Chebyshev orthogonal polynomials

The Chebyshev procedure consists of expanding the quantum operators in terms of orthogonal Chebyshev polynomials. It is considered to be an efficient and reliable method, since the convergence of the expansion is guaranteed. According to the method, the time-dependent hamiltonian is treated as a constant operator within the time slice dt. Thus, the time propagator is expanded in a series of Chebyshev polynomials for a time t within dt as... 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

Variation of the Orthogonal Polynomial. Among the Jacobi polynomials, the Chebyshev polynomials of the first kind (y = 8 =... 

The Chebyshev polynomials of the first kind have long been recognized as an effective basis for fitting non-periodic ftinctions.[7] In many aspects they resemble the Fourier basis for periodic systems. This special type of classical orthogonal polynomials can be generated by the following three-term recurrence relationship [8]... 

As mentioned in Chapter 1 of Vol. 2 (Buzzi-Ferraris and Manenti, 2010b), the orthogonal polynomial that best fits the selection of the P support points used to build the interpolating polynomial is the P-order Chebyshev polynomial. 

By means of the discrete orthogonality property for Chebyshev polynomials... 

The idea behind the DVR method [8-11] is to use a representation in terms of localized functions obtained by transformation from a global basis [12], Usually, bases constructed from orthogonal polynomials, noted F x), which are solution of one dimensional problems such as the particle in a box (Chebyshev polynomials) or the harmonic oscillator (Hermite polynomials), are used. These polynomial bases verify the general relationship... 

This is one of the variants of the FEMs. The essence of orthogonal collocation (OC) is that a set of orthogonal polynomials is fitted to the unknown function, such that at every node point, there is an exact fit. The points are called collocation points, and the set of polynomials is chosen suitably, usually as Jacobi or Chebyshev polynomials. The optimal choice of collocation points is to make them the roots of the polynomials. There are tables of such roots, and thus point placements, in Appendix A. The notable things here are the small number of points used (normally, about 10 or so will do), their uneven spacing, crowding closer both at the electrode and (perhaps strangely) at the outer limit, and the fact that the outer limit is always unity. This is discussed below. 

The extrema cubes are generated using volume reflection spectral analysis technology [9, 10], where each seismic trace is locally reconstructed using the orthogonal basis defined by the Chebyshev polynomials. Thus, locally the seismic trace is represented as... 

The performance of DORTHO, the double precision version of ORTHO, was a very pleasant surprise to the writers. Up through the fitting of fifteenth degree polynomials the first six digits of each coefficient were the same whether monomials or Chebyshevs were used for a coordinate system. This means that the internal orthogonalization scheme built into ORTHO and DORTHO functioned very effectively and that the double-precision arithmetic avoided meaningful roundoff errors. From this, we conclude that... 

Equation (29) may be extrapolated outside the experimental range-limitation of the value of x is required only for the fitting procedure. The fitting of experimental results by orthogonal polynomials and their transformation to the Chebyshev form by computer is straightforward the experimental points should be evenly spaced throughout the temperature range, but in practice little difficulty arises if this condition is not fully met. 

Chebyshev orthogonal polynomial series along the nonperiodic (shear) direction ... 

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