Chebyshev approximation problem

The problem [Eq. (15)] is a minimax optimization problem. For the case (as it is here) where the approximating function depends linearly on the coefficients, the optimization problem [Eq. (15)] has the form of the Chebyshev approximation problem and has a known solution (Murty, 1983). Indeed, it can be easily shown that with the introduction of the dummy variables z, z, z the minimax problem can be transformed to the following linear program (LP) ... 

Minimax, or Chebyshev, approximation utilizes a nonlinear criterion, and hence raises problems that are much more difficult. For a given function f(x), the problem is to find a function P(x) in a certain class such that... 

Antilinear operator, antiunitary, 688 Antiunitary operators, 727 A-operation, 524 upon Dirac equation, 524 Approximation, 87 methods, successive minimax (Chebyshev), 96 problem of, 52 Arc, 258... 

This procedure is also known as uniform or Chebyshev approximation. We have the introduce the single auxiliary variable s > 0 to translate the minimization of (1.83) into the problem... 

In ref 139 the authors presented variable-stepsize Chebyshev-type methods for the integration of second-order initial-value problems. More specifically, Panovsky and Richardson in ref. 140 presented a method based on Chebyshev approximations for the numerical solution of the problem y" = f(x,y), with constant stepsize. In ref. 141 Coleman and Booth analyzed the method developed in ref 140 and proposed the convenience to design a variable stepzesize methods of Chebyshev-type. The development of the new methods is based on the test equation ... 

In ref. 164 the authors consider a new BDF fourth-order method for solving stiff initial-value problems, based on Chebyshev approximation. The authors prove that the developed method may be presented as a Runge-Kutta method having stage order four. They examine the stability properties of the method and they presented a strategy for changing the step size based on embedded pair of the Runge-Kutta schemes. 

The Chebyshev-collocation method. This example problem uses the Chebyshev -collocation method to approximate u(x,t) in a domain x e [-1,1] as a solution to the PDE... 

Given that the apparent rate constants for complex reaction systems might strongly deviate from Lindemann-type rate expressions, the concept of correcting the Lindemann expression for k(T, p) leads to problems. Based on this conclusion, Venkatesh et al. [110] proposed the use of a purely mathematical approximation, Chebyshev polynomials, to represent the temperature and pressure dependences of apparent rate constants. Briefly, a Chebyshev polynomial of degree i— is defined as... 

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