Definition
In general, the orthogonal polynomials Pn with respect to a weight W:R→R+ on the real line are defined by
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:
The standard normalisation (also called standardization) is detailed below.
Read more about this topic: Classical Orthogonal Polynomials
Famous quotes containing the word definition:
“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 (18421910)
“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 possessafter many mysterieswhat one loves.”
—François, Duc De La Rochefoucauld (16131680)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lensif we are unaware that women even have a historywe live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)