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:

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)

    “According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
    —Ana Castillo (b. 1953)

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)