• In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced...
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  • Thumbnail for Chebyshev polynomials
    The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}...
    58 KB (11,026 words) - 12:31, 26 June 2025
  • include the discrete Chebyshev polynomials, Charlier polynomials, Krawtchouk polynomials, Meixner polynomials, dual Hahn polynomials, Hahn polynomials, and Racah...
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  • coefficients of Chebyshev polynomials of the first kind. Other discrete Chebyshev transforms involve related grids and coefficients of Chebyshev polynomials of the...
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  • Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special...
    15 KB (2,079 words) - 21:55, 22 May 2025
  • named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred...
    67 KB (12,073 words) - 00:35, 29 June 2025
  • Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859...
    68 KB (12,148 words) - 20:43, 19 June 2025
  • form" Chebyshev norm Discrete Chebyshev polynomials Discrete Chebyshev transform Chebyshev rational functions Chebyshev–Gauss quadrature Chebyshev–Markov–Stieltjes...
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  • Chebyshev polynomials, and fast DCT algorithms (below) are used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials,...
    101 KB (11,972 words) - 11:01, 5 July 2025
  • multiple dimensions Discrete Chebyshev polynomialspolynomials orthogonal with respect to a discrete measure Favard's theorem — polynomials satisfying suitable...
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  • Thumbnail for Bernstein polynomial
    4}\delta ^{-2}n^{-1}.} (Chebyshev's inequality) It follows that the polynomials fn tend to f uniformly. Bernstein polynomials can be generalized to k...
    26 KB (4,491 words) - 17:53, 1 July 2025
  • Thumbnail for Fast Fourier transform
    fast algorithms for discrete cosine or sine transforms (e.g. fast DCT used for JPEG and MPEG/MP3 encoding and decoding), fast Chebyshev approximation, solving...
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  • wavelets are examples of orthogonal functions with discrete ranges. Legendre and Chebyshev polynomials provide orthogonal families for the interval [−1...
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  • Thumbnail for Approximation theory
    a polynomial of degree N. One can obtain polynomials very close to the optimal one by expanding the given function in terms of Chebyshev polynomials and...
    16 KB (2,319 words) - 16:40, 3 May 2025
  • than fixed in advance. Discrete Chebyshev transforms (on the 'roots' grid and the 'extrema' grid of the Chebyshev polynomials of the first kind). This...
    7 KB (1,001 words) - 15:55, 27 May 2025
  • derived at the Chebyshev Gauss–Lobatto collocation points by using the discrete orthogonal relationship of the Chebyshev polynomials. Then, using two...
    3 KB (415 words) - 17:58, 26 October 2021
  • Thumbnail for Butterworth filter
    {\displaystyle s_{n}} . The polynomials are normalized by setting ω c = 1 {\displaystyle \omega _{c}=1} . The normalized Butterworth polynomials then have the general...
    29 KB (4,141 words) - 23:58, 13 March 2025
  • terms of Chebyshev polynomials. Equivalently, they employ a change of variables x = cos ⁡ θ {\displaystyle x=\cos \theta } and use a discrete cosine transform...
    24 KB (4,366 words) - 20:34, 30 June 2025
  • Thumbnail for Window function
    parameter α sets the Chebyshev norm of the sidelobes to −20α decibels. The window function can be calculated from W0(k) by an inverse discrete Fourier transform...
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  • Thumbnail for Interpolation
    Chebyshev polynomials. Linear interpolation uses a linear function for each of intervals [xk,xk+1]. Spline interpolation uses low-degree polynomials in...
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  • non-uniform discrete Fourier transform (NUDFT or NDFT) of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier...
    15 KB (2,481 words) - 15:53, 18 June 2025
  • Remez algorithm (category Polynomials)
    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...
    16 KB (2,701 words) - 22:07, 19 June 2025
  • Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in...
    4 KB (609 words) - 15:27, 25 March 2023
  • Thumbnail for Cubic equation
    polynomials in r1, r2, r3, and a. The proof then results in the verification of the equality of two polynomials. If the coefficients of a polynomial are...
    68 KB (10,311 words) - 18:51, 6 July 2025
  • Thumbnail for Parks–McClellan filter design algorithm
    Thomas Parks in 1972, is an iterative algorithm for finding the optimal Chebyshev finite impulse response (FIR) filter. The Parks–McClellan algorithm is...
    14 KB (1,798 words) - 19:43, 13 December 2024
  • Thumbnail for Gaussian filter
    Detector used in image processing. Butterworth filter Comb filter Chebyshev filter Discrete Gaussian kernel Elliptic filter Gaussian blur Gaussian pyramid...
    19 KB (2,885 words) - 13:34, 23 June 2025
  • It is one of several graph polynomials studied in algebraic graph theory. Several different types of matching polynomials have been defined. Let G be...
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  • mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on...
    4 KB (801 words) - 03:25, 15 February 2023
  • discrete Chebyshev polynomial, and the corresponding eigenvalue, P ℓ ( cos ⁡ β ) {\displaystyle P_{\ell }(\cos \beta )} , is the Legendre polynomial....
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  • scaling the H ( s ) {\displaystyle H(s)} polynomials to the inverse of that frequency. To scale the polynomials, simply append ω c {\displaystyle \omega...
    18 KB (2,911 words) - 00:23, 24 May 2025