem, is shown to hold for the finite Fourier-Haar series , provided that the variable exponent satisfies the condition . Here, is the modulus of continuity in defined in terms of Steklov functions. If the function lies in the Sobolev space with variable exponent , it is shown that . Methods for estimating the deviation for at a given point are also examined. The value of is calculated in the case when , where .Bibliography: 17 titles.

Absolute convergence of multiple Fourier–Haar series

Abstract. The class of functions of s variables H α / s ( s ) $H_{\alpha /s}^{(s)}$ is defined. For the functions of this class, Sász's theorem for s-dimensional Fourier–Haar series is proved. When α = 1 $\alpha =1$ , it is shown that the obtained theorem is best possible for general complete orthonormal systems.

On the convergence of multiple Haar series

We prove that the rectangular and spherical partial sums of the multiple Fourier–Haar series of an individual summable function may behave differently at almost every point, although it is known that they behave in the same way from the point of view of almost-everywhere convergence in the scale of integral classes: is the best class in both cases. We also find optimal additional conditions under which the spherical convergence of a multiple Fourier–Haar series (general Haar series, lacunary series) follows from its convergence by rectangles, and prove that these conditions are indeed optimal.

On partial sums of Fourier-Haar series with arbitrary dilation factor

In this paper, we use the following notation: N and R, respectively, are the sets of natural and real numbers; 〈A,B〉 is the inner product in Rd; C is the space of continuous 1-periodic functions; and ω(f, t) is the modulus of continuity of a function f . Consider the classical Haar system [1] with natural numbering. For any continuous 1-periodic function, its Fourier series with respect to a given system converges uniformly. The following estimate of the deviation of partial sums is well known:

On the divergence of multiple Fourier-Haar series

It is proved that for any dimension n ≥ 2, L(ln+ L) n−1 is the widest integral class in which the almost everywhere convergence of spherical partial sums of multiple Fourier-Haar series is provided. Moreover,it is shown that the divergence effects of rectangular and spherical general terms of multiple Fourier-Haar series can be achieved simultaneously on a set of full measure by an appropriate rearrangement of values of arbitrary summable function f not belonging to L(ln+ L) n−1.