Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

Approximations in $L^p $ and Chebyshev Approximations

Abstract: Introduction. One of the difficulties of the analysis of Chebyshev approximations for functions of several variables lies in the fact that the best approximation is not always unique [1]. Consequently it can be useful to define a reasonable particular unique Chebyshev approximation. In [1] J. R. Rice defines a unique "strict approximation," but only for functions defined on a finite set. In this paper, we show that the strict approximation is the limit of the best approximation in LP when p -* oo. Unfortunately, one cannot generalize this result for functions defined on an infinite set. A continuous function H(x) will be constructed such that the coefficient a, of the best approximation in 1'), epr, on the interval (-2, 21 does not converge for p -* o; this example has been extended in [4] for a continuous function F(x, y) defined on a rectangle: the coefficients of the best linear approximation in LP (i.e., apx + bpy + cp) do not converge for p -* 00. Approximations on a finite set. Let f(x), gq(x), *. gn(x) be functions defined on the finite set E = { xl, * **, xm}. We suppose that the matrix qi g(xj) it has rank n. Using the same notations as in [11, for any set of parameters al, *, an,

Canonical piecewise-linear approximation of smooth functions

Abstract: This paper deals with the approximation of smooth functions using canonical piecewise-linear functions. The developing of tools in the field of analysis and control of nonlinear systems based on this kind of functions, as well as its efficiency in the representation of electronic devices, motivates the development of useful methods to obtain accurate approximations. A recursive method is proposed to obtain simultaneously all the parameters required and its convergence is studied. In addition, an iterative method to introduce new partitions on the domain, when the error obtained is not satisfactory, is described. This method takes advantage of the partitions already found to reduce the total number of parameters that the algorithm has to handle.

Nonlinear Effects in Transport Theory

Abstract: A method for the formal statistical description of transport processes beyond the linear approximation is discussed. External forces which maintain a deviation from equilibrium are introduced into the Liouville equation, and solution of this equation yields an ensemble characteristic of the non‐equilibrium state. The transport relations are obtained with the aid of the resulting ensemble. For simplicity, the discussion is restricted to heat flow, but the generalization to the processes of viscosity and diffusion is straightforward.