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The Chebyshev polynomials T n are polynomials with the largest possible leading coefficient whose absolute value on the interval [−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.
The discrete Chebyshev polynomial () is a polynomial of degree n in x, for =,,, …,, constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function = = (), with () being the Dirac delta function.
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 ...
Hermite polynomials were defined by Pierre-Simon Laplace in 1810, [1] [2] though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. [3] Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new. [4]
Chebyshev's equation is the second order ... where the coefficients obey the ... that function is a polynomial of degree p and it is proportional to the ...
The rational term may be obtained by manually performing the polynomial arithmetic, or with the short cut below which is a solution derived from polynomial arithmetic and uses binomial coefficients. The algorithm is extremely efficient if the Binomial coefficients are implemented from a look-up table of pre-calculated values.
A typical example of a Chebyshev space is the subspace of Chebyshev polynomials of order n in the space of real continuous functions on an interval, C[a, b]. The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum absolute difference between the polynomial and the function.
The Chebyshev nodes of the second kind, also called the Chebyshev extrema, are the extrema of the Chebyshev polynomials of the first kind, which are also the zeros of the Chebyshev polynomials of the second kind. Both of these sets of numbers are commonly referred to as Chebyshev nodes in literature. [1] Polynomial interpolants constructed from ...