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Measures, orthogonal polynomials, and continued fractions
 

Summary: Measures, orthogonal polynomials,
and continued fractions
Michael Anshelevich
November 7, 2008
MEASURES AND ORTHOGONAL POLYNOMIALS.
a positive measure on R.
A linear functional [P] = P(x) d(x),
: R[x] R.
Positive:
[P(x)2] 0.
Inner product P, Q = [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