Approximation of Functions by Riesz-Zygmund, Borel and Abel-Poisson Means in Weighted Smirnov Classes with Variable Exponent

In this paper the approximation of functions by linear means of Fourier series in weighted variable exponent Lebesgue spaces was studied. This result was applied to the approximation of the functions by linear means of Faber series in Smirnov classes with variable exponent defined on simply connected domain of the complex plane.

The approximation properties of various classical methods of linear summation of Fourier series in weighted spaces of Orlicz type with variable exponent are considered. In particular, in terms of approximation by such methods the constructive characterizations for classes of functions whose moduli of smoothness do not exceed some majorant are obtained.

In weighted Orlicz type spaces ${\\mathcal S}_{_{\\scriptstyle \\mathbf\np,\\,\\mu}}$ with a variable summation exponent, the direct and inverse\napproximation theorems are proved in terms of best approximations of functions\nand moduli of smoothness of fractional order. It is shown that the constant\nobtained in the inverse approximation theorem is in a certain sense the best.\nSome applications of the results are also proposed. In particular, the\nconstructive characteristics of functional classes defined by such moduli of\nsmoothness are given. Equivalence between moduli of smoothness and certain\nPeetre $K$-functionals is shown in the spaces ${\\mathcal S}_{_{\\scriptstyle\n\\mathbf p,\\,\\mu}}$.\n

We consider the space formed by -periodic real measurable functions for which the integral exists and is finite, where , , is a -periodic measurable function (a variable exponent). If , then the space can be endowed with the structure of Banach space with the norm 0: \\int_{-\\pi}^{\\pi}\\biggl|\\frac{f(x)}{\\alpha}\\biggr|^{p(x)}\\,dx\\le1\\biggr\\}. \\end{equation*} ?> In the space we distinguish a subspace of Sobolev type. We investigate the approximation properties of the de la Vallée-Poussin means for trigonometric Fourier sums for functions in the space . In particular, we prove that if the variable exponent satisfies the Dini-Lipschitz condition and if , then the de la Vallée-Poussin means with satisfy where is a modulus of continuity of the function defined in terms of the Steklov functions. It is proved that if , , and the Dini-Lipschitz condition holds, then where stands for the best approximation to by trigonometric polynomials of order . Bibliography: 19 titles.

This work deals with basic approximation problems such as direct, inverse and simultaneous theorems of trigonometric approximation of functions of weighted Lebesgue spaces with a variable exponent on weights satisfying a variable Muckenhoupt A p(·) type condition. Several applications of these results help us transfer the approximation results for weighted variable Smirnov spaces of functions defined on sufficiently smooth finite domains of complex plane ℂ.

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.

In the present work we investigate the approximation of the functions by the Zygmund means in variable exponent Lebesgue spaces. Here the estimate which is obtained depends on sequence of the best approximation in Lebesgue spaces with variable exponent. Also, these results were applied to estimates of approximations of Zygmund sums in Smirnov classes with variable exponent defined on simply connected domains of the complex plane.

In this article various types of weighted variable exponent Hardy and Smirnov classes of analytic functions in simple and doubly connected domains are introduced and studied. In particular, a wide class of those domains are revealed in which the functions from the above-mentioned classes are representable by the Cauchy-type integrals with densities of weighted variable exponent Lebesgue spaces. On the basis of these results, a solution of the Dirichlet problem in explicit form in a ring for harmonic functions, real parts of the functions of variable exponent Smirnov classes is given.

The Riemann–Hilbert problem Re [a(t)Φ+(t)] = b(t) is studied in Smirnov classes with variable exponent in a domain D. Both simply and doubly connected domains with piecewise smooth boundary Γ are considered under the assumption that Γ consists of simple smooth arcs A k A k+1, the tangents of which at the points A k form angles πν k , 0 ≤ ν k ≤ 2, k = 1, . . . , n, A n+1 = A 1 . Various conditions (necessary, necessary and sufficient) are found for the problem to be solvable and Noetherian. In some cases, the index is calculated. In the case of solvability, the solutions in a simply connected domain are constructed in an explicit form. Special attention is given to the Dirichlet problem, i.e. the case a(t) ≡ 1. Bibliography: 21 titles.

In this work, the inverse problem of approximation theory in the variable exponent Smirnov classes of analytic functions, defined on the Jordan domains with a Dini-smooth boundaries, is studied. First, for this purpose, an inverse theorem in the variable exponent Lebesgue spaces of periodic functions is obtained. Later, using the special linear operators, this inverse theorem to the variable exponent Smirnov classes of analytic functions is moved.

Let be a multiply connected domain with boundary , where , , are simple closed rectifiable curves such that lie outside one another, but all of them lie inside . The paper introduces the Smirnov classes with variable exponent , where , , are given positive measurable functions on . The properties of functions from these classes are established, in particular: an expansion theorem, representability by a Cauchy integral, and generalizations of Smirnov’s and Tumarkin’s theorems related to simply connected domains for multiply connected domains. Also, the question of belonging of Cauchy-type integrals with a density from to the class is investigated.

We found order estimates for the upper bounds of the deviations of analogue of Zygmund?s sums on the classes of (?;?)-differentiable functions in the metrics of generalized Lebesgue spaces with variable exponent.

In Avikainen (2009, On irregular functionals of SDEs and the Euler scheme. Finance Stoch., 13, 381–401) the author showed that, for any $p,q \in [1,\infty )$, and any function $f$ of bounded variation in $\mathbb{R}$, it holds that $ \mathbb{E}[|f(X)-f(\widehat{X})|^{q}] \leq C(p,q) \mathbb{E}[|X-\widehat{X}|^{p}]^{\frac{1}{p+1}} $, where $X$ is a one-dimensional random variable with a bounded density, and $\widehat{X}$ is an arbitrary random variable. In this article we will provide multi-dimensional versions of this estimate for functions of bounded variation in $\mathbb{R}^{d}$, Orlicz–Sobolev spaces, Sobolev spaces with variable exponents and fractional Sobolev spaces. The main idea of our arguments is to use the Hardy–Littlewood maximal estimates and pointwise characterizations of these function spaces. We apply our main results to analyze the numerical approximation for some irregular functionals of the solution of stochastic differential equations.

In generalized Lebesgue spaces with variable exponent, we determine the orders of the best approximations in the classes of (ψ; β)-differentiable 2π-periodic functions, deduce an analog of the well-known Bernstein inequality for the (ψ; β)-derivative, and apply this inequality to prove the inverse theorems of approximation theory in these classes.

The problem of expansion in powers is generalized into decomposition of positive integers in the sequence of degrees of different orders, the con-ditions of decomposition are determined, and the algorithm for decomposi-tion is constructed. The algorithm is based on two procedures: 1) achieve-ment a minimum of residual at each algorithm step; 2) speeding of decom-position through expanding the local base by reducing decomposition in-dex, which ensures finiteness of algorithm. The algorithm has such effi-ciency factors as high rate of decomposition, ease of implementation, availability of different options for the decomposition of numbers as in ex-tended, narrowed, sparse bases, which protects the encoded information from external influences. The algorithm can be used to encode large amounts of digital information under basic systems of small dimensions. Decomposition of positive integers into a sequence of powers is opti-mal and correct. Optimality of decomposition follows from the condition that at each step of algorithm the minimum value of disjunction in the space of mixed parameters x∈N,y∈Ris achieved. Correctness of algo-rithm is due to the fact that when the disjunction is reduced, the algorithm expands the basis of decomposition by reducing the degree indicators by one. By switching from a discrete model to a continuous model by replac-ing the degrees with power functions, we obtain a smooth approximation of the ill-conditioned function in the neighborhood of decomposition. The construction of posinomial polynomials on the basis of smooth polynomi-als is one of the promising directions of integration of ill-conditioned non-differentiable functions and smooth replacement of variables in the catas-trophe theory.Posinomials (functions with a variable exponent) predict the step of splitting the integration interval into parts, since they determine the loga-rithmic rate of change of an arbitrary monotonic function. The method of decomposition of positive integers provides an optimal decomposition into the sum of powers, and therefore the transition from a discrete model to a continuous model in the neighborhood of decomposition by replacing powers with power functions as well as allows to achieve the high accura-cy of approximation.