Summary: Measures, orthogonal polynomials,
and continued fractions
Michael Anshelevich
November 7, 2008
MEASURES AND ORTHOGONAL POLYNOMIALS.
µ a positive measure on R.
A linear functional µ[P] = P(x) dµ(x),
µ : R[x] R.
Positive:
µ[P(x)2] 0.
Inner product P, Q = µ[PQ].
Gram-Schmidt 1, x, x2, x3, . . .
monic orthogonal polynomials
{P0 = 1, P1, P2, P3, . . .}.
1
Theorem. (Favard, Stone, etc.) For some i R, i 0
xPn = Pn+1 + nPn + nPn-1.
2nd order recursion relation.
Two independent solutions {Pn, Qn}.
Initial conditions

Source: Anshelevich, Michael - Department of Mathematics, Texas A&M University

Collections: Mathematics