Spectral measures corresponding to orthogonal polynomials with unbounded recurrence coefficients
The author studies sequences of polynomials that satisfy recursion formulas of the form \[ P_1(\lambda)=1, \quad P_2(\lambda) = (\lambda-b_1)/a_1, \quad P_ n(\lambda) = [(\lambda - b_{n-1}) P_{n-1} (\lambda) - a_{n-2} P_{n-2} (\lambda)] /a_{n-1}, \quad n=3,4,\ldots, \] with \(a_ n\) and \(b_ n\) real and \(a_ n>0\). It is known that if \(\{a_ n\}\) and \(\{b_ n\}\) are bounded or if lim \(a_ n=\infty\) and \(\sum^{\infty}_{n=1}1/a_ n=\infty\), then there exists a unique measure \(\mu\), supported on the real line, with respect to which the polynomials are orthogonal. Properties of \(\mu\) have been studied in the past by various authors. In the present paper the author continues this study for the case that lim \(a_ n=\infty\), \(\sum^{\infty}_{n=1}1/a_ n=\infty\), and lim \(b_ n=b\). The plan of the paper is to associate with the system of polynomials a tridiagonal matrix which represents an unbounded selfadjoint operator C in \(\ell^ 2\). C is unitarily equivalent to the multiplication oprator in \(L^ 2(\mu)\), where \(\mu\) is the measure mentioned above. Additional conditions are imposed on \(\{a_ n\}\) and \(\{b_ n\}\) which provide information about the spectral measure E(\(\lambda)\) of C and hence about \(\mu\). Following is an example of the type of result that is proved: Let \(d_ n=| a_ n-a_{n-1}|\). Suppose that \(\sum^{\infty}_{n=2}[a^ 2_ n-a^ 2_{n-1}]^- < \infty\), \(\sum^\infty_{n=2} | d_ n-d_{n-1}| < \infty,\) and \(\sum^\infty_{n=2} a_{n-1} | b_ n-b_{n-1}| < \infty\). (Here, \([x]^- = ([x]-x)/2\).) Then, \(\mu\) has an absolutely continuous part with support sp(C). The case \(b_ n=0\) has been previously studied by the author [Proc. Am. Math. Soc. 100, No.3, 457-463 (1987); SIAM J. Math. Anal. 19, No.4, 939-943 (1988)].
unitarily equivalent to the multiplication oprator
scientific article; zbMATH DE number 4111372