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In the present paper, we establish a theorem on best approximation of a function g 2 Bλ q (Lr) of its Fourier series. Our main theorem generalizes some known results of this direction of work. Thus, the results of [10], [26] and [27] become the particular case of our main Theorem 3.1. [ABSTRACT FROM AUTHOR]

We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √n lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk [1993], Goldreich and Goldwasser [2000], and Aharonov and Regev [2003]. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier series over the lattice. This technique might be useful elsewhere -- we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev [2003]. An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in Aharonov and Regev [2003]. This route to proving purely classical results might be beneficial elsewhere. [ABSTRACT FROM AUTHOR]

The problem of obtaining degree of approximation for the 2π-periodic functions in the weighted Lipschitz class W(Lp; ξ(t)) (p ≥ 1) by Riesz means of the Fourier series have been studied by various investigators under Lp-norm. Recently, Deepmala and Piscoran [Approximation of signals(functions) belonging to certain Lipschitz classes by almost Riesz means of its Fourier series, J. Inequal. Appl., (2016), 2016:163. DOI 10.1186/s13660-016-1101-5] obtained a result on degree of approximation for weighted Lipschitz class by Riesz means. In this note, we extend this study to the weighted Lp-norm which in turn improves some of the previous results. We also derive some corollaries from our result. [ABSTRACT FROM AUTHOR]

Fourier extension is an approximation scheme in which a function on an arbitrary bounded domain is approximated using classical Fourier series on a bounding box. On the domain of interest the Fourier series exhibits redundancy, and it has the mathematical structure of a frame rather than a basis. It is not trivial to construct approximations in this frame using function evaluations in points that belong to the domain only, but one way to do so is through a discrete least squares approximation. The corresponding system is extremely ill-conditioned, due to the redundancy in the frame, yet its solution via a regularized singular value decomposition is known to be accurate to very high (and nearly spectral) precision. Still, this computation requires Q (N3) operations. In this paper we describe an algorithm to compute such Fourier extension frame approximations in only Q (N2 log2 N) operations for general 2D domains. The cost improves to Q (N log2 N) operations for simpler tensor-product domains. The algorithm exploits a phenomenon called the plunge region in the analysis of time-frequency localization operators, which manifests itself here as a sudden drop in the singular values of the least squares matrix. It is known that the size of the plunge region scales like Q (logN) in 1D problems. In this paper we show that for most 2D domains in the fully discrete case the plunge region scales like Q (N logN), proving a discrete equivalent of a result that was conjectured by Widom for a related continuous problem. The complexity estimate depends on the Minkowski or box-counting dimension of the domain boundary, and as such it is larger than Q (N logN) for domains with fractal shape. [ABSTRACT FROM AUTHOR]

The problem of approximation of electrocardiograms has a great practical importance and it is constantly in the field of research. Solution of this problem allows us to automate and computerize the process of diagnosis and to detect timely deviations from normal characteristics of cardiograms of heartbeat in the early stages of appearance and development of a disease. There are different methods for approximation of cardiograms. All these methods have their drawbacks. The article deals with new methods of approximation of cardiograms, which do not have any disadvantages of the known methods. The developed mathematical models are based on new methods of approximation of step functions, and these methods can significantly reduce errors of the approximation of cardiograms. In this paper new methods of approximation of step functions with estimation of errors of the approximation are supposed. The proposed methods do not have any drawbacks of traditional expansions of step functions in Fourier series and can be used in problems of mathematical modeling of a wide class of processes and systems, in particular, for solution of medical and biological problems. [ABSTRACT FROM AUTHOR]

In this work the approximation of functions by linear means of Fourier series in reflexive weighted Orlicz spaces with Muckenhoupt weights is studied. This result is applied to the approximation of functions by linear means of Faber series in weighted Smirnov-Orlicz classes defined on simply connected domain of the complex plane. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the least upper bounds of the approximations of functions from the classes $$ {W}_{\beta}^r{H}^{\alpha } $$ by Weierstrass integrals in the uniform metric. [ABSTRACT FROM AUTHOR]

Degree of approximation of functions of different classes has been studied by several researchers by different summability methods. In the proposed paper, we have established a new theorem for the approximation of a signal (function) belonging to the W(Lr, ξ(t))-class by (..., pn, qn)(E, s)-product summability means of a Fourier series. The result obtained here, generalizes several known theorems. [ABSTRACT FROM AUTHOR]

In this paper, we have proved the degree of approximation of functions belonging to L[0, ∞) by Harmonic-Euler means of its Fourier-Laguerre series at x = 0. The aim of this paper is to concentrate on the approximation properties of the functions in L[0, ∞) by Harmonic-Euler means of its Fourier-Laguerre series associated with the function f. [ABSTRACT FROM AUTHOR]

The results corresponding to some theorems of S. Lal [Appl. Math. Comput. 209 (2009), 346-350] and the results of the authors [In: Banach Center Publ. 95, Polish Acad. Sci. Inst. Math., Warsaw, 2011, pp. 339-351] are shown. The same degrees of pointwise approximation as in mentioned papers by weaker assumptions on considered functions and examined summability methods are obtained. From presented pointwise results the estimation on norm approximation are derived. Some special cases as corollaries are also formulated. [ABSTRACT FROM AUTHOR]

An approximation of functions in Müntz spaces M Λ , L 1 of L 1 functions is studied with the help of Fourier series. Müntz spaces are considered with the Müntz condition and the gap condition imposed. It is proved that up to an isomorphism such spaces are contained in Weil–Nagy's class. Moreover, existence of Schauder bases in Müntz spaces M Λ , L 1 is investigated. [ABSTRACT FROM AUTHOR]

The aim of the paper is to determine the degree of approximation of functions by matrix means of their Fourier series in a new space of functions introduced by Das, Nath, and Ray. In particular, we extend some results of Leindler and some other results by weakening the monotonicity conditions in the results obtained by Singh and Sonker for some classes of numerical sequences introduced by Mohapatra and Szal and present new results by using matrix means. [ABSTRACT FROM AUTHOR]

In application, one often expands the functions f on [0, 1]² into Fourier sine series and uses few Fourier sine coe±cients to reconstruct f. In this paper, we give a decomposition formula of Fourier sine coe±cients. Based on it, we discuss hyperbolic cross approximations of Fourier sine series and Fourier sine expansion with simple polynomial factors. In the end of this paper, we consider the three-dimensional case. [ABSTRACT FROM AUTHOR]

We prove the results generalized and improved of some theorems of X. Z. Krasniqi [] on pointwise approximation of integrable functions and their conjugate by matrix means of Fourier series and its conjugate. [ABSTRACT FROM AUTHOR]

The paper is primarily devoted to the problem of smooth interpolation of data in 1D. In addition to the exact interpolation of data at nodes, we are also concerned with the smoothness of the interpolating curve and its derivatives. The interpolating curve is defined as the solution of a variational problem with constraints. The system of functions exp ( - i kx ) , k being an integer, is taken for the basis of the space where we measure the smoothness of the result. It also generates the functions used for the interpolation itself. Choosing different norms when measuring the smoothness, we arrive at different interpolating functions. We also mention the problem of smooth curve fitting (data smoothing). We discuss the proper choice of different norms for this way of approximation and present the results of several 1D numerical examples that show the advantages and drawbacks of smooth interpolation. [ABSTRACT FROM AUTHOR]

The equivalence of the error of approximation by Fourier means and of general moduli of smoothness, provided that their generators are equivalent, is established. [ABSTRACT FROM AUTHOR]

We compute the values of the best approximations for the Cauchy kernel on the real axis ℝ by some subspaces from L(ℝ) . This result is applied to the evaluation of the sharp upper bounds for pointwise deviations of certain interpolation operators with interpolation nodes in the upper half plane and certain linear means of the Fourier series in the Takenaka-Malmquist system from the functions lying in the unit ball of the Hardy space H , 2 ≤ p < ∞. [ABSTRACT FROM AUTHOR]

Questions dealing with the approximation of functions from the classes C H by Poisson integrals are studied. The Kolmogorov-Nikol'skii problem for Poisson integrals for the classes C H is solved in the uniform metric. [ABSTRACT FROM AUTHOR]

We study the Walsh series expansion of multivariate functions in Lp (1 ≤ p ≤ ∞) and, in particular, in Lip(α, p). The rate of uniform approximation by T-transformation of rectangular partial sums of double Walsh to these functions is investigated. By extending the concepts of rest (head) bounded variation series, which was introduced by Leindler (2004), we generalize the related results of Móricz and Rhoades (1996), Nagy (2012). Our results can be applied to many summability methods, including the Norlund summability and weighted summability. [ABSTRACT FROM AUTHOR]

A new estimate for the degree of approximation of a function ... ∊ Hω class by (Np · E¹) means of its Fourier series has been determined. Here, we extend the results of Singh and Mahajan (2008) which in turn generalize the result of Lal and Yadav (2001). Some corollaries have also been deduced from our main theorem. [ABSTRACT FROM AUTHOR]

Sharp Jackson-Stechkin type inequalities in which the modulus of continuity of mth order of functions is defined via the Steklov function are obtained. For the classes of functions defined by these moduli of continuity, exact values of various n-widths are derived. [ABSTRACT FROM AUTHOR]

In this paper, new best approximations of the function f E Zr(w), (r = 1) class by Matrix-Euler means (Δ.E1) of its Fourier Series have been determined. [ABSTRACT FROM AUTHOR]

We generalize a theorem of Leindler while the series is under the condition of MVBVS ∗, and investigate the relation between the best approximation of functions and its Fourier coefficients under the given $L_{2\pi}^{p}$ norm. In the second theorem, we add a new condition to get the same result as in Theorem B. [ABSTRACT FROM AUTHOR]

In the present paper, two quite new reults on the degree of approximation of a function f belonging to the class Lip(α, r), 1 ≤ r < ∞ and the weighted class W(Lr, ξ(t)), 1 ≤ r < ⊂ by (C, 2)(E, 1) product operators have been obtained. The results obtained in the present paper generalize various known results on single operators. [ABSTRACT FROM AUTHOR]

The pointwise estimates of the deviation T f(·)− f(·) in terms of moduli of continuity −w·f and w·f there are proved. Analogical results on norm approximation with remarks and corollaries are also given. In the results there are used the essentially weaker conditions than these in [Mittal, M. L.: J. Math. Anal. Appl., 220, 434-450 (1998) Theorem 1, p. 437]. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper Lal and Yadov obtained a theorem on the degree of approximation for a function belonging to the Lipschitz class Lipα using the product of the Cesàro and Euler means of order one of its Fourier series. In this paper we extend this result to any regular Hausdorff matrix for the same class of functions. [Copyright &y& Elsevier]

We consider the problem of determining a pair of functions (u, f) satisfying the two-dimensional backward heat equation [image omitted] with a homogeneous Cauchy boundary condition, where ϕ and g are given approximately. The problem is severely ill-posed. Using an interpolation method and the truncated Fourier series, we construct a regularized solution for the source term f and provide Holder-type error estimates in both L2 and H1 norms. Numerical experiments are provided. [ABSTRACT FROM AUTHOR]

Modified Fourier series have recently been introduced as an adjustment of classical Fourier series for the approximation of nonperiodic functions defined on $ d$ In this paper we extend Eckhoff's method to the convergence acceleration of multivariate modified Fourier series. By suitable augmentation of the approximation basis we demonstrate how to increase the convergence rate to an arbitrary algebraic order. Moreover, we illustrate how numerical stability of the method can be improved by utilising appropriate auxiliary functions. In the univariate setting it is known that Eckhoff's method exhibits an auto-correction phenomenon. We extend this result to the multivariate case. Finally, we demonstrate how a significant reduction in the number of approximation coefficients can be achieved by using a hyperbolic cross index set. [ABSTRACT FROM AUTHOR]

We study the nonlinear problem of mean-square approximation of a real finite nonnegative continuous function of two variables by the modulus of a double Fourier integral depending on two parameters. The solution of this problem is reduced to the solution of a nonlinear two-dimensional integral equation of the Hammerstein type. Numerical algorithms for determination of branching lines and branched solutions of equation are constructed and substantiated. Some numerical examples are given. [ABSTRACT FROM AUTHOR]

A regularization method is proposed for the polynomial approximation of a function from its approximated values in a fixed family of nodes. As a regularization parameter we consider the number of nodes. We present explicit expressions for the optimal number of nodes in terms of the original error of the approximated values of the function. These problems appear frequently in studying inverse problems and when a smoothing technique should be applied to a series of numerical data. We obtain estimation of the approximation error by means of discrete versions of a convolution operators with polynomial kernels, and we observe the differences between the use of positive and non positive kernels. Some numerical examples are provided to illustrate the efficiency and computational performance of the method. They also help us to compare different criteria for the construction of polynomial approximations. [ABSTRACT FROM AUTHOR]

Recently, new polynomial approximation formulas were proposed for the reconstruction of compactly supported piecewise smooth functions from Fourier data. Formulas for zero and first degree polynomials were presented. For higher degree approximations, polynomial formulas become extremely complicated to be handled. In this paper we solve this problem by introducing spline approximations. The new approach can be used in the same way as the polynomial one but producing computable formulas for any degree of approximation in Fourier reconstruction. We present general error estimates and numerical It experiments. [ABSTRACT FROM AUTHOR]

We give direct and inverse approximation theorems for Dirichlet series in Smirnov spaces over convex polygons. We estimate the degree of convergence and the regularity of the functions with moduli of arbitrary order k. Moreover, we consider the influence of differentiability conditions on the rate of pproximation and vice versa. This work extends results by Yu. I. Mel'nik and gives an example on the improvement by our results. [ABSTRACT FROM AUTHOR]

An endoprimeR-moduleMis a nonzero module whose nonzero fully invariant submodules are faithful over End(MR). Such modules have prime endomorphism rings. The class of endoprime modules properly contains the class of prime modules in the sense of Zelmanowitz [5], and the class of prime modules with condition (*) of Wisbauer [4]. While over commutative rings, endoprime modules are prime, in general, prime and endoprime are different conditions. A nonzero rightR-moduleMis a direct sum of isomorphic simple modules if and only if each nonzero element ofσ [M] is an endoprime module. Various properties of endoprime modules including their Morita invariance, are obtained, and the structure of endomorphism rings of certain endoprime modules are determined. [ABSTRACT FROM AUTHOR]

We obtain asymptotic equalities for upper bounds of the deviations of operators generated by ?-methods (defined by a collection ?={?s(·)} of functions continuous on [0; 8) and depending on a real parameter s) on classes of (?, ß)-differentiable functions defined on the real axis. [ABSTRACT FROM AUTHOR]

Methods of the theory of functions and functional [1-3] are used widely for solving the different problems of digital signal processing. Among the classical approximation tools (algebraic and trigonometric polynomials, splines), the non-classical approximation methods based on the theory of atomic functions (AF) [4-7] should be mentioned. In this work, for the first time AFs are used for solving such spectral problems of approximation theory as summing the Fourier series and approximation by Levitan's polynomials. Examples of numerical realization of the proposed algorithms are presented in connection with solving the problems of an antenna synthesis [8-10]. [ABSTRACT FROM AUTHOR]

In this paper, we state and prove some new inequalities related to the rate of Lp approximation by Cesàro means of the quadratic partial sums of double Vilenkin-Fourier series of functions from Lp. [ABSTRACT FROM AUTHOR]

The article presents information density estimation and Bump-Hunting by the penalized liklihood method. The argument that a sign pattern in the coefficients in the Hermite expansion of a function slows the convergence is ingenious, but there is another reason for the better approximation by Fourier series than for Hermite series that is probably more important.