Fine Structure of the Zeros of Orthogonal Polynomials, I. a Tale of Two Pictures
نویسنده
چکیده
FINE STRUCTURE OF THE ZEROS OF ORTHOGONAL POLYNOMIALS, I. A TALE OF TWO PICTURES BARRY SIMON Dedicated to Ed Saff on the occasion of his 60th birthday Abstract. Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large . Motivated by two plots, we look at the finer structure for the case of random Verblunsky coefficients and for what we call the BLS condition: . In the former case, we describe results of Stoiciu. In the latter case, we prove asymptotically equal spacing for the bulk of zeros.
منابع مشابه
Domain of attraction of normal law and zeros of random polynomials
Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...
متن کاملFine Structure of the Zeros of Orthogonal Polynomials, Iii. Periodic Recursion Coefficients
We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings inverse to the density of zeros. Zeros away from the a.c. spectrum have limit points mod p and only finitely many of them.
متن کاملRecurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials
Suppose that for an arbitrary function $f(x,y)$ of two discrete variables, we have the formal expansions. [f(x,y)=sumlimits_{m,n=0}^{infty }a_{m,n},P_{m}(x)P_{n}(y),] $$ x^{m}P_{j}(x)=sumlimits_{n=0}^{2m}a_{m,,n}(j)P_{j+m-n}(x),$$ we find the coefficients $b_{i,j}^{(p,q,ell ,,r)}$ in the expansion $$ x^{ell }y^{r},nabla _{x}^{p}nabla _{y}^{q},f(x,y)=x^{ell }y^{r}f^{(p,q)}(x,y) =sumli...
متن کاملAnalysis of High-order Approximations by Spectral Interpolation Applied to One- and Two-dimensional Finite Element Method
The implementation of high-order (spectral) approximations associated with FEM is an approach to overcome the difficulties encountered in the numerical analysis of complex problems. This paper proposes the use of the spectral finite element method, originally developed for computational fluid dynamics problems, to achieve improved solutions for these types of problems. Here, the interpolation n...
متن کاملFine Structure of the Zeros of Orthogonal Polynomials, Iv. a Priori Bounds and Clock Behavior Yoram Last and Barry Simon
We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrödinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.
متن کامل