One-sided Approximation Research Articles

Best one sided approximation of unbounded functions trigonometrical polynomials for one dimensional weighted-space

The aim of studying this research is to find the best one sided trigonometrical approximation of unbounded function of one variable in weighted space. Moreover we estimated the degree of the best one-sided trigonometrical approximation by averaged modulus of smoothness in same space.

Optimal Hermite-Fejér interpolation of algebraic polynomials and the best one-sided approximation on the interval [−1,1

In this paper, we give the relations of the optimal Hermite-Fejér interpolation and the best one-sided approximation to the smooth function classes BC2r,+, r∈N in weighted space L1,ω([−1,1]), with a positive, continuous and integrable weight function ω on (−1,1). We proved that the Hermite-Fejér interpolation based on the set of the zeros of some orthogonal polynomials is optimal respectively in the space L1,ω([−1,1]) and the space L∞([−1,1]) and gave the exact constants of the approximation errors of these Hermite-Fejér interpolation. We also obtained the exact constant of the approximation errors of the optimal Hermite-Fejér interpolation when the endpoints of the interval [−1,1] are included in the considered sets of the nodes.

Best One-Sided Multiplier Approximation by Operators

The aim of studying this research is to find the best one-sided multiplier approximation of unbounded function in 퐿푝,휓푛(푋)−space,푋=[0,1],푝≥1by using type of operators 하푛(푓),픾풏(푓)by means of operators of algebraic polynomials as well as to show the relationship between the multiplier averaged modules of smoothness (휏-modules) and variation of unbounded functions in퐿푝,휓푛(푋)−spac

Zipper Fractal Functions with Variable Scalings

Zipper fractal interpolation function (ZFIF) is a generalization of fractal interpolation function through an
 improved version of iterated function system by using a binary parameter called a signature. The signature
 allows the horizontal scalings to be negative. ZFIFs have a complex geometric structure, and they can
 be non-differentiable on a dense subset of an interval I. In this paper, we construct k-times continuously
 differentiable ZFIFs with variable scaling functions on I. Some properties like the positivity, monotonicity,
 and convexity of a zipper fractal function and the one-sided approximation for a continuous function by a
 zipper fractal function are studied. The existence of Schauder basis of zipper fractal functions for the space
 of k-times continuously differentiable functions and the space of p-integrable functions for p ∈ [1,∞) are
 studied. We introduce the zipper versions of full Müntz theorem for continuous function and p-integrable
 functions on I for p ∈ [1,∞).

Exact Constants of the Best One Sided Approximations of the Sum Analytic Functions from Different Classes

The task of obtaining the exact values of the best approxima-tions by trigonometric polynomials of continuous or summable functions originates from the works of P. L. Chebyshev, who, back in the 1950s, posed the problem of finding the polynomial, that de-viates the least from a given continuous function. Subsequently this direction in the theory of approximation got further development thanks to the works of K. Weierstrass, D. Jackson, S. N. Bernstein, Valle-Poussin and others. On this time there is an increase attention to problems of one-sided approximation of individual functions and their classes in the metric space L. Problems of this content arise up in number theory, coding theory, and other areas of math-ematics. The first results of this direction were obtained in the 1880th by A.A. Markov and T.Y. Stieltijes. In the future, these studies were continued in the works of J. Karamata (1930), G.Freud and T. Hanelius (mid-20th century). General issues related to the problem of the best approximation of classes of functions by trigonometric polynomials: the existence of the best approximation polynomial, its characteristic properties, one sided approximations are detailed in many works, in particular, for example, in the book of M. P. Korneychuk [1], the works of T.Ganelius [4] , V.G. Doronin, A.A. Ligun [5].The exact constants of the best one sided approximation of the sum ofthe majorant functions of the classes that allow analytical extension into a strip of fixed width and of functions harmonic in a circle of radius 1 have been found in this work.

Local Compatibility Boundary Conditions for High-Order Accurate Finite-Difference Approximations of PDEs

We describe a new approach to derive numerical approximations of boundary conditions for high-order accurate finite-difference approximations. The approach, called the Local Compatibility Boundary Condition (LCBC) method, uses boundary conditions and compatibility boundary conditions derived from the governing equations, as well as interior and boundary grid values, to construct a local polynomial, whose degree matches the order of accuracy of the interior scheme, centered at each boundary point. The local polynomial is then used to derive a discrete formula for each ghost point in terms of the data. This approach leads to centered approximations that are generally more accurate and stable than one-sided approximations. Moreover, the stencil approximations are local since they do not couple to neighboring ghost-point values which can occur with traditional compatibility conditions. The local polynomial is derived using continuous operators and derivatives which enables the automatic construction of stencil approximations at different orders of accuracy. The LCBC method is developed here for problems governed by second-order partial differential equations, and it is verified for a wide range of sample problems, both time-dependent and time-independent, in two space dimensions and for schemes up to sixth-order accuracy.

Contracting differential equations in weighted Banach spaces

Geodesic contraction in vector-valued differential equations is readily verified by linearized operators which are uniformly negative-definite in the Riemannian metric. In the infinite-dimensional setting, however, such analysis is generally restricted to norm-contracting systems. We develop a generalization of geodesic contraction rates to Banach spaces using a smoothly-weighted semi-inner product structure on tangent spaces. We show that negative contraction rates in bijectively weighted spaces imply asymptotic norm-contraction, and apply recent results on asymptotic contractions in Banach spaces to establish the existence of fixed points. We show that contraction in surjectively weighted spaces verifies non-equilibrium asymptotic properties, such as convergence to finite- and infinite-dimensional subspaces, submanifolds, limit cycles, and phase-locking phenomena. We use contraction rates in weighted Sobolev spaces to establish existence and continuous data dependence in nonlinear PDEs, and pose a method for constructing weak solutions using vanishing one-sided Lipschitz approximations. We discuss applications to control and order reduction of PDEs.

FRACTAL DIMENSION OF MULTIVARIATE α-FRACTAL FUNCTIONS AND APPROXIMATION ASPECTS

In this paper, we explore the concept of dimension preserving approximation of continuous multivariate functions defined on the domain [Formula: see text] (q-times) where [Formula: see text] is a natural number). We establish a few well-known multivariate constrained approximation results in terms of dimension preserving approximants. In particular, we indicate the construction of multivariate dimension preserving approximants using the concept of [Formula: see text]-fractal interpolation functions. We also prove the existence of one-sided approximation of multivariate function using fractal functions. Moreover, we provide an upper bound for the fractal dimension of the graph of the [Formula: see text]-fractal function. Further, we study the approximation aspects of [Formula: see text]-fractal functions and establish the existence of the Schauder basis consisting of multivariate fractal functions for the space of all real valued continuous functions defined on [Formula: see text] and prove the existence of multivariate fractal polynomials for the approximation.

One-Sided Multiplier Approximation of Unbounded Functions

The main objective of this article is to study the degree of best one-sided multiplier approximation of unbounded functions ,( ),[ 1,1]npgLY Y= −by means of the average modulus of smoothness by using sequences of algebraic polynomials Pn of degree less than n, nr+ 1, also in this search we shall prove a direct theorem by sequences Pn and some results.

Non-symmetric approximations of functional classes by splines on the real line

Let $S_{h,m}$, $h>0$, $m\in {\mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $k\in {\mathbb Z}$.
 We obtain exact values of the best $(\alpha, \beta)$-approximations by spaces $S_{h,m}\cap L_1({\mathbb R})$ in the space $L_1({\mathbb R})$ for the classes $W^r_{1,1}({\mathbb R})$, $r\in {\mathbb N}$, of functions, defined on the whole real line, integrable on ${\mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({\mathbb R})$.
 These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({\mathbb R})$ by splines from $S_{h,m}\cap L_1({\mathbb R})$ (case $\alpha=\beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({\mathbb T})$ of $2\pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({\mathbb T})$ by periodic polynomial splines of minimal deficiency.
 As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({\mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({\mathbb T})$.

Approximation properties of periodic multivariate quasi-interpolation operators

We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions φ˜j and trigonometric polynomials φj. The class of such operators includes classical interpolation polynomials (φ˜j is the Dirac delta function), Kantorovich-type operators (φ˜j is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on φ˜j and φj, we obtain upper and lower bound estimates for the Lp-error of approximation by quasi-interpolation operators in terms of the best and best one-sided approximation, classical and fractional moduli of smoothness, K-functionals, and other terms.

One-sided approximation of some classes of singular integrals with regard to point position on the interval

We obtain asymptotically exact estimates of approximation of functions from some classes of singular integrals by algebraic polynomials with regard to point position on the interval.

The best one-sided approximation of classes $W^r H^{\omega}$ with regard to point location on the interval

The estimation of the best one-sided pointwise approximation to the classes $W^r H^{\omega}$ with the remainder better than one of the previous estimation is obtained.

The best one-side approximation of classes of integrals of fractional order with regard to position of point on interval

We obtain the asymptotic estimations for the best one-sided point-wise approximation to the classes $W_{\infty}^r$, $r > 1$ (in case of fractional $r$) by algebraic polynomials.

A High-Efficient Finite Difference Method for Flexible Manipulator with Boundary Feedback Control

The paper presents a high-efficient finite difference method for solving the PDE model of the single-link flexible manipulator system with boundary feedback control. Firstly, an abstract state-space model of the manipulator is derived from the original PDE model and the associated boundary conditions of the manipulator by using the velocity and bending curvature of the flexible link as the state variables. Then, the second-order implicit Crank-Nicolson scheme is adopted to discretize the state-space equation, and the second-order one-sided approximation is used to discretize the boundary conditions with excitations and feedback control. At last, the state-space equation combined with the boundary conditions of the flexible manipulator is transformed to a system of linear algebraic equations, from which the response of the flexible manipulator can be easily solved. Numerical simulations are carried out to simulate the manipulator under various excitations and boundary feedback control. The results are compared with ANSYS to demonstrate the accuracy and high efficiency of the presented method.

H-MAGMA, inheriting a shaky statistical foundation, yields excess false positives.

All data used in the manuscript are available at https://github.com/ryurko/HMAGMA-comment. Supplementary Fig. 1. Comparison of covariance approximations for two-sided tests (green-solid) versus Brown's one-sided approximation described in MAGMA paper (red-dashed) and the code implemented in MAGMA with r2 (cyan-dashed). Supplementary Fig. 2. Comparison of the type 1 error rate control, plus/minus two standard errors, at target a = 0:05 (denoted by dashed red line) by gene-level method for seven example genes of different sizes. Supplementary Fig. 3. Comparison of multiple testing family-wise error rate (FWER), plus/minus two standard errors, at target a = 0:05 (denoted with dashed red line) by gene-level method. Supplementary Fig. 4. a, Comparison of the proportion of simulations with zero to six false positives for each method, excluding MAGMA: paper, using the Bonferroni correction at target a = 0:05 (plus/minus two standard errors). Supplementary Fig. 5. Comparison of gene-set analysis type 1 error, plus/minus two-standard errors, at target a = 0:05 (denoted with dashed red line) by gene-level method for each considered gene-set analysis method (from left to right) MAGMA: gene-set, Fisher's combination test, and Stouffer's z-test. Supplementary Table 1. Comparison of number of BH discoveries (union across adult and fetal Hi-C data annotations) with H-MAGMA versus corrected Monte Carlo based approach. Supplementary Fig. 6. Comparison of null histograms by gene-set analysis method (columns) and methods for computing gene-level p-values (rows) for gene-sets of size G = 45 genes. Red dashed lines indicate expected 2.5% and 97.5% quantiles for uniform p-values across 200,000 simulations. Supplementary Fig. 7. Comparison of distributions for MAGMA gene-set analysis standardized test statistic by methods for computing gene-level p-values. Red density curve displays the expected t(G 1) distribution, with the red vertical dashed line denoting the center at zero. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

The pointwise estimation of the one-sided approximation of the class $\breve{W}_\infty^r,\; 0 < r < 1$

The pointwise estimation of the one-sided approximation of the class $\breve{W}_\infty^r,\; 0<r<1$, is established.

Best One-Sided Approximation in the Mean of the Characteristic Function of an Interval by Algebraic Polynomials

Let υ be a weight on (−1, 1), i.e., a measurable integrable nonnegative function nonzero almost everywhere on (−1, 1). Denote by Lυ(−1, 1) the space of real-valued functions f integrable with weight υ on (−1, 1) with the norm $$f = \int\limits_{ - 1}^1 {f(x)v(x)dx.} $$. We consider the problems of the best one-sided approximation (from below and from above) in the space Lυ(−1, 1) to the characteristic function of an interval (a, b), −1 < a < b < 1, by the set of algebraic polynomials of degree not exceeding a given number. We solve the problems in the case where a and b are nodes of a positive quadrature formula under some conditions on the degree of its precision as well as in the case of a symmetric interval (−h, h), 0 < h < 1, for an even weight υ.

On the nonsymmetric approximation of continuous functions in the integral metric

In the paper, an exact estimate of the best nonsymmetric approximation in the integral metric by the constants of continuous functions that belong to the classes $H^\omega[a,b]$ is proved. Taking into account Babenko's theorem on the connection of nonsymmetric approximation with the usual best approximation in the integral metric and the best one-sided approximations, from the proved result we obtain the exact estimate for the usual best approximation obtained by N.P. Korneichuk, and the exact estimate for the best one-sided approximation obtained by V.G. Doronin and A.A. Ligun.

Error-functions in double-sided Taylor's approximations

In this paper we introduce the error-functions for one-sided and double-sided Taylor's approximations of real analytic functions. We illustrate the application of error-functions in the process of generalization of one trigonometric inequality.