Remainder Formula Research Articles

FACTORIZATIONS AND UNIFIED HARDY INEQUALITIES ON HOMOGENEOUS LIE GROUPS

In this note we obtain Hardy and critical Hardy inequalities with any homogeneous quasi-norm in unified way. Actually, we show a sharp remainder formula for these results. In particular, our identity implies corresponding Hardy and critical Hardy inequalities with any homogeneous quasi-norm for the radial derivative operator, thus yielding improved versions of corresponding classical counterparts. Moreover, we discuss extensions of these results in the setting of Folland and Stein’s homogeneous Lie groups. Such a more general setting is convenient for the distillation of those results of harmonic analysis depending only on the group and dilation structures, which is one of our motivations working in the setting. Our approach based on the factorization method of differential operators introduced by Gesztesy and Littlejohn. As an application, we show Caffarelli-Kohn-Nirenberg type inequalities with more general weight. Because of the freedom in the choice of any homogeneous quasi-norm, our results give new insights already in both anisotropic ℝn and isotropic ℝn .

Sharp remainder of the Poincaré inequality for Baouendi–Grushin vector fields

In this note, we establish a sharp remainder formula for the Poincaré inequality for the Baouendi–Grushin vector fields. We give a simple proof for it without using the variational principle. As an application, we obtain a blow-up result for solutions to the Dirichlet initial-boundary value problem for the Baouendi–Grushin heat operator.

“Smooth rigidity” and Remez-type inequalities

If a \((d+1)\)-smooth function f(x) on \([-1,1]\), with \(\mathrm{max\,}_{[-1,1]}|f(x)|\ge 1,\) has \(d+1\) or more distinct zeroes on \([-1,1]\), then \(\mathrm{max\,}_{[-1,1]}|f^{(d+1)}(x)|\ge 2^{-d-1}(d+1)!\). This follows from the polynomial interpolation of f at its zeroes, with Lagrange’s remainder formula. This is one of the simplest examples of what we call “smooth rigidity”: certain geometric properties of zero sets of smooth functions f imply explicit lower bounds on the high-order derivatives of f. In dimensions greater than one, the powerful one-dimension tools such as Lagrange’s remainder formula, and divided finite differences, are not directly applicable. Still, the result above implies, via line sections, rather strong restrictions on zeroes of smooth functions of several variables (Yomdin in Proc AMS 90(4):538–542, 1984). In the present paper we study the geometry of zero sets of smooth functions, and significantly extend the results of Yomdin (1984), including into consideration, in particular, finite zero sets (for which the line sections usually do not work). Our main goal is to develop a pure multi-dimensional approach to smooth rigidity, based on polynomial Remez-type inequalities (which compare the maxima of a polynomial on the unit ball, and on its subset). Very informally, one of our main results is that the smooth rigidity of a zeroes set Z is approximately the reciprocal Remez constant of Z.

A higher order Faber spline basis for sampling discretization of functions

This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber–Schauder basis (Faber, 1908) except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebel’s monograph (Triebel, 2012) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylor’s remainder formula to the dual Chui–Wang wavelets. As a first step we explicitly determine these dual wavelets which may be of independent interest. Using this new basis we provide sampling characterizations for Besov and Triebel–Lizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear Faber–Schauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the Faber–Schauder situation.

Simultaneous approximation of Birkhoff interpolation and the associated sharp inequalities

This paper gives a kind of sharp simultaneous approximation error estimation of Birkhoff interpolation [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is the Birkhoff interpolation based on [Formula: see text] pairs of numbers [Formula: see text] with its P[Formula: see text]lya interpolation matrix to be regular. First, based on the integral remainder formula of Birkhoff interpolation, we refer the computation of [Formula: see text] to the norm of an integral operator. Second, we refer the values of [Formula: see text] and [Formula: see text] to two explicit integral expressions and the value of [Formula: see text] to the computation of the maximum eigenvalue of a Hilbert–Schmidt operator. At the same time, we give the corresponding sharp Wirtinger inequality [Formula: see text] and sharp Picone inequality [Formula: see text].

Phase unwrapping simulation of dual-frequency combined analog encoding structured light

In order to improve the measurement accuracy of phase shift method, we present a novel dual-frequency combined analog encoding phase unwrapping model and 3D measurement method. This engineered phase unwrapping model is built based on two phase principal values of one pixel from two different periodic fringe images. By calculating analytic solution from remainder formula, this model has simple and fast advantages; and by adding one rounding process, measurement error can be limited in some extent. Furthermore, based on cosine phase shift, 3D measurement method of dual-frequency combined analog encoding phase unwrapping is presented. Then theoretical analysis, simulation and actual experiment are used to verify the proposed model and method in this paper.

Exact remainder formula for the Young inequality and applications

Exact remainder formula for the Young inequality and applications

Taylor's Expansion Revisited: A General Formula for the Remainder

We give a new approach to Taylor's remainder formula, via a generalization of Cauchy's generalized mean value theorem, which allows us to include the well-known Schölomilch, Lebesgue, Cauchy, and the Euler classic types, as particular cases.

A two-dimensional matrix Padé-type approximation in the inner product space

By introducing a bivariate matrix-valued linear functional on the scalar polynomial space, a general two-dimensional (2-D) matrix Padé-type approximant ( B M P T A ) in the inner product space is defined in this paper. The coefficients of its denominator polynomials are determined by taking the direct inner product of matrices. The remainder formula is developed and an algorithm for the numerator polynomials is presented when the generating polynomials are given in advance. By means of the Hankel-like coefficient matrix, a determinantal expression of B M P T A is presented. Moreover, to avoid the computation of the determinants, two efficient recursive algorithms are proposed. At the end the method of B M P T A is applied to partial realization problems of 2-D linear systems.

A remainder formula of numerical differentiation for the generalized Lagrange interpolation

Through exploiting the generalized Lagrange interpolation for continuous piecewise smooth functions, a remainder term of numerical differentiation is given. Also, the error estimates for the numerical differentiation formula are presented.

On the convergence of derivatives of Bernstein approximation

By differentiating a remainder formula of Stancu, we derive both an error bound and an asymptotic formula for the derivatives of Bernstein approximation.

Continued fraction algorithm for matrix exponentials

A recursive rational algorithm for matrix exponentials was obtained by making use of the generalized inverse of a matrix in this paper. On the basis of the n-th convergence of Thiele-type continued fraction expansion, a new type of the generalized inverse matrix-valued Pade approximant (GMPA) for matrix exponentials was defined and its remainder formula was proved. The results of this paper were illustrated by some examples.

On Bivariate Hermite Interpolation with Minimal Degree Polynomials

A Newton-type approach is used to deal with bivariate polynomial Hermite interpolation problems when the data are distributed in the intersections of two families of straight lines, as a generalization of regular grids. The interpolation operator is degree-reducing and the interpolation space is a minimal degree space. Integral remainder formulas are given for the Lagrange case, then extended to the Hermite case, and finally used to obtain error estimates.

Simultaneous Approximations for Functions in Sobolev Spaces by Derivatives of Polyharmonic Cardinal Splines

We prove in this paper that functions in Sobolev spaces and their derivatives can be approximated by polyharmonic splines and their derivatives in Lp(Rn) norms. Of particular interest are the remainder formulas of such approximations and the order of convergence by the derivatives of cardinal polyharmonic interpolational splines.

New Methods for High-Dimensional Verified Quadrature

Conventional verified methods for integration often rely on the verified bounding of analytically derived remainder formulas for popular integration rules. We show that using the approach of Taylor models, it is possible to devise new methods for verified integration of high order and in many variables. Different from conventional schemes, they do not require an a-priori derivation of analytical error bounds, but the rigorous bounds are calculated automatically in parallel to the computation of the integral. The performance of various schemes are compared for examples of up to order ten in up to eight variables. Computational expenses and tightness of the resulting bounds are compared with conventional methods.

Polynomial interpolation of minimal degree

Minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which Lagrange interpolation with respect to the given points is uniquely solvable and degree reducing. This is a generalization of the concept of least interpolation introduced by de Boor and Ron. This paper investigates the behavior of Lagrange interpolation with respect to these spaces, giving a Newton interpolation method and a remainder formula for the error of interpolation. Moreover, a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view.

On multivariate Hermite interpolation

We study the problem of Hermite interpolation by polynomials in several variables. A very general definition of Hermite interpolation is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.

On Multivariate Lagrange Interpolation

Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formula for Lagrange interpolation of degree n of a function f, which is a sum of integrals of certain $(n + 1)$st directional derivatives of f multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.

Remainder Formulas Involving Generalized Fibonacci and Lucas Polynomials

Remainder Formulas Involving Generalized Fibonacci and Lucas Polynomials

On multivariate Lagrange interpolation

Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formula for Lagrange interpolation of degree n of a function f, which is a sum of integrals of certain ( n + 1 ) (n + 1) st directional derivatives of f multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.