Peano Kernel Research Articles

A note on the remainder in the approximation of functions by some positive linear operators

In this note we give representations for the remainder in approximations formulas generated by positive linear operators which preserve some functions including linear functions. We show that in these cases the Peano kernel from the integral representation of the remainder has the same sign on the definition domain. Applications to the iterates of some positive linear operators and quadrature formulas are given.

Some local fractional Maclaurin type inequalities for generalized convex functions and their applications

In this paper, we establish a new local fractional integral identity involving three point by the use of Peano kernel approach. Using this identity we derive some new local fractional integrals inequalities of Maclaurin-type for functions whose local fractional derivatives are generalized convex functions. In order to show the effectiveness of the obtained results, we apply them in numerical integration and to special means.

Collocation boundary value methods for auto-convolution Volterra integral equations

In this paper, we analyze the collocation boundary value solutions of the second-kind auto-convolution Volterra integral equations. In contrast to classical collocation methods, the full discretization of auto-convolution Volterra integral equations via the boundary value technique leads to a nonlinear system. Thus we reconsider the solvability of the proposed scheme and the boundedness of the numerical solution. Moreover, with the help of the Peano kernel theorem, the convergence order is developed, which is demonstrated by several numerical examples.

Ostrowski type inequalities and some selected quadrature formulae

Some selected Ostrowski type inequalities and a connection with numerical integration are studied in this survey paper, which is dedicated to the memory of Professor D. S. Mitrinovic, who left us 25 years ago. His significant influence to the development of the theory of inequalities is briefly given in the first section of this paper. Beside some basic facts on quadrature formulas and an approach for estimating the error term using Ostrowski type inequalities and Peano kernel techniques, we give several examples of selected quadrature formulas and the corresponding inequalities, including the basic Ostrowski's inequality (1938), inequality of Milovanovic and Pecaric (1976) and its modications, inequality of Dragomir, Cerone and Roumeliotis (2000), symmetric inequality of Guessab and Schmeisser (2002) and asymmetric inequality of Franjic (2009), as well as four point symmetric inequalites by Alomari (2012) and a variant with double internal nodes given by Liu and Park (2017).

A generalization of the peano kernel and its applications

Based on the q-Taylor Theorem, we introduce a more general form of the Peano kernel (q-Peano) which is also applicable to non-differentiable functions. Then we show that quantum B-splines are the q-Peano kernels of divided differences. We also give applications to polynomial interpolation and construct examples in which classical remainder theory fails whereas q-Peano kernel works

A Comparison and Error Analysis of Error Bounds

In this paper, we present an error analysis with the help of Ostrowski type inequalities for n-times differentiable mappings by using n-times peano kernel. A comparison is also presented which shows that obtained error bounds are better than the previous error bounds.

Error analysis of the extended Filon-type method for highly oscillatory integrals

We investigate the impact of adding inner nodes for a Filon-type method for highly oscillatory quadrature. The error of Filon-type method is composed of asymptotic and interpolation errors, and the interplay between the two varies for different frequencies. We are particularly concerned with two strategies for the choice of inner nodes: Clenshaw–Curtis points and zeros of an appropriate Jacobi polynomial. Once the frequency omega is large, the asymptotic error dominates, but the situation is altogether different when omega ge 0 is small. In the first regime, our optimal error bounds indicate that Clenshaw–Curtis points are always marginally better, but this is reversed for small omega ; then, Jacobi points enjoy an advantage. The main tool in our analysis is the Peano Kernel theorem. While the main part of the paper addresses integrals without stationary points, we indicate how to extend this work to the case when stationary points are present. Numerical experiments are provided to illustrate theoretical analysis.

Inequalities between remainders of quadratures

It is well-known that in the class of convex functions the (nonnegative) remainder of the Midpoint Rule of approximate integration is majorized by the remainder of the Trapezoid Rule. Hence the approximation of the integral of a convex function by the Midpoint Rule is better than the analogous approximation by the Trapezoid Rule. Following this fact we examine remainders of certain quadratures in classes of convex functions of higher orders. Our main results state that for 3-convex (5-convex, respectively) functions the remainder of the 2-point (3-point, respectively) Gauss–Legendre quadrature is non-negative and it is not greater than the remainder of Simpson’s Rule (4-point Lobatto quadrature, respectively). We also check 2-point Radau quadratures for 2-convex functions to demonstrate that similar results fail to hold for convex functions of even orders. We apply the Peano Kernel Theorem as a main tool of our considerations.

On Interpolation Approximation: Convergence Rates for Polynomial Interpolation for Functions of Limited Regularity

The convergence rates on polynomial interpolation in most cases are estimated by Lebesgue constants. These estimates may be overestimated for some special points of sets for functions of limited regularities. In this paper, by applying the Peano kernel theorem and Wainerman's lemma, new formulas on the convergence rates are considered. Based upon these new estimates, it shows that the interpolation at strongly normal pointsystems can achieve the optimal convergence rate, the same as the best polynomial approximation. Furthermore, by using the asymptotics on Jacobi polynomials, the convergence rates are established for Gauss-Jacobi, Jacobi-Gauss-Lobatto or Jacobi-Gauss-Radau pointsystems. From these results, we see that the interpolations at the Gauss-Legendre, Legendre-Gauss-Lobatto pointsystem, or at strongly normal pointsystems, has essentially the same approximation accuracy compared with those at the two Chebyshev piontsystems, which also illustrates the equally accuracy of the Gauss and Clenshaw-Curtis quadrature. In addition, numerical examples illustrate the perfect coincidence with the estimates, which means the convergence rates are optimal.

TOWARDS A MORE GENERAL TYPE OF UNIVARIATE CONSTRAINED INTERPOLATION WITH FRACTAL SPLINES

Recently, in [Electron. Trans. Numer. Anal. 41 (2014) 420–442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current paper is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.

On a corrected Fejér quadrature formula of the second kind

We consider an interpolatory quadrature formula having as nodes the zeros of the nth degree Chebyshev polynomial of the second kind, on which the Fejer formula of the second kind is based, and the additional points $$\pm \tau _{c}=\pm \cos \frac{\pi }{2(n+1)}$$±?c=±cos?2(n+1). The new formula is shown to have positive weights given by explicit formulae. Furthermore, we determine the precise degree of exactness, and we obtain optimal error bounds for this formula either by Peano kernel methods or by Hilbert space techniques for analytic functions and $$1\le n\le 40$$1≤n≤40. In addition, the convergence of the quadrature formula is shown not only for Riemann integrable functions on $$[-1,1]$$[-1,1], but also for functions having monotonic singularities at $$\pm $$±1. The new formula has essentially the same rate of convergence as, and it is therefore an alternative to, the well-known Clenshaw-Curtis formula.

A generalized Peano Kernel Theorem for distributions of exponential decay

The Peano Kernel Theorem (PKT) is a classical representation theorem in numerical integration. The idea is that if T is a quadrature rule that exactly integrates polynomials up to degree n−1 on a bounded interval [a,b], then there exists a kernel K, depending only on T, such thatE(f)=T(f)−∫abf(t)dt=∫abK(t)f(n)(t)dt, whenever f∈Cn([a,b]). In this work, we generalize the PKT from the class of linear functionals on Cn([a,b]) to the class of Laplace transformable tempered distributions of exponential decay. In particular, it is not necessary that the functionals being approximated have compact support. The generalized result is proven using an approach that provides a formula for computing the kernel K in the Fourier domain, which can be more computationally tractable and efficient in many cases. We conclude with examples of how the generalized Peano Kernel Theorem can be used for error analysis in signal processing.

Product integration rules for Chebyshev weight functions with Chebyshev abscissae

We study two product integration rules, one for the Chebyshev weight of the first-kind based on the Chebyshev abscissae of the second-kind, and another one constructed the other way around, i.e., relative to the Chebyshev weight of the second-kind and based on the Chebyshev abscissae of the first-kind. The new rules are shown to have positive weights given by explicit formulae. Furthermore, we determine the precise degree of exactness and we compute the variance of the quadrature formulae, we examine their definiteness or nondefiniteness, and we obtain error bounds for these formulae either asymptotically optimal by Peano kernel methods or for analytic functions by Hilbert space techniques. In addition, the convergence of the quadrature formulae is shown not only for Riemann integrable functions on [−1,1], but also, by generalizing a result of Rabinowitz, for functions having a monotonic singularity at one or both endpoints of [−1,1].

The weighted (0,1,…,m−2,m)-interpolation technique based on the roots of the classical orthogonal polynomials and application in deriving new quadrature rules

We investigate the problem of weighted (0,1,…,m−2,m)-interpolation (m≧2, \({m\in \mathbb{N}}\)) on the roots of the classical orthogonal polynomials. The necessary and sufficient conditions for the existence and uniqueness of this problem are established. Meanwhile, the explicit representation (characterization) of the weighted (0,1,…,m−2,m)-interpolation is given. As an application we obtain a Birkhoff type quadrature formula which is exact for the polynomials of degree at most mn. Also we investigate the error function of the new (0,1,…,m−2,m)-Birkhoff type weighted quadrature formulae using Peano kernel theorem. Some numerical examples are given to support the results.

The fractional Cubic Spline Interpolation without using the derivative values

The paper introduces a function value based fraction of cubic spline interpolation, which is used for studying the curves and surfaces. The interpolation function has a simple and explicit mathematical representation, convenient both in practical application and in theoretical studies. It should be mentioned that the interpolating surfaces are in the interpolating region under the condition that the interpolation is only based on the function values. Moreover, properties and views are shown in matrix notation, and then the error is calculated. 2000 Mathematics Subject Classifications: 68U05, 65D05, 65D07, 65D18. Keywords: Fractional spline; fractional interpolation: spline interpolation function, Peano Kernel theorem

A study of the uniform accuracy of univariate thin plate spline interpolation

The usual power function error estimates do not capture the true order of uniform accuracy for thin plate spline interpolation to smooth data functions in one variable. In this paper we propose a new type of power function and we show, through numerical experiments, that the error estimate based upon it does match the expected order. We also study the relationship between the new power function and the Peano kernel for univariate thin plate spline interpolation.

Error estimate and extrapolation of a quadrature formula derived from a quartic spline quasi-interpolant

In this paper we analyze a quadrature rule based on integrating a C3 quartic spline quasi-interpolant on a bounded interval which has been introduced in Sablonniere (Rend. Semin. Mat. Univ. Pol. Torino 63(3):107–118, 2005). By studying the sign structure of its associated Peano kernel we derive an explicit formula of the quadrature error with an approximation order O(h6). A comparison of this rule with the composite Boole’s and the three-point Gauss-Legendre rules is given. We also compare the Nystrom methods associated with the above quadrature formulae for solving the linear Fredholm integral equation of the second kind. Then, by combining the proposed rule with composite Boole’s rule, we construct a new quadrature rule of order O(h7). All the obtained results are illustrated by several numerical tests.

Filtering error estimates and order of accuracy via the Peano Kernel Theorem

The Peano Kernel Theorem is introduced and a frequency domain derivation is given. It is demonstrated that the application of this theorem yields simple and accurate formulas for estimating the error introduced into a signal by filtering it to reduce noise. The concept of the order of accuracy of a filter is introduced and used as an organizing principle to compare the accuracy of different filters.

Design of Fractional Delay Filter, Differintegrator, Fractional Hilbert Transformer, and Differentiator in Time Domain With Peano Kernel

In this paper, we propose a fractional delay filter, an integer-order differintegrator, a fractional Hilbert transformer, and a fractional differintegrator. Through the time-domain analysis on the desired input and output signals of a linear time-invariant system, we derive a set of linear equations, which can be solved to obtain the coefficients of the desired filter. We also show that the difference between the desired output signal and the actual output of the system can be represented as the convolution of the derivative of the input signal and the Peano kernel.

Modified product cubature formulae

In the univariate case, there is a well-developed theory on the error estimation of the quadrature formulae for integrands from the Sobolev classes of functions. It is based on the Peano kernel representation of linear functionals, which yields sharp error bounds for the quadrature remainder. The product cubature formulae are the usual tool for the approximation of a double integral over a rectangular domain. In this paper we suggest a modification of the product cubature formulae, based on blending interpolation of bivariate functions. Besides the usual point evaluations, the modified cubature formulae involve few line integrals. Our approach allows application of the Peano kernel theory for derivation of error bounds for both standard cubature formulae and their modifications. Sufficient conditions for the definiteness of the modified product cubature formulae are given, and some classes of integrands are specified, for which a product cubature formula is inferior to its modified version.