Trace formula for orthogonal polynomials with asymptotically 2-periodic recurrence coefficients
In this interesting paper, the author considers orthonormal polynomials with asymptotically periodic recurrence coefficients. Let \(\mu\) be a probability measure on \([-1,1]\) and let \(p_n(x)\) denote the \(n\)th orthonormal polynomial for \(\mu\), so that there is the three term recurrence relation \[ xp_n(x)= a_{n+1} p_{n+1}(x)+ b_n p_n(x)+ a_n p_{n-1}(x),\quad n\geq 1, \] with the initial conditions \(p_{-1}(x)= 0\); \(p_0(x)= 1\); \(a_0= 0\). Suppose that we are given two-periodic sequences \(\{a^0_n\}\), \(\{b^0_n\}\), that is \[ a^0_{n+2}= a^0_n;\quad b^0_{n+2}= b^0_n,\quad n\geq 0. \] We say that the recurrence coefficients \(\{a_n\}\), \(\{b_n\}\) are asymptotically two-periodic if \[ \lim_{n\to\infty} | a_n- a^0_n|= 0= \lim_{n\to\infty} | b_n- b^0_n|. \] The author extends a classical trace formula of P. Nevai and others, that dealt with recurrence coefficients that have a limit (and lie in the so-called Nevai class \(M\)) to asymptotically two-periodic recurrence coefficients. This was a problem posed by P. Nevai. The exact formulation involves a fair number of auxiliary quantities, but has roughly the following form: let \[ \sum^\infty_{n=0} (| a_{n+2}- a_n|+| b_{n+2}- b_n|)< \infty;\quad \sum^\infty_{n=0} (| b_{n-1}+ b_n- b^0_0- b^0_1|)< \infty. \] Then \[ \sum^\infty_{n=0} (A_n p^2_n(x)+ B_n p_n(x) p_{n+1}(x)+ C_n p_n(x) p_{n+2}(x)+ D_n p_n(x) p_{n+3}(x))= F(x), \] in a certain set that is, essentially, the support of \(\mu\). Here \(F(x)\) is an explicitly given function involving \(\mu'(x)\) and the recurrence coefficients, while \(A_n\), \(B_n\), \(C_n\), \(D_n\) are explicit polynomials involving \(\{a_n\}\), \(\{a^0_n\}\), \(\{b_n\}\), \(\{b^0_n\}\).
asymptotically two-periodic recurrence coefficients
scientific article; zbMATH DE number 1197040
scientific article