By Theodore S Chihara, Mathematics
Topics comprise the illustration theorem and distribution features, persevered fractions and chain sequences, the recurrence formulation and houses of orthogonal polynomials, certain capabilities, and a few particular platforms of orthogonal polynomials. various examples and routines, an in depth bibliography, and a desk of recurrence formulation complement the text.
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Extra info for An introduction to orthogonal polynomials
A(j)u(λ, j)u(λ, j + 1) a(0)(φ+ (λ) − φ(λ)) 36 2. 64) v(λ, n) = u(λ, n) 1 >0 −a(j)u(λ, j)u(λ, j + 1) j=n solves τ u = λu (cf. 51)) and equals u+ (λ, n) up to a constant multiple since limn→∞ v(λ, n)/u(λ, n) = 0. By reflection we obtain a corresponding minimal positive solution u− (λ, n) near −∞. Let us summarize some of the results obtained thus far. 9. Suppose a(n) < 0, λ ≤ σ(H) and let u(λ, n) be a solution with u(λ, n) > 0, ±n ≥ 0. Then the following conditions are equivalent. (i). u(λ, n) is minimal near ±∞.
Let us summarize some of the results obtained thus far. 9. Suppose a(n) < 0, λ ≤ σ(H) and let u(λ, n) be a solution with u(λ, n) > 0, ±n ≥ 0. Then the following conditions are equivalent. (i). u(λ, n) is minimal near ±∞. (ii). 65) ±n ≥ 0, for any solution v(λ, n) with v(λ, n) > 0, ±n ≥ 0. (iii). 66) lim n→±∞ u(λ, n) = 0. v(λ, n) for one solution v(λ, n) with v(λ, n) > 0, ±n ≥ 0. (iv). 67) j∈±N 1 = ∞. −a(j)u(j)u(j + 1) Recall that minimality says that for a solution u(λ, n) with u(λ, 0) = 1, u(λ, 1) = φ(λ) to be positive on N, we need φ(λ) ≥ φ+ (λ).
By construction H+ = U ˜ −1 H ˜U ˜ is a bounded Jacobi operator associU ated with a(n), b(n). That ρ+ is the spectral measure of H+ follows from (using 44 2. 119) ˜ δ1 , PΛ (H) ˜ U ˜ δ1 = δ1 , PΛ (H+ )δ1 = U χΛ (λ)dρ+ (λ) R for any Borel set Λ ⊆ R. , C(N + 1) = 0) and we get a finite Jacobi matrix with dρ+ as spectral measure. 98) with m = 1). 122) G+ (z, n, m) = R s(λ, n)s(λ, m) dρ+ (λ). λ−z The Jacobi operator H can be treated along the same lines. Since we essentially repeat the analysis of H+ we will be more sketchy.