#### Zero distribution of orthogonal polynomials직교다항식의 근의 분포

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In [5], Karlin and Szego conjectured : If \${P_n(x)}^∞_{n=0}\$ is an orthogonal polynomial system and \${P``_{n}(x)}^∞_{n=1}\$ is a Sturm sequence, then \${P_n(x)}^∞_{n=0}\$ is essentially (that is, after a linear change of variable) a classical orthogonal polynomial system of Jacobi, Laguerre, or Hermite. Here, we prove that for any orthogonal polynomial system \${P_n(x)}^∞_{n=0}\$, \${P``_{n}(x)}^∞_{n=1}\$ is always a Sturm sequence. Thus, in particular, the above conjecture by Karlin and Szeg\$ö\$ is false. And zero distribution of the polynomials \${S_{n}(x)}^∞_{n=0}\$ which are orthogonal with respect to the discrete Sobolev inner product type. ＜f,g> = ∫_If(x)g(x)dμ(x)+\$Nf^{(r)}(c)g^{(r)}(c)\$ where I is a real interval, with I = [a,b], a,b ∈ R, N ≥ 0, r ≥ 1, c ≥ b and let μ(x) be an arbitrary distribution function on I. \$S_{n}(x)\$ has n real, simple zeros. The location of these zeros is given in relation to the position of the zeros of some classical polynomials (i.e., polynomials with respect to an inner product with N=0).
Kwon, Kil-Hunresearcher권길헌researcher
Description
한국과학기술원 : 수학과,
Publisher
한국과학기술원
Issue Date
1995
Identifier
98711/325007 / 000933045
Language
eng
Description

학위논문(석사) - 한국과학기술원 : 수학과, 1995.2, [ [ii], 20 p. ]

URI
http://hdl.handle.net/10203/42398