L-orthogonal polynomials associated with related measures (original) (raw)
Abstract
A positive measure ψ defined on [a, b] such that its moments μ n = b a t n dψ(t) exist for n = 0, ±1, ±2,. .. , can be called a strong positive measure on [a, b]. When 0 a < b ∞, the sequence of polynomials {Q n } defined by b a t −n+s Q n (t) dψ(t) = 0, s = 0, 1,. .. ,n − 1, exist and they are referred here as L-orthogonal polynomials. We look at the connection between two sequences of L-orthogonal polynomials {Q (1) n } and {Q (0) n } associated with two closely related strong positive measures ψ 1 and ψ 0 defined on [a, b]. To be precise, the measures are related to each other by (t − κ) dψ 1 (t) = γ dψ 0 (t), where (t − κ)/γ is positive when t ∈ (a, b). As applications of our study, numerical generation of new L-orthogonal polynomials and monotonicity properties of the zeros of a certain class of L-orthogonal polynomials are looked at.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.
References (21)
- M.I. Bueno Cachadina, A. Deaño, E. Tavernetti, A new algorithm for computing the Geronimus transformation with large shifts, Numer. Algorithms 54 (2010) 101-139.
- A. Bultheel, C. Díaz-Mendoza, P. González-Vera, R. Orive, Orthogonal Laurent polynomials and quadrature formulas for unbounded intervals: I. Gauss- type formulas, Rocky Mountain J. Math. 33 (2003) 585-608.
- T.S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its Applications Series, Gordon and Breach, New York, 1978.
- D.K. Dimitrov, A. Sri Ranga, Monotonicity of zeros of orthogonal Laurent polynomials, Methods Appl. Anal. 9 (2002) 1-12.
- A.K. Common, J.H. McCabe, The symmetric strong moment problem, J. Comput. Appl. Math. 67 (1996) 327-341.
- S.C. Cooper, W.B. Jones, W.J. Thron, Asymptotics of orthogonal L-polynomials for log-normal distributions, Constr. Approx. 8 (1992) 59-67.
- D.K. Dimitrov, M.V. de Mello, F.R. Rafaeli, Monotonicity of zeros of Jacobi-Sobolev-type orthogonal polynomials, Appl. Numer. Math. 60 (2010) 236-276.
- W.L. Ferrar, Integral Calculus, Oxford Univ. Press, London, 1958.
- M.E.H. Ismail, Monotonicity of zeros of orthogonal polynomials, in: D. Stanton (Ed.), q-Series and Partitions, Springer-Verlag, New York, 1989, pp. 177- 190.
- W.B. Jones, O. Njåstad, W.J. Thron, Two point Padé expansions for a family of analytic functions, J. Comput. Appl. Math. 9 (1983) 105-123.
- W.B. Jones, W.J. Thron, Two-point Padé tables and T-fractions, Bull. Amer. Math. Soc. 83 (1977) 388-390.
- W.B. Jones, W.J. Thron, H. Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc. 261 (1980) 503-528.
- A. Markov, Sur les racines de certain équations (second note), Math. Ann. 27 (1886) 177-182.
- J.H. McCabe, A formal extension of the Padé table to include two point Padé quotients, J. Inst. Math. Appl. 15 (1975) 363-372.
- P.I. Pastro, Orthogonal polynomials and some q-beta integrals of Ramanujan, J. Math. Anal. Appl. 112 (1985) 517-540.
- F.R. Rafaeli, Zeros de polinômios ortogonais na reta real, Tese de Doutorado at the Universidade Estadual de Campinas, SP, Brazil, 2010.
- A. Sri Ranga, Another quadrature rule of highest algebraic degree of precision, Numer. Math. 68 (1994) 283-294.
- A. Sri Ranga, Symmetric orthogonal polynomials and the associated orthogonal L-polynomials, Proc. Amer. Math. Soc. 123 (1995) 3135-3141.
- A. Sri Ranga, W. Van Assche, Blumenthal's theorem for Laurent orthogonal polynomials, J. Approx. Theory 117 (2002) 255-278.
- D.V. Widder, Advance Calculus, Dover, New York, 1989.
- A. Zhedanov, The "classical" Laurent biorthogonal polynomials, J. Comput. Appl. Math. 98 (1998) 121-147.