(M,N)-coherent pairs of order(m,k)and Sobolev orthogonal

Abstract

a b s t r a c tA pair of regular linear functionals (U, V) is said to be a (M, N)-coherent pair of order (m, k) if their corresponding sequences of monic orthogonal polynomials {P n (x)} n≥0 and {Q n (x)} n≥0 satisfy a structure relation such as To be more precise, let us assume that m ≥ k. If m = k then U and V are related by a rational factor (in the distributional sense); if m > k then U and V are semiclassical and they are again related by a rational factor. In the second part of this work we deal with a Sobolev type inner product defined in the linear space of polynomials with real coefficients, P, aswhere λ is a positive real number, m is a positive integer number and (μ 0 , μ 1 ) is a (M, N)-coherent pair of order m of positive Borel measures supported on an infinite subset of the real line, meaning that the sequences of monic orthogonal polynomials {P n (x)} n≥0 and {Q n (x)} n≥0 with respect to μ 0 and μ 1 , respectively, satisfy a structure relation as above with k = 0, a i,n and b i,n being real numbers fulfilling the above mentioned conditions. We generalize several recent results known in the literature in the framework of Sobolev orthogonal polynomials and their connections with coherent pairs (introduced in [A. Iserles, P.E. Koch, S.P. Nørsett, J.M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products J. Approx. Theory 65 (2) (1991) 151-175]) and their extensions. In particular, we show how to compute the coefficients of the Fourier expansion of functions on an appropriate Sobolev space (defined by the above inner product) in terms of the sequence of Sobolev orthogonal polynomials {S n (x; λ)} n≥0 .

A matrix approach for the semiclassical and coherent orthogonal polynomials

“…In addition, they showed that the linear functionals U and V satisfy the relation rðxÞU .ðxÞV, where rðxÞ and .ðxÞ non zero polynomials such that degðrðxÞÞ 3 and degð.ðxÞÞ 1. Finally, in a recent work by de Jesús et al [5], the more general case for coherence was characterized. They studied the structure relation …”

(M,N)-Coherent pairs of linear functionals and Jacobi matrices

“…In [9], it was proved that if ðU; VÞ is an ðM; NÞ coherent pair of order m; m P 1, of positive definite linear functionals such that l 0 and l 1 are their corresponding positive Borel measures, then the SMOP fP n ðxÞg nP0 and fS n ðx; kÞg nP0 with respect to U and h ; i k , respectively, satisfy the algebraic relation (1.2). So, using the notation e a i;n ðn þ 1Þ m ðn i þ 1Þ m a i;n ; n P 0;…”

“…Indeed, when the linear functionals U and V are positive definite and l 0 and l 1 are the positive Borel measures associated with them, respectively, de Jesus, Marcellán, Petronilho, and Pinzón Cortés [9] showed that the ðM; NÞ coherence relation of order m (1.1), m P 1, implies…”

“…The algebraic relations in (1.2) allow to study analytic properties of Sobolev orthogonal polynomials (see [9]). In this paper, we will study the ðM; NÞ coherence relations from a matrix point of view.…”

“…The most general notion of coherent pair was introduced and studied by de Jesus, Marcellán, Petronilho, and Pinzón Cortés in [9], which includes, as a particular case, the concept of ðM; NÞ coherent pair of order m.…”