Search results
Results from the WOW.Com Content Network
The Chebyshev polynomials form a complete orthogonal system. The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in f(x) and its derivatives. At a discontinuity, the series will converge to ...
One can obtain polynomials very close to the optimal one by expanding the given function in terms of Chebyshev polynomials and then cutting off the expansion at the desired degree. This is similar to the Fourier analysis of the function, using the Chebyshev polynomials instead of the usual trigonometric functions.
Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials. Equivalently, they employ a change of variables x = cos θ {\displaystyle x=\cos \theta } and use a discrete cosine transform (DCT) approximation for ...
This means that all their local minima and maxima have values of −1 and +1, that is, the polynomials are "level". Because of this, expansion of functions in terms of Chebyshev polynomials is sometimes used for polynomial approximations in computer math libraries.
The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A. A. Markov and T. J. Stieltjes.
In numerical analysis Chebyshev–Gauss quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind:
Maybe you’ve already got a ready-made pet first-aid kit that includes all the essentials, like a tick remover, saline, and pet wrap.
In applied mathematics, a discrete Chebyshev transform (DCT) is an analog of the discrete Fourier transform for a function of a real interval, converting in either direction between function values at a set of Chebyshev nodes and coefficients of a function in Chebyshev polynomial basis. Like the Chebyshev polynomials, it is named after Pafnuty ...