Orthogonal polynomials

In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative "weight function" w precisely if

Related Topics:
Mathematics - Polynomials

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:int_{x_1}^{x_2} f(x)g(x)w(x),dx=0.

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In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as

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:langle f,g angle=int_{x_1}^{x_2} f(x)g(x),w(x),dx

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then the orthogonal polynomials are simply orthogonal vectors in this inner product space.

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A polynomial sequence pn(x) for n = 0, 1, 2, ... , where pn(x) has degree n, is said to be a sequence of orthogonal polynomials with respect to a "weight function" w when any two of them are orthogonal with respect to that weight function, i.e.,

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:langle p_n, p_m angle=int_{x_1}^{x_2} p_n(x) p_m(x),w(x),dx=0 mbox{whenever} n eq m.

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The sequence of orthogonal polynomials can be successively constructed by carrying out the Gram-Schmidt process with the sequence of powers x^k, ; k ge 0, where the positive-definite inner product langle p,q angle on the space of polynomials is given by the integral above.

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