A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula
Nenhuma Miniatura disponível
Data
2014-08-01
Autores
Castillo, K. [UNESP]
Costa, M. S.
Ranga, A. Sri [UNESP]
Veronese, D. O.
Título da Revista
ISSN da Revista
Título de Volume
Editor
Elsevier B.V.
Resumo
The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formulaRn+1(Z) = [(1 + ic(n+i))z + (1 - ic(n+1))]R-n(z) - 4d(n+1)zR(n-1)(z), n >= 1,with R-0(z) = 1 and R-1(z) = (1 + ic(1))z + (1 - ic(1)), where {c(n)}(n=1)(infinity) is a real sequence and {d(n)}(n=1)(infinity) is a positive chain sequence. We establish that there exists a unique nontrivial probability measure mu on the unit circle for which {R-n(z) - 2(1 - m(n))Rn-1(Z)} gives the sequence of orthogonal polynomials. Here, {m(n)}(n=0)(infinity) is the minimal parameter sequence of the positive chain sequence {d(n)}(n=1)(infinity). The element d(1) of the chain sequence, which does not affect the polynomials R-n, has an influence in the derived probability measure mu and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if {M-n}(n=0)(infinity) is the maximal parameter sequence of the chain sequence, then the measure mu is such that M-0 is the size of its mass at z = 1. An example is also provided to completely illustrate the results obtained.
Descrição
Palavras-chave
Szegö polynomials, Kernel polynomials, Para-orthogonal polynomials, Chain sequences, Continued fractions
Como citar
Journal Of Approximation Theory. San Diego: Academic Press Inc Elsevier Science, v. 184, p. 146-162, 2014.