Ratio Asymptotics, Hessenberg Matrices, and Weak Asymptotic Measures

Abstract

We discuss the relationship between ratio asymptotics for general orthogonal polynomials and the asymptotics of the associated Bergman shift operator. More specifically, we consider the case in which a measure is supported on an infinite compact subset of the complex plane. We show that there is a straightforward connection between the corresponding orthonormal polynomials exhibiting ratio asymptotics and the corresponding Bergman shift operator being asymptotically Toeplitz. We also discuss a connection to the weak asymptotics of the measures derived from the orthonormal polynomials.

Determining system poles using row sequences of orthogonal Hermite–Padé approximants

“…The results of this paper remain valid when E is a compact set whose complement is connected, provided that the sequence of orthonormal polynomials (p n ), n ≥ 0, and second type functions (s n ), n ≥ 0, relative to the measure µ, supp(µ) ⊂ E, satisfy (9) and (10), respectively, inside C \ E, and the sequence of leading coefficients (κ n ), n ≥ 0, fulfill (11). On the right hand side of (9) and (10) one should place e g(z,∞) , where g(z, ∞) denotes Green's function relative to the region C \ E. The problem with stating the results with this degree of generality is related with the zeros that the second type functions s n may have in Co(E) \ E. For example, if E is made up of two intervals symmetric with respect to the origin and µ is any measure supported on E symmetric with respect to the origin then s n has a zeros at z = 0 for all even n. In this case, no matter how good the measure is, there are problems in proving (18) at ξ = 0 if this point happens to be a system pole. In this example, this can be avoided requiring that 0 is not a system pole of F. But, in a more general configuration, this is hard to guarantee in terms of the data since the zeros of s n in Co(E) \ E may have a quite exotic behavior.…”

Relative asymptotics for general orthogonal polynomials

“…Lemma 2.3 implies that the coefficient ((π n+j M µ π n+j ) m ) n+j,n+j appearing in (14) can be written as a finite sum of products of elements of this form. Therefore, the Laurent coefficients in (14) converge as n → ∞ through N , and hence we have the desired ratio asymptotics for the monic orthogonal polynomials. However, convergence to a right limit also implies convergence of the ratio κ n+j−1 κ −1 n+j to a (j-dependent) limit as n → ∞ through N .…”

“…Since we are assuming that each of these limits is non-zero, we know that each of the Laurent coefficients appearing in (14) converges as n → ∞ through N . If we examine the m = 1 term in (14), we also conclude that the following limits exist:…”

“…For more on right limits, we refer the reader to [15] and [20,Chapter 7]. The recent publications [14,15] shed some light on the relationship between right limits of the matrix M µ and ratio asymptotics for general orthonormal polynomials under the added condition that…”

Positive functionals and Hessenberg matrices

“…All presented from the traditional point of view of the functional analyst or function theorist. Hessenberg matrices however are cultivated by rather disjoint groups of mathematicians, notably in numerical analysis [6] and approximation theory [10].…”