A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view. It is well-kn...

Line spectral frequencies provide an alternate parameterization of the analysis and synthesis filters used in linear predictive coding (LPC) of speech. In this paper, a new method of converting between the direct form predictor coefficients and line spectral frequencies is presented. ,The system polynomial for the analysis filter is converted to two even-order symmetric polynomials with interla...

Knowledge of a truncated Fourier series expansion for a discontinuous 2^-periodic function, or a truncated Chebyshev series expansion for a discontinuous nonperiodic function defined on the interval [-1, 1], is used in this paper to accurately and efficiently reconstruct the corresponding discontinuous function. First an algebraic equation of degree M for the M locations of discontinuities in e...

We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the pap...

Abstract We discuss optimal design problems for a popular method of series estimation in regression problems. Commonly used design criteria are based on the generalized variance of the estimates of the coefficients in a truncated series expansion and do not take possible bias into account. We present a general perspective of constructing robust and efficient designs for series estimators which ...

We describe a method for the numerical evaluation of normalized versions of the associated Legendre functions P−μ ν and Q −μ ν of degrees 0 ≤ ν ≤ 1, 000, 000 and orders −ν ≤ μ ≤ ν on the interval (−1, 1). Our algorithm, which runs in time independent of ν and μ, is based on the fact that while the associated Legendre functions themselves are extremely expensive to represent via polynomial expan...

Abstract In this study, the spectral perturbation method and relaxation are used to solve nonlinear differential equations of an unsteady MHD flow in presence thermal radiation heat generation. The SPM is mainly based on series expansion, generating approximation coupled with Chebyshev method. numerical results generated using were compared those found literature, two good agreement.

Keywords: Set-valued mapping Chebyshev center Uniformly convex Locally uniformly convex Chebyshev fixed point a b s t r a c t The existence of a continuous Chebyshev selection for a Hausdorff continuous set-valued mapping is studied in a Banach space with some uniform convexity. As applications, some existence results of Chebyshev fixed point for condensing set-valued mappings are given, and th...

Neutral stability limits for wake flow behind a flat plate is studied using spectral method. First, Orr-Sommerfeld equation was changed to matrix form, covering the whole domain of solution. Next, each term of matrix was expanded using Chebyshev expansion series, a series very much equivalent to the Fourier cosine series. A group of functions and conditions are applied to start and end points i...