Orthogonal Polynomials On Ellipses And Their Recurrence Relations

Abstract In this note we study the connection between orthogonal polynomials on an ellipse and orthogonal Laurent polynomials on the unit circle relative to some multiplicative measures and then establish the recurrence relations for orthogonal polynomials on an ellipse. The matrix representation of the operator of multiplication by coordinate function is obtained


Introduction
Orthogonal polynomials on ellipses have been studied to a much less extent than their counterparts on the real line or even those on the unit circle.In recent decades there has been some work concerning orthogonal polynomials on ellipses.In a study of the invariant subspaces of a three-diagonal Toeplitz operator T , Duren [3,4] used a sequence tp n pλqu of orthogonal polynomials with respect to a measure ωpλq| dλ|, where ωpλq ≥ 0 and | dλ| is the arc-length measure on some ellipse E T , to describe the lattice of invariant subspaces of the three-diagonal operator T. Such polynomials were obtained as part of the computation of the point spectrum of the operator T and they turn out to satisfy the three-term recurrence equation (1) p n`1 pλq " λ p n pλq ´b p n´1 pλq.
In [5], Duren proved that, relative to a measure ωpλq | dλ| on an analytic curve C in which ωpλq is a non-negative function, if the corresponding orthogonal polynomials satisfy a tree-term recurrence equation of the form (2) p n`1 pλq " pα n λ `βn qp n pλq `γn p n´1 pλq, then the curve C is an ellipse.
The orthogonal polynomials with respect to the harmonic measure along the boundary of a Caratheodory domain were studied by Dovgoshei [2] and were proved to satisfy the three-term recurrence equation ( 2) if and only if the curve is an ellipse.
More recently, Putinar and Stylianopoulos proved in [7] that, under some natural hypothesis, ellipses are the most general curves associated with a finite-term recurrence.We need to recall some terminology.A sequence of orthogonal polynomials with respect to a measure µ satisfies a finite-term recurrence if for every k ≥ 0, there exists an N pkq ≥ 0 such that a k,n :" xλp n pλq, p k pλqy µ " 0, n ≥ N pkq, while the sequence satisfies an pN `1q-term recurrence if Obviously, polynomials satisfying an pN `1q-term recurrence also satisfy a finite-term recurrence.
Thus Putinar and Stylianopoulos proved that for certain domains Ω, the corresponding Bergman orthogonal polynomials (that is, orthogonal polynomials with respect to the area measure), as well as Szegö polynomials (orthogonal polynomials with respect to the arc-length measure on BΩ), satisfy a finite-term recurrence if and only if Dirichlet's problem for Ω with polynomial data on BΩ has a polynomial solution.As a consequence, they proved that if Bergman orthogonal polynomials on a Caratheodory domain Ω satisfy a three-term recurrence relation, then Ω must be an ellipsoid (that is a domain whose boundary is an ellipse), as well as: if BΩ is a subset of tpx, yq P R 2 : ψpx, yq " 0u, where ψ is a polynomial with bounded zero set and if the Bergman orthogonal polynomials satisfy a finite-term recurrence, then the domain is an ellipsoid.A result along the same line was obtained by Khavinson and Stylianopoulos in [6], namely if Bergman orthogonal polynomials on a domain Ω with "nice" boundary satisfy an pN `1q-term recurrence with N ≥ 2, then that the domain is an ellipsoid and N " 2.
In the present note, we begin a study of orthogonal polynomials with respect to a measure supported by an ellipse, which satisfies a certain multiplicativity property.The main goal of this note is to obtain the recurrence equations of such polynomials with respect to a finite positive definite Borel measure by studying their connection with the Laurent polynomials on the unit circle with respect to a corresponding measure via a certain transformation.
It is well known that orthogonal polynomials on the real line with respect to a non-negative measure µ satisfy a three-term recurrence equation and the matrix representation of the operator of multiplication by coordinate function gives rise to a symmetric three-diagonal matrix (traditionally called a Jacobi matrix), which is a bounded self-adjoint operator.In the case of the unit circle, the polynomials are dense in L 2 pµq if the measure µ satisfies a certain property (Szegö condition), and thus the multiplication operator by coordinate function restricted to P 2 pµq (the closure of the polynomials in L 2 pµq-norm) is a subnormal operator whose matrix representation is a Hessenberg (a matrix whose only nonzero sub-diagonal is the one right below main diagonal and all entries of that sub-diagonal are equal to 1).During last decade it has been proven that the sequence of monic Laurent polynomials obtained by applying the Gramm-Schmidt procedure to the sequence 1, 1 z , z, 1 z 2 , z 2 , . . .satisfies a five-term recurrence equation and the operator of multiplication by coordinate function has a "staircase" matrix representation, called CMV matrix, (e.g., cf.[1], [8], [10]).Our goal here is to obtain the equivalent of the CMV matrix for orthogonal polynomials on ellipses.

Recurrence equations and matrix representation
For r ą 0, let φ r pzq :" z `r z , T " tz : |z| " 1u and let E r :" φ r pTq be an ellipse of foci ˘2? r and x-intercepts ˘p1 `rq.For a finite non-negative Borel measure ξ with infinite support on E r , let P n pλq, n P N be the sequence of unique monic polynomials such that the degree of P n pλq is n and xP n pλq, P m pλqy ξ " µ ´2 n δ nm , µ n ą 0, where xf pλq, gpλqy ξ " ş Er f pλq ¨gpλq dξpλq.We can assume that ξpE r q " 1 since monic orthogonal polynomials will be the same with respect to the normalized measure.Substituting λ P E r with z `r z , z P T, there exists a finite non-negative Borel measure µ ξ on T (it will be denoted in what follows by µ) such that xP n pλq, P m pλqy ξ " xΦ 2n pzq, Φ 2m pzqy µ , where and of course xhpzq, kpzqy µ :" ş T hpzq ¨kpzq dµpzq.Thus, Φ 2n pzq is a Laurent polynomial in which the exponents of z vary between ´n and n.We define for n ≥ 1 where Φ 2n p¨q is the Laurent polynomial Φ 2n p¨q in which the complex conjugate is applied only to its coefficients; thus for z P T, Φ 2n´1 pzq " Φ 2n pzq V. Lauric and consequently, xΦ 2n pzq, Φ 2m pzqy µ " xΦ 2n´1 pzq, Φ 2m´1 pzqy µ " µ ´2 n δ nm .
We will be interested in measures µ that will give rise to a sequence tΦ k pzqu 8  k"0 of orthogonal Laurent polynomials.
Proposition 1.The sequence tΦ k pzqu 8 k"0 is orthogonal with respect to the measure µ if and only if the measure µ is multiplicative on the sequence tΦ 2n pzqu 8  n"0 , that is, ż Proof.Assume first that the measure µ is multiplicative on the sequence tΦ 2n pzqu 8 n"0 .This is equivalent to xΦ 2n pzq, Φ 2m´1 pzqy µ " 0, for n ≥ 0, m ≥ 1.Since the sequence tΦ 2n pzqu 8  n"0 is orthogonal (by construction) and the measure µ is non-negative, it implies the entire sequence tΦ k pzqu 8 k"0 is orthogonal.
Conversely, if the sequence tΦ k pzqu 8 k"0 is orthogonal, then A question that arises naturally is whether such measures exist.Obviously, Dirac measures δ z , z P T are multiplicative measures.On the other hand, since the conversion between the measure ξ on the ellipse and the corresponding measure µ on the unit circle preserves the multiplicativity property, then the measure, say µ 0 , that arises from the harmonic measure ξ 0 , is another example of such measure.
An interesting and useful question is to describe all measures that satisfy such multiplicative property, but it is not the purpose of the note.
The remainder of this note will be a narrative construction in which we obtain the recurrence equations and a CMV-type of matrix representation of the operator of multiplication by coordinate function and will be concluded with a formal statement.
The assumption that we will make in the remainder of the note is that the measure µ is multiplicative in the sense stated in Proposition 1.
We define a sequence of standard Laurent polynomials that will be used to construct some subspaces, as follows: Let H 0 :" _tl 0 u " _tΦ 0 u, H 1 :" _tl 0 , l 1 u " _tΦ 0 , Φ 1 u, and in general and where the symbol _ denotes the linear span generated by finitely many vectors.Since Φ 2n pzq arises from a monic polynomial of degree n in variable λ, one can write α n´k,n l n´k pzq.
Remark.With some standard arguments, the multipicativity property of a measure µ on the sequence of the orthogonal Laurent polynomials generated by µ can be extended to the entire space L 2 pµq.
With the notation used above, we summarize this note with the following.
Theorem.The matrix representation of the operator M l 1 : L 2 pµq Ñ L 2 pµq defined by pM l 1 hqpzq " l 1 pzq hpzq with respect to the orthonormal basis tφ k pzqu k≥0 that arises from a multiplicative measure µ on on L 2 pµq, has the following form: ¨α0,0 Of course, the same representation is valid for the operator M λ : L 2 pξq Ñ L 2 pξq with respect to the orthonormal basis p that arises from a multiplicative measure ξ on E r .