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21: 1.16 Distributions
§1.16(i) Test Functions
The closure of the set of points where ϕ 0 is called the support of ϕ . If the support of ϕ is a compact set (§1.9(vii)), then ϕ is called a function of compact support. A test function is an infinitely differentiable function of compact support. …
22: 18.33 Polynomials Orthogonal on the Unit Circle
Let μ be a probability measure on the unit circle of which the support is an infinite set. … This states that for any sequence { α n } n = 0 with α n and | α n | < 1 the polynomials Φ n ( z ) generated by the recurrence relations (18.33.23), (18.33.24) with Φ 0 ( z ) = 1 satisfy the orthogonality relation (18.33.17) for a unique probability measure μ with infinite support on the unit circle. …
23: 18.2 General Orthogonal Polynomials
Nevai (1979, p.39) defined the class 𝒮 of orthogonality measures with support inside [ 1 , 1 ] such that the absolutely continuous part w ( x ) d x has w in the Szegő class 𝒢 . … If d μ 𝐌 ( a , b ) then the interval [ b a , b + a ] is included in the support of d μ , and outside [ b a , b + a ] the measure d μ only has discrete mass points x k such that b ± a are the only possible limit points of the sequence { x k } , see Máté et al. (1991, Theorem 10). … for x , y in the support of the orthogonality measure and z such that the series in (18.2.41) converges absolutely for all these x , y . …
24: Bibliography M
  • A. Máté, P. Nevai, and W. Van Assche (1991) The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.
  • 25: Bibliography K
  • C. Kormanyos (2011) Algorithm 910: a portable C++ multiple-precision system for special-function calculations. ACM Trans. Math. Software 37 (4), pp. Art. 45, 27.
  • 26: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    For 𝒟 ( T ) we can take C 2 ( X ) , with appropriate boundary conditions, and with compact support if X is bounded, which space is dense in L 2 ( X ) , and for X unbounded require that possible non- L 2 eigenfunctions of (1.18.28), with real eigenvalues, are non-zero but bounded on open intervals, including ± . …
    27: Errata
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