Generalized Taylor's Theorem

Garret Sobczyk

Home/Archive/arXiv

arXiv · 0710.0143

Generalized Taylor's Theorem

Garret Sobczyk

View original source ↗Read source document ↗Export metadata ↓

Abstract

The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given multiplicities. Taken together with a simple expression for the remainder, this theorem becomes a powerful tool for approximation and interpolation in numerical analysis. We also have a corresponding theorem for rational approximation.

Explore related subjects

math.NA cs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Garret Sobczyk. 2007-09-30. Generalized Taylor's Theorem. https://arxiv.org/abs/0710.0143

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A survey of sampling discretization of integral and uniform norms

This paper surveys recent developments in the sampling discretization of integral and uniform norms for functions in general finite-dimensional spaces. These results generalize the classical Marcinkiewicz-Zygmund inequalities for trigonometric and algebraic polynomials, which play a crucial role in Fourier analysis, interpolation, and approximation theory. We focus on the problem in the broad context of finite-dimensional subspaces, where norms defined by general probability measures are approximated by their discrete counterparts. The primary emphasis is on results closely related to the authors' recent research. A key objective is to highlight the main ideas and techniques that form the foundation of the proofs in this area. This survey serves as a complement to three recently published survey papers on sampling discretization \cite{DPTT, KKLT, LMT}.

math.NA↗

arXiv2026

A conservative reduced collision model for the Boltzmann equation

The evaluation of the quadratic collision operator is one of the major computational bottlenecks in solving the Boltzmann equation owing to the high dimensionality and the nonlinear structure of the collision term. In this work, a conservative reduced collision (RC) model is developed within the discrete velocity framework. The bilinear collision operator is approximated through a kernel tensor preassembled with the problem-dependent reduced bases for the distribution function and the collision term. Consequently, online evaluation of the quadratic collision operator is reduced to tensor contractions. Exact discrete conservation of mass, momentum, and energy is enforced by constructing the collision basis in the orthogonal complement of the space spanned by the discrete collision invariants. Several numerical experiments validate the efficiency of this RC model.

math.NA↗

arXiv2026

A hybridization-derived interior penalty Nitsche method for heterogeneous and anisotropic diffusion

We develop an interior penalty Nitsche finite element formulation for heterogeneous and anisotropic diffusion problems, which is derived from an interior-penalty stabilized Lagrange multiplier method. A distinctive feature of the derived method is that it simultaneously combines two popular stabilization strategies: (i) a boundary penalty mechanism similar to Nitsche's methods and (ii) an interior penalty mechanism as for Continuous Interior Penalty (CIP) methods. Furthermore, the normal component of the numerical flux is single-valued across the mesh skeleton and available in closed form. We establish the equivalence between the hybridized Lagrange multiplier formulation and its condensed Interior Penalty Nitsche (IPN) form, and we deliver the stability and convergence analysis of the latter. For sufficiently regular solutions, we establish energy-norm error estimates for the three symmetry variants. Under a piecewise elliptic regularity assumption on the adjoint problem, the $L^2$ estimate is optimal for the symmetric variant and exhibits a half-order loss for the incomplete and nonsymmetric variants; its constant may depend on the diffusion coefficients through the adjoint regularity constant. This single-valued numerical normal flux, available in closed form, provides a starting point for locally conservative $H(\mathrm{div})$-conforming flux reconstruction, left for future work. Numerical experiments validate the theoretical findings.

math.NA↗