Stieltjes Transformation - Relationships To Orthogonal Polynomials

Relationships To Orthogonal Polynomials

The correspondence defines an inner product on the space of continuous functions on the interval I.

If {Pn} is a sequence of orthogonal polynomials for this product, we can create the sequence of associated secondary polynomials by the formula

It appears that is a Padé approximation of Sρ(z) in a neighbourhood of infinity, in the sense that

Since these two sequences of polynomials satisfy the same recurrence relation in three terms, we can develop a continued fraction for the Stieltjes transformation whose successive convergents are the fractions Fn(z).

The Stieltjes transformation can also be used to construct from the density ρ an effective measure for transforming the secondary polynomials into an orthogonal system. (For more details see the article secondary measure.)

Read more about this topic:  Stieltjes Transformation

Famous quotes containing the words relationships to:

    With only one life to live we can’t afford to live it only for itself. Somehow we must each for himself, find the way in which we can make our individual lives fit into the pattern of all the lives which surround it. We must establish our own relationships to the whole. And each must do it in his own way, using his own talents, relying on his own integrity and strength, climbing his own road to his own summit.
    Hortense Odlum (1892–?)