Optimal Approximation of Periodic Analytic Functions with Integrable Boundary Values

We investigate the two-point in time boundary value problem for the partial differential equations of the second-order with one spatial variable and constant coefficients. The problem is considered in in the spaces of functions which Fourier coefficients are characterized by exponential behavior on the Cartesian product of the time interval and spatial domain $\mathbb{R}/2\pi\mathbb{Z}$. The correct solvability of the problem is established, the formulas for solutions are presented, the kernel is described and the smoothness of the solution is established in the spaces of functions that are periodic in one spatial variable. We have established the conditions which are close to the necessary conditions of solvability of the problem in scale of spaces of functions with exponentially increasing (or decreasing) Fourier coefficients.We also found the asymptotic estimates demonstrating the absence of the problem of small denominators, which arises of many spatial variables and makes the boundary value problem incorrect. We have established sufficient conditions of the finite-dimensionality of the kernel of the problem and found upper bounds for its dimension. The results are obtained under the condition of minimum smoothness on the right-hand sides of two-point conditions, which is close to the necessary condition.

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

Series expansion of the period function and representations of Hecke operators

The purpose of this work is to study remotely almost periodic functions defined on the semi-axis ℝ + := [ 0 , + ∞ ) {\mathbb{R}_{+}:=[0,+\infty)} (respectively, on the real axis ℝ := ( - ∞ , + ∞ ) {\mathbb{R}:=(-\infty,+\infty)} ) with values in the Banach space 𝔅 {\mathfrak{B}} . We establish a relation between remotely almost periodic movements in the shift dynamical system ( C ( 𝕋 , 𝔅 ) , 𝕋 {(C(\mathbb{T},\mathfrak{B}),\mathbb{T}} , σ ) {\sigma)} ( 𝕋 = ℝ + {\mathbb{T}=\mathbb{R}_{+}} or ℝ {\mathbb{R}} ) in the space of continuous functions C ⁢ ( 𝕋 , 𝔅 ) C(\mathbb{T},\mathfrak{B}) and remotely almost periodic functions. Using this connection, we prove some important analytical and topological properties of remotely almost periodic functions.

The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schrodinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Schrodinger equation at the zero energy level is a smooth M-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schrodinger operator on the space of (anti)periodic functions. The second part of the paper is devoted to the construction of such operators with the help of a generalization of the Novikov-Veselov construction.

Extremal solutions and Green's functions of higher order periodic boundary value problems in time scales

Approximation of analytic functions in Korobov spaces

In this work, we propose a method for the derivation of prolate spheroidal wave functions (PSWFs) and Slepian functions on continuous and disjoint intervals on the real number line. The proposed method uses Fourier series to obtain a closed-form approximation for Slepian functions on the real line. With this closed-form expression, Slepian functions can be evaluated at arbitrary points in the region of interest with high accuracy. The conventional method uses properties of the Slepian concentration problem to evaluate PSWFs on finite number of points in an interval. The conventional method is computationally expensive and does not allow for easy storage. By approximating an interval containing regions of interest as periodic, we express the Slepian concentration problem as a finite dimensional problem using the Fourier series domain. Solutions to the Slepian concentration problem in this form are Fourier series coefficients corresponding to the Slepian functions. Reconstruction in Fourier series basis, scaling and subsequent truncation provides the closed-form expression for the Slepian problem. Upon comparison with PSWFs obtained by the conventional method, we find negligible difference.

УДК 517.5 Якщо неперервна на дійсній осі 2 π -періодична функція f змінює свій знак у 2 s , s ∈ ℕ , точках y i : - π ≤ y 2 s < y 2 s - 1 < … < y 1 < π , а для інших i ∈ ℤ точки y i визначаються періодично, то для кожного натурального n , більшого за деяку сталу N ( k , y i ) , що залежить тільки від k ∈ ℕ і min i = 1 , … ,2 s { y i - y i + 1 } , знайдено тригонометричний поліном P n порядку не вищого за n такий, що P n має скрізь той самий знак, що і f , за винятком, можливо, маленьких околів точок y i : ( y i - π / n , y i + π / n ) , і ‖ f - P n ‖ ≤ c ( k , s ) ω k ( f , π / n ) , де c ( k , s ) — стала, що залежить тільки від k і s , ω k ( f , ⋅ ) — модуль гладкості k -го порядку функції f і ‖ ⋅ ‖ — max-норма.

Korovkin type approximation theorems are useful tools to check whether a given sequence ( L n ) n ≥ 1 of positive linear operators on C[0,1] of all continuous functions on the real interval [0,1] is an approximation process. That is, these theorems exhibit a variety of test functions, which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, x and x 2 in the space C[0,1] as well as for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line. In this paper, we use the notion of statistical summability (C,1) to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence.MSC:41A10, 41A25, 41A36, 40A30, 40G15.

The independent increments process, also known as (positive) Levy process, with its associated Levy measure has emerged as an important vehicle in the development of a group of processes such as neutral to the right, gamma, extended gamma, beta, and beta-Stacy processes that are discussed cohesively in this chapter. After a brief review of the Levy process and its generalization the completely random measure and of the Poisson Process which is used in its construction, we present a detail treatment of the neutral to the right process defined on the real line since all other processes discussed here can be classified as of the neutral to the right type. It is shown to have the desirable conjugacy property which facilitates the Bayesian analysis of statistical data even in the case of right censored data which the Dirichlet process fails to do. We devote a sizable effort to the derivation of the posterior distribution under this prior and simulation procedures necessary thereof, since it carries over to other neutral to the right type processes as well. Thereafter we introduce a simple case of Levy process, namely the gamma process, in which the independent increments are assumed to be distributed as gamma distribution, followed by its convolution with a known function, the extended gamma process, to serve as a prior on the space of hazard functions.We discuss next the beta process, which was developed to place a prior on the space of cumulative hazard functions defined on the real line, and also indirectly on the space of distribution functions, along with its important properties. The Levy measure of the beta process when viewed on an arbitrary space is shown to be useful in generating a prior process known as the Indian buffet process, which serves as a prior on the space of sparse binary matrices encountered in featural modeling. Further generalizations of the beta process, such as the stable and kernel beta processes, are also considered in the same context. A special case of the Levy process—the log-beta process, in which the increments are assumed to have a beta-Stacy distribution, is used in developing the beta-Stacy process presented next, and noted that the latter not only generalizes the Dirichlet process but is also conjugate with respect to data which may include right censored observations as well. Finally, we present the derivation of three special processes, namely the Chinese restaurant, the Indian buffet, and infinite gamma-Poisson processes, which have emerged to be very useful in areas outside the statistical field, such as machine learning, information retrieval, and featural modeling. We also discuss in the chapter various properties, special characteristics, essential features and limitations, as well as challenges in applications of the above mentioned processes.

Regular Level Sets of Averages of Nemytskiı Operators Are Contractible

The present work is devoted to the construction of optimal quadrature formulas for the approximate calculation of the integrals ∫02πeiωxφ(x)dx in the Sobolev space H˜2m. Here, H˜2m is the Hilbert space of periodic and complex-valued functions whose m-th generalized derivatives are square-integrable. Here, firstly, in order to obtain an upper bound for the error of the quadrature formula, the norm of the error functional is calculated. For this, the extremal function of the considered quadrature formula is used. By minimizing the norm of the error functional with respect to the coefficients, an optimal quadrature formula is then obtained. Using the explicit form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula is calculated. The convergence of the constructed optimal quadrature formula is investigated, and it is shown that the rate of convergence of the optimal quadrature formula is O(hm) for |ω|<N and O(|ω|−m) for |ω|≥N. Finally, we present numerical results of comparison for absolute errors of the optimal quadrature formula with the exp(iωx) weight in the case m=2 and the Midpoint formula. There, one can see the advantage of the optimal quadrature formulas.

In this study, our main goal is to obtain approximation properties of convolution operators for multivariables via a special method which is not included in any other methods given before, also known as $P$-statistical convergence. We present the $P$-statistical rate of this approximation and provide examples of convolution operators. It is noteworthy to express that one can not approximate $f$ by earlier results for our examples. Therefore, our results fill an important gap in the existing literature. Furthermore, we also present a $P$-statistical approximation result in the space of periodic continuous functions of period $2\pi$ for short $C^*$.

We consider the problem of the best polynomial approximation of 2 -periodic functions in the space L2 in the case where the error of approximation En1 .f / is estimated via the k th-order modulus of continuityk.f / in which the Steklov operator Shf is used instead of the operator of transla- tion Thf.x/D f.xCh/: For the classes of functions defined by using the indicated characteristic of smoothness, we determine the exact values of various n-widths. 1. In the solution of some problems of the approximation theory of functions of real variable, it is cus- tomary to use various modifications of the classic definition of the modulus of continuity (see, e.g., (1-11)). In numerous cases, this is explained by the specific conditions of the analyzed problems and enables one to get the results illustrating the profound nature of these problems. Thus, for the approximation of nonperiodic functions by algebraic polynomials, Potapov and his colleagues proposed various modifications of the classic definition of the modulus of continuity based on the use of different averaging operators instead of the operator of translation T h f.x/Df.xCh/ (see, e.g., (3, 4)). In the case of approximation of 2 -periodic functions, Abilov and Abilova used the Steklov function S h .f / instead of the operator of translation Thf (5). A similar approach was applied by Kokilashvili and Yildirir in (8) and by the authors in (10). In the present paper, we continue our investigations in this direction. Let L2 L2.a0;2c/ be the space of 2 -periodic Lebesgue measurable functions with the norm