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In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials , consisting of the Hermite polynomials , the Laguerre polynomials and ...
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets for wavelet transform analysis; probability, such as the Edgeworth series, as well as in connection with Brownian motion; combinatorics, as an example of an Appell sequence, obeying the umbral ...
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections ...
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike , laureate of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy , they play important roles in various optics branches such as beam optics and imaging.
Several orthogonal polynomials, including Jacobi polynomials P (α,β) n and their special cases Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials, Zernike polynomials can be written in terms of hypergeometric functions using (, + + +; +;) =!
Let (()) = be a sequence of orthogonal polynomials defined on the interval [,] satisfying the orthogonality condition () =,, where () is a suitable weight function, is a constant depending on , and , is the Kronecker delta.
In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel and Jean Gaston Darboux .
It is an extension of the Rademacher system of orthogonal functions. [2] Walsh functions, the Walsh system, the Walsh series, [3] and the fast Walsh–Hadamard transform are all named after the American mathematician Joseph L. Walsh. They find various applications in physics and engineering when analyzing digital signals.