Classical Orthogonal Polynomials - Definition

Definition

In general, the orthogonal polynomials Pn with respect to a weight W:R→R+ on the real line are defined by

\begin{align} &\deg P_n = n~, \quad n = 0,1,2,\ldots\\ &\int P_m(x) \, P_n(x) \, W(x)\,dx = 0~, \quad m \neq n~. \end{align}

The relations above define Pn up to multiplication by a number. Various normalisations are used to fix the constant, e.g.

The classical orthogonal polynomials correspond to the three families of weights:

\begin{align} \text{(Jacobi)}\quad &W(x) = \begin{cases} (1 - x)^\alpha (1+x)^\beta~, & -1 \leq x \leq 1 \\ 0~, &\text{otherwise} \end{cases} \\ \text{(Hermite)}\quad &W(x) = \exp(- x^2) \\ \text{(Laguerre)}\quad &W(x) = \begin{cases} x^\alpha \exp(- x)~, &\quad x \geq 0 \\ 0~, &\text{otherwise} \end{cases} \end{align}

The standard normalisation (also called standardization) is detailed below.

Read more about this topic:  Classical Orthogonal Polynomials

Famous quotes containing the word definition:

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
    —William James (1842–1910)