Peano Kernels Research Articles

A new lower bound for the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{L}^2$$\\end{document}-norm of the Caputo fractional derivative

We prove a novel, tight lower bound for the norm in L2[0,T]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{L}^2[0,T]$$\\end{document} of the Caputo fractional derivative. It is based on continuous linear functionals, Peano kernels, and the Gaussian hypergeometric function.

A new version of Ostrowski type integral inequalities for different differentiable mapping

In this paper, improved and generalized version of Ostrowski’s type inequalities is established. The parameters used in the peano kernels help us to obtain previous results. The obtained bounds are then applied to numerical integration.

A performance comparison of the zero-finding by extended interval Newton method for Peano monosplines

Peano monosplines are a special class of Peano kernels. Their zeros are essential for deriving multiple error bounds for numerical quadrature regarding the weight function w(x)≡1. As the performance of their zero-finding can influence the numerical results of Peano error constants, this article aims to give a concrete inspection of the improvements in deriving the zeros of Peano monosplines by the extended interval Newton method.

A variable step size numerical method based on fractional type quadratures for linear integro-differential equations

In this work a discretization in time of variable step size of the convolution equation u(t)[email protected]!0^t(t-s)^@a^-^1u(s)ds, based on a fractional type quadrature is studied. The convergence is directly proved thanks to a suitable representation of the error by means of Peano kernels. Practical illustrations showing the efficiency of our numerical scheme are provided.

Bivariate Product Cubature Using Peano Kernels for Local Error Estimates

The error estimates of automatic integration by pure floating-point arithmetic are intrinsically embedded with uncertainty. This in critical cases can make the computation problematic. To avoid the problem, we use product rules to implement a self-validating subroutine for bivariate cubature over rectangular regions. Different from previous self-validating integrators for multiple variables (Storck in Scientific Computing with Automatic Result Verification, pp. 187---224, Academic Press, San Diego, [1993]; Wolfe in Appl. Math. Comput. 96:145---159, [1998]), which use derivatives of specific higher orders for the error estimates, we extend the ideas for univariate quadrature investigated in (Chen in Computing 78(1):81---99, [2006]) to our bivariate cubature to enable locally adaptive error estimates by full utilization of Peano kernels theorem. The mechanism for active recognition of unreachable error bounds is also set up. We demonstrate the effectiveness of our approach by comparing it with a conventional integrator.

B-splines and Hermite–Padé approximants to the exponential function

This paper is the continuation of a work initiated in [P. Sablonnière, An algorithm for the computation of Hermite–Padé approximations to the exponential function: divided differences and Hermite–Padé forms. Numer. Algorithms 33 (2003) 443–452] about the computation of Hermite–Padé forms (HPF) and associated Hermite–Padé approximants (HPA) to the exponential function. We present an alternative algorithm for their computation, based on the representation of HPF in terms of integral remainders with B-splines as Peano kernels. Using the good properties of discrete B-splines, this algorithm gives rise to a great variety of representations of HPF of higher orders in terms of HPF of lower orders, and in particular of classical Padé forms. We give some examples illustrating this algorithm, in particular, another way of constructing quadratic HPF already described by different authors. Finally, we briefly study a family of cubic HPF.

Verified computed Peano constants and applications in numerical quadrature

Peano kernels are critical for the error estimates of numerical approximation of linear functionals. In the literature, considerable efforts have been devoted to the numerical computation or estimation of Peano constants, however, their discussions have mainly focused on the Peano kernels of some specific higher orders related to the degrees of the underlying error functionals. Limiting to such higher orders either requires certain smoothness of the approximated functions, which is not always fulfilled, or is not always the optimal choice for error estimates, even if the approximated functions are sufficiently smooth. Practical considerations in computing Peano constants for the full range of orders are given in this paper. Numerical examples are also given to demonstrate that much better error bounds can be derived while Peano constants of lower orders are considered.

Computing Interval Enclosures for Definite Integrals by Application of Triple Adaptive Strategies

In addition to the well-known and widely-used adaptive strategies for region subdivision and the choice of local quadrature rules, there still exists a third possibility for doing numerical integration adaptively, when interval computation is considered. Based on the Peano's kernels theorem and interval arithmetic, we are able to estimate the truncation error of a quadrature rule by means of different derivatives of the integrand, where the available orders of the derivatives depend on the degree of smoothness of the integrand and the exactness degree of the underlying quadrature rule. We classify the methods as the adaptive orders strategies, if they make use of the derivatives of different orders to improve each single local error estimation. In this paper, alternatives for adaptive error estimation are discussed. Moreover, a practical way for realization of the optimal error estimation is suggested. Numerical results for integrands of different classes as well as numerical comparisons of different methods are given.

A Definiteness Theory for Cubature Formulae of Order Two

Let Ω ⊂ ℝd be a compact convex set of positive measure. A cubature formula will be called positive definite (or a pd-formula, for short) if it approximates the integral ∫Ω f(x) dx of every convex function f from below. The pd-formulae yield a simple sharp error bound for twice continuously differentiable functions. In the univariate case (d = 1), they are the quadrature formulae with a positive semidefinite Peano kernel of order two. As one of the main results, we show that there is a correspondence between pd-formulae and partitions of unity on Ω. This is a key for an investigation of pd-formulae without employing the complicated multivariate analogue of Peano kernels. After introducing a preorder, we establish criteria for maximal pd-formulae. We also find a lower bound for the error constant of an optimal pd-formula. Finally, we describe a phenomenon which resembles a property of Gaussian formulae.

Pointwise error bounds for orthogonal cardinal spline approximation

For orthogonal cardinal spline approximation, closed form expressions of the reproducing kernel and the Peano kernels in terms of exponential splines are proved. Concrete and sharp pointwise error bounds are deduced for low degree splines.

Double integral inequalities based on multi-branch Peano kernels

The Ostrowski inequality in one dimension has been known for about seventy years. In the last two or three years an Ostrowski type inequality in two dimensions has been developed. In this paper we extend these ideas in two dimensions to obtain double integral inequalities that are based on multi-branch Peano kernels. Mathematics subject classification (2000): 26D15, 41A55.

On Optimal Error Bounds for Derivatives of Interpolating Splines on a Uniform Partition

Based on Peano kernel technique, explicit error bounds (optimal for the highest order derivative) are proved for the derivatives of cardinal spline interpolation (interpolating at the knots for odd degree splines and at the midpoints between two knots for even degree splines). The results are based on a new representation of the Peano kernels and on a thorough investigation of their zero distributions. The bounds are given in terms of Euler–Frobenius polynomials and their zeros.

Refinements of the peano kernel theorem

It is shown that by starting with a general form of the Peano kernel theorem which makes no reference to the interchange of linear functionals and integrals, the most general results can be obtained in an elementary manner. In particular, we classify how the Peano kernels become increasingly smooth and satisfy boundary (or equivalently moment) conditions as the linear functionals they represent become continuous on wider classes of functions. These results are then used to give new representations of the continuous duals of .

Peano Kernels of Non-Integer Order

We consider the representation of error functionals in numerical quadrature by the Peano kernel method. It is easily observed that the usual expressions for Peano kernels of order s still make sense if s is not a natural number. In this paper, we discuss how to interpret these Peano kernels, we state their main properties, and we compare them to the (classical) Peano kernels of integer order.

Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals

We show that, if\(f \in C^k[-1,1]\) (\(k \ge 2\)), the error term of every modified positive interpolatory quadrature rule for Cauchy principal value integrals of the type\(\mathop{\int\!\!\!\!\!\! -}\nolimits_{-1}^1 w(x){f(x) \over x-\lambda} dx\) ,\(\lambda \in (-1,1)\) , fulfills\(R_n[f; \lambda] = O(n^{-k} \ln n)\) uniformly for all\(\lambda \in (-1,1)\) , and hence it is of optimal order of magnitude in the classes\(C^k[-1,1]\) (\(k=2,3,4,\ldots\)). Here, \(w\) is a weight function with the property\(0 \le w(x) \sqrt{1-x^2} \le C\) . We give explicit upper bounds for the Peano-type error constants of such rules. This improves and completes earlier results by Criscuolo and Mastroianni (Calcolo 22 (1985), 391–441 and Numer. Math. 54 (1989), 445–461) and Ioakimidis (Math. Comp. 44 (1985), 191–198). For the special case of the Gaussian rule, we show that the restriction\(k \ge 2\) can be dropped. The results are based on a new representation of the Peano kernels of these formulae via the Peano kernels of the underlying classical quadrature formulae. This representation may also be useful in connection with some different problems.

Peano kernel behaviour and error bounds for symmetric quadrature formulas

For symmetric quadrature formulas, sharper error bounds are generated by a formulation of the Peano Kernel theory which fully exploits the symmetry properties. Formulas which use nodes outside the integration interval and/or derivative data are taken into consideration. We prove also that, for any symmetric formula, the Peano Kernels of a sufficiently large degree do not change sign in a suitable interval. This result allows a finite expansion of the truncation error for any regular integrand function.

Gaussian quadrature formulae-second peano kernels, nodes, weights and Bessel functions

The central results of this paper are bounds for the second Peano kernels of the Gaussian quadrature formulae. Hence, for these quadrature formulae, we derive asymptotically optimal error constants for classes of functions with a bounded second derivative or with a first derivative of bounded variation, or for a class of convex functions. To obtain these bounds, we first prove inequalities related to MacMahon's expansion and some further results on the Bessel functionJ o , as well as some “trapezoid theorems for the weights of Gaussian formulae” (cf. Davis and Rabinowitz [6]) with explicit bounds for the error term.

Estimates for Sard's best formulas for linear functionals on CS[ a, b

Let J:C s[a, b]→ R be a bounded linear functional on the space of s times continuously differentiable functions ( s⩾0), and let Y=( y i, n ) be a triangular matrix of nodes satisfying a= y 0, n < y 1, n ⋯ < y n, n = b. Then one may approximate J by linear functionals Q n of the form Q n[ƒ]=Σ n−m 2 i=m 1a i,nf(y n,n) where m 1, m 2 ϵ {0, 1}. Among these, we consider the best formulas Q B n in the sense of Sard. For certain classes of nodes (which include, e.g., equidistant nodes, and the nodes of the Gauss quadrature formulas), and for arbitrary J, we give estimates for the weights a B i,n of Q B n , for the corresponding Peano kernels, and for the approximation error, including the error for interpolation by natural splines. By choosing special J's, estimates for best formulas for numerical integration, interpolation and differentiation are obtained, and also exponential decay of the fundamental natural splines is proved.

Hermite-Lagrange Two-Point Interpolation Via Peano Kernels

In a recent paper (JAT 53, 1988, pp. 17–25) Berger and Tasche present a constructive approach to Hermite-Lagrange two-point interpolation using Taylor's formula in the theory of right-invertible operators. In this paper we alternatively propose the argumentation via Peano kernels to obtain explicit representation formulas for this special kind of interpolation.

Convergence of LMM when the solution is not smooth

LMM (linear multistep methods) are popular for the solution of the initial value problem for a system of ordinary differential equations. In the classical theory it is assumed that the solution is as smooth as necessary. LMM are constructed to be of about as high order as possible, subject to stability, so it is not obvious that they will provide reasonable results when the the solution is not as smooth as anticipated. It is shown that they do, the order is just reduced. The discretization error constants are investigated using Peano kernels. In practice, solutions seem to be piecewise smooth. It is shown then that the order of convergence is two higher than might be expected. Piecewise smooth solutions commonly arise when data is fitted with piecewise polynomial functions. An example of the propagation of sound in the ocean illustrates this and confirms the theory presented.