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1. Orthogonal polynomials: A system of polynomials $ \{ P _ {n} \} $ which satisfy the condition of orthogonality $$ \int\limits _ { a } ^ { b } P _ {n} ( x) P _ {m} ( x) h( x) dx = 0,\ \ n \neq m, $$ whereby the degree of every polynomial $ P ... (Mathematics) [100%] 2024-01-08
2. Orthogonal polynomials: 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 ... (Set of polynomials where any two are orthogonal to each other) [100%] 2024-12-01 [Orthogonal polynomials] [Articles containing proofs]...
3. Orthogonal polynomials on a complex domain: The general name for polynomials orthogonal on the circle, over a contour or over an area. Unlike the case of orthogonality in a real domain, the polynomials of the three kinds of systems mentioned can have imaginary coefficients and are ... (Mathematics) [57%] 2024-01-12
4. Classical orthogonal polynomials: The general term for Jacobi polynomials; Hermite polynomials; and Laguerre polynomials. These systems of orthogonal polynomials have the following properties in common: 1) The weight function $ \phi ( x) $ on the interval of orthogonality $ ( a , b ) $ satisfies the Pearson differential equation ... (Mathematics) [81%] 2023-09-10
5. Classical orthogonal polynomials: In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications in such ... (Type of orthogonal polynomials) [81%] 2023-07-14 [Articles containing proofs] [Orthogonal polynomials]...
6. Fourier series in orthogonal polynomials: A series of the form $$\sum_{n=0}^\infty a_nP_n\label{1}\tag{1}$$ where the polynomials $\{P_n\}$ are orthonormal on an interval $(a,b)$ with weight function $h$ (see Orthogonal polynomials) and the coefficients $\{a_n\}$ are calculated from the ... (Mathematics) [63%] 2023-10-12