PARA-ORTHOGONAL POLYNOMIALS IN FREQUENCY ANALYSIS. 1. Introduction. By a trigonometric signal we mean an expression of the form.

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 33, Number 2, Summer 2003 PARA-ORTHOGONAL POLYNOMIALS IN FREQUENCY ANALYSIS LEYLA DARUIS, OLAV NJÅSTAD AND WALTER VAN ASSCHE 1. Introduction. By a trigonometric signal we mean an expression of the form I ( (1.1) x(m) = αj e imω j + α j e imω ) j, j=1 and we assume α j = α j,andω j = ω j (0,π)forj =1, 2,...,I. The constants α j represent amplitudes, the quantities ω j are frequencies, andm is discrete time. The frequency analysis problem is to determine the numbers {α j,ω j : j =1, 2,...,I}, andn 0 =2I when values {x(m) : m = 0, 1,..., N 1} (observations) are known. The Wiener-Levinson method, formulated in terms of Szegőpoly- nomials, can briefly be described as follows (the original ideas of the method can be found in [12, 20]). An absolutely continuous measure ψ N is defined on [, π] (or on the unit circle T through the transformation θ z = e iθ ) by the formula dψ N (1.2) dθ = 1 N 1 2π x(m)e 2. imθ m=0 HereN is an arbitrary natural number. The measure gives rise to a positive definite inner product which determines a sequence {Φ n (ψ N,z): n = 0, 1, 2...} of monic orthogonal polynomials (Szegő polynomials). All the zeros of Φ n (ψ N,z) lie in the open unit disk. Let ϕ n (ψ N,z) be the orthonormal polynomials (with positive leading coefficient κ N n ) with respect to ψ N.Thenwehave (1.3) ϕ n (ψ N,z)=κ N n Φ n (ψ N,z), Received by the editors onseptember 30, The first author was partially supported by Laguna University under contract and by Ministerio de Ciencia y Technología del Gobierno Español under contract BF The third author s research is supported by INTAS and research grant G of FWO-Vlaanderen. Copyright c 2003 Rocky Mountain Mathematics Consortium 629

2 630 L. DARUIS, O. NJÅSTAD, W. VAN ASSCHE where ( n 1/2 (1.4) κ N n = (1 Φ j (ψ N, 0) )) 2. j=1 For the basic theory of Szegő polynomials, see e.g., [1, 2, 7, 18, 19]. Let {ζ k : k =1, 2,...,n 0 } be a numbering of the so-called frequency points {e iω j : j = ±1, ±2,...,±I}, andsetλ k = α j 2 for ζ k = e iω j. Let ψ be the discrete measure defined by n 0 (1.5) ψ(θ) = λ k δ(e iθ ζ k ). k=1 Then the measures ψ N /N converge in the weak sense to ψ (see [6, 15]). Forafixeddegreen, n n 0, every subsequence of {Φ n (ψ N,z):N = 1, 2,...} contains a subsequence converging to a polynomial of the form (1.6) P n (z) =Q n n0 (z) I (z e iω j )(z e iω j ), j=1 where Q n n0 (z) is a polynomial of degree n n 0. It follows that n 0 of the zeros of Φ n (ψ N,z), closest to the frequency points, converge to these frequency points (see, e.g., [6, 8, 9, 15]). Furthermore, for every n there is a constant K n < 1 such that n n 0 of the zeros of Φ n (ψ N,z) are contained in the disk { z K n } for all N, (see[13, 15] andalso[14] where more general orthogonal rational functions are used in frequency analysis problems). These properties make it possible to determine the number n 0 of frequency points and to localize these frequency points from the behavior of the zeros of Φ n (ψ N,z)asN increases. For a survey on the use of Szegő polynomials in frequency analysis, see [11]). See also [17] where a matrix approach is discussed. In this paper we shall sketch a different approach to the frequency analysis problem, which uses zeros of para-orthogonal polynomials instead of zeros of orthogonal polynomials. A para-orthogonal polynomial is a polynomial of the form (1.7) B n (ψ N,τ,z)=Φ n (ψ N,z)+τΦ n(ψ n,z), τ T,

3 PARA-ORTHOGONAL POLYNOMIALS 631 where Φ n(z) = z n Φ n (1/ z) is the reversed polynomial. For convenience we shall suppress the τ in all notation when we are considering a fixed value of τ. The polynomial B n (ψ N,z)hasn simple zeros z1 N,z2 N,...,zn N, all lying on T. The following convergence result will be fundamental in the sequel. A proof can be found in [10]. Theorem 1.1. Let {N k : k =1, 2,...} be an arbitrary subsequence of the sequence of natural numbers, let τ be an arbitrary point on T, and let n n 0. Then there exists a subsequence {N k(ν) } and a polynomial W n n0 (z) of degree n n 0 such that (1.8) lim ν B n(ψ Nk(ν),z)=W n n0 (z) where ζ k are the frequency points. n 0 (z ζ k ), It follows that some of the zeros z1 N,z2 N,...,zn N of B n (ψ Nk(ν),z) converge to the frequency points, and the rest converge to zeros of W n n0 (z). A frequency point may also be a zero of W n n0 (z). We shall occasionally write B n (z) for the polynomial W n n0 (z) n 0 k=1 (z ζ k). We will use the weights in the Szegő quadrature formula for Szegő polynomials to distinguish the frequency points from the zeros of W n n0 (z). These quadrature weights will also determine the numbers λ k = α j 2 for ζ k = e iω j and therefore provide useful estimates of the modulus of the amplitudes in the signal. 2. General results The zeros z1 N,...,zn N of B n (ψ N,z) are nodes in a Szegő quadrature formula for Szegő polynomials with respect to the measure ψ N (θ)/n.theweightsλ N k are given by k=1 (2.1) λ N k = 1 N L N k (e iθ ) dψ N (θ), k =1, 2,...,n, where (2.2) L N (z z1 N k (z) = ) (z zn k 1 )(z zn k+1 ) (z zn n ) (zk N zn 1 ) (zn k zn k 1 )(zn k zn k+1 ) (zn k zn n )

4 632 L. DARUIS, O. NJÅSTAD, W. VAN ASSCHE are the fundamental polynomials of Lagrange interpolation (see [1, 2, 7]). The weights may also be expressed as (see [3, 5]) (2.3) λ N k = ( n 1 ) 1 ϕ j (ψ N,zk N ) 2. j=0 For more information on Szegő quadrature formulas, see [1, 2, 3, 5, 7]. In the following we shall always assume that n n 0. The following general result holds. Theorem 2.1. Let λ N k, k =1, 2,...,n,andλ m, m =1, 2,...,n 0, be defined as in the foregoing. Then (2.4) lim N k=1 n λ N k = n 0 m=1 λ m. Proof. On the one hand, since ψ N (θ)/n ψ(θ), we have lim N 1 π dψ N (θ) = dψ(θ) = N n 0 m=1 λ m. On the other hand, since the Szegő quadrature is exact for the function f(θ) 1, we have 1 π n dψ N (θ) = λ N k N for all N. From this the result follows. We note that the above result is valid irrespective of whether the sequence {B n (ψ N,z):N =1, 2,...} converges or not, or of whether the zeros of the limiting polynomials B n (z) of convergent sequences are simple or not. For the sake of completeness, we give a proof of a convergence result, which will be used when convergence of weights belonging to individual zeros is discussed. k=1

5 PARA-ORTHOGONAL POLYNOMIALS 633 Theorem 2.2. Assume that the sequence {f p : p = 1, 2,...} of continuous functions on T converges uniformly to f on T. Then 1 π (2.5) lim f p (e iθ ) dψ p (θ) = f(e iθ ) dψ(θ). p p Proof. Let ε>0. Since ψ p /p converges to ψ in the weak topology, we have 1 π f(e iθ ) dψ p (θ) f(e iθ ) dψ(θ) p < ε 2 for p sufficiently large. Since f p converges uniformly to f we have 1 π [f p (e iθ ) f(e iθ )] dψ p (θ) p < ε 2 for p sufficiently large, since all the measures ψ p /p have finite total mass 1 π n 0 dψ p (θ) = λ k. p k=1 It follows that 1 π f p (e iθ ) dψ p (θ) f(e iθ ) dψ(θ) p <ε for p sufficiently large, which completes the proof. Simple zeros in the limit. If the zeros ζ 1,...,ζ n0,ζ n0 +1,...,ζ n of B n (z) =W n n0 (z) n 0 k=1 (z ζ k) in Theorem 1.1 are distinct, then we set (z ζ 1 ) (z ζ m 1 )(z ζ m+1 ) (z ζ n ) (3.1) Λ m (z) = (z m ζ 1 ) (z m ζ m 1 )(z m ζ m+1 ) (z m ζ n ). We note that Λ m (ζ j )=δ m,j, and hence (3.2) (3.3) Λ m (e iθ ) dψ(θ) =λ m, for m =1, 2,...,n 0, Λ m (e iθ ) dψ(θ) =0, for m = n 0 +1,...,n.

6 634 L. DARUIS, O. NJÅSTAD, W. VAN ASSCHE We can now prove the following result. Theorem 3.1. Let the situation be as in Theorem 1.1, andassume that the zeros of the limiting polynomial B n (z) = B n (ψ Nk(ν),z) are distinct and that lim ν z N k(ν) m = ζ m.then (3.4) (3.5) lim ν λn k(ν) m = λ m, for m =1, 2,...,n 0, lim ν λn k(ν) m =0, for m = n 0 +1,...,n. Proof. Since z N k(ν) m converges to ζ m as ν, we conclude that (e iθ )convergestoλ m (e iθ ) uniformly for θ [, π]. Thus by L N k(ν) m letting L N k(ν) m playtheroleoff p,λ m theroleoff, andψ Nk(ν) the role of ψ p in Theorem 2.2, we find that lim ν 1 N k(ν) L N k(ν) m (e iθ ) dψ Nk(ν) (θ) = Λ(e iθ ) dψ(θ). Taking into account (2.1) (2.2) and (3.2) (3.3), we conclude that (3.4) (3.5) hold. Corollary 3.2. Assume that every subsequence {N k : k =1, 2,...} has a subsequence {N k(ν) : ν =1, 2,...} such that the limiting polynomial B n ({N k(ν) },z) has n distinct zeros. Then (3.6) (3.7) lim N λn m = λ m for m =1, 2,...,n 0, lim N λn m =0 for m = n 0 +1,...,n. Thus the n 0 zeros of B n (ψ N,z) corresponding with the eventually largest weights in the quadrature formula approach the frequency points ζ 1,...,ζ n0. Proof. It follows from Theorem 3.1 and the assumption in Corollary 3.2 that every subsequence {λ N k m : k =1, 2,...} has a subsequence

7 PARA-ORTHOGONAL POLYNOMIALS 635 {λ N k(ν) m : ν =1, 2,...} which converges to λ m for m =1, 2,...,n 0, and to 0 for m = n 0 +1,...,n. A general convergence property for sequences then ensures that the whole sequence {λ N m : N =1, 2,...} converges to λ m or 0, respectively. It follows from Corollary 3.2 that if, in particular, the sequence {B n (ψ N,z) : N = 1, 2,...} itself converges to a polynomial B n (z) with distinct zeros, then (3.6) (3.7) holds. Example 3.1. We consider the signal ( (3.8) x(m) =α e πmi/2 + e mi/2), α > 0. The frequency points are ζ 1 = i and ζ 2 = i. By using the Szegő recursion formulas (see, e.g., [7, 18]) it can be shown that, for n = 3, 4, 5, the para-orthogonal polynomials B n (ψ N,z) converge to the polynomial (3.9) B n (τ,z)=(z n 2 + τ)(z i)(z + i). Let n =3andτ = 1. Then in addition to the frequency points ζ 1 and ζ 2, the polynomial B n ( 1,z) has the zero ζ 3 = 1. By a suitable ordering of the zeros zk N, k =1, 2, 3, of B 3(ψ N,z)wehavezk N ζ k as N. Then λ N 1 α 2, λ N 2 α 2,andλ 3 0asN. Observations of the limiting behavior of zeros and weights will here indicate that n 0 = 2, with two frequency points ±i. For the amplitude we can conclude that the modulus is α. Example 3.2. With the signal as in Example 3.1, with α = 1, n =4andτ = i we get the following numerical results. First, we have computed the modulus of the zeros of the Szegő polynomials ϕ 4 (ψ N,z) for some values of N (Table 1). Observe that two of the zeros have modulus close to 1 and the remaining two zeros have small modulus. The zeros with modulus close to 1 approach the frequency points ±i.

8 636 L. DARUIS, O. NJÅSTAD, W. VAN ASSCHE TABLE 1. Modulus of the zeros of Szegő polynomials. N =21 N =41 N =81 N = 201 N = E E E E 02 Next, we have computed the quadrature weights λ N m in the Szegő quadrature formula for some values of N (Table 2). Observe that two of the weights are close to 1 (which is α 2 ) and the remaining weights are small. The weights near 1 correspond to the zeros near the frequency points ±i. TABLE 2. Quadrature coefficients of the Szegő quadrature. N =21 N =41 N =81 N = 201 N = E E E E E E E E E E-03 Example 3.3. As a numerical example we take N = observations from the signal 4 (3.10) x(m) = (A k cos(mω k )+B k sin(mω k )) + Z m k=1 which contains some noise Z m, which we have taken to be white noise with variance (i.e., all Z m are uncorrelated random variables). We have taken the following values for the parameters:

9 PARA-ORTHOGONAL POLYNOMIALS 637 k ω k A k B k This was obtained by taking the two main frequencies ω 1,ω 2 of the sound of some flute, to which harmonics were added (ω 3 = ω 1 /2and ω 4 = ω 2 /2). The resulting sound can be heard as bill.wav 1. The zeros of the para-orthogonal polynomial of degree 100 (with τ = 1) were computed as the eigenvalues of a unitary Hessenberg matrix following [4, 5] and the weights of the Szegő quadrature were obtained from the first components of the corresponding eigenvectors [5, 16]. Figure 1 gives the weights λ k as a function of θ k,wheree iθ k = zk N. The values of the largest weights and the corresponding θ k are given in Table * * * * * * * * ************************************************** FIGURE 1. Weights for the Szegő quadrature for 100 nodes.

10 638 L. DARUIS, O. NJÅSTAD, W. VAN ASSCHE TABLE 3. The largest weights inthe Szegő quadrature. θ λ (A 2 k + B2 k )/4 ± ± ± ± Observe that the signal (3.10) can be written as (3.11) x(m) = 4 ( Ak ib k e imω k + A ) k + ib k e imω k + Z m, 2 2 k=1 so that α k = α k =(A k ib k )/2. According to Corollary 3.2, the weights corresponding to the zeros near the frequency points converge to α k 2 =(A 2 k + B2 k )/4. 4. Zeros that are not frequency points. In this section we shall show that for zeros of W n n0 ({N k(ν) },z) which are not frequency points, the weights associated with the corresponding zeros of the paraorthogonal polynomials tend to zero. We recall the connection between the monic orthogonal polynomials Φ n (ψ N,z) and the orthonormal polynomials ϕ n (ψ N,z) given in (1.3) (1.4), as well as the expression (2.3) for the weights λ N k. We shall also make use of the fact that for n<n 0 we have (4.1) lim N Φ n(ψ N,z)=Φ n (ψ, z), with ψ given by (1.5), (see [8, 15]) and that (4.2) lim N Φ n 0 (ψ N, 0) =1 (see, e.g., [7]). In the following theorem, we assume the same situation as in Theorem 1.1.

11 PARA-ORTHOGONAL POLYNOMIALS 639 Theorem 4.1. Let n > n 0, and suppose that the sequence {z N k(ν) m : ν = 1, 2,...} of zeros of B n (ψ Nk(ν),z) converge to a zero ζ m of W n n0 ({N k(ν) },z) which is not a frequency point. Then (4.3) lim ν λn k(ν) m =0. Proof. Weobservethat ϕ n0 (ψ Nk(ν),z N k(ν) m 2 is a term in the denominator of (2.3). From (1.3) (1.4) we get m ) 2 (4.4) ϕ n0 (ψ Nk(ν),z N k(ν) m ) 2 = Φ n 0 (ψ Nk(ν),z N k(ν) n0 j=1 (1 Φ j(ψ Nk(ν), 0) 2 ). The numerator Φ n0 (ψ Nk(ν),z N k(ν) m tends to Φ n0 (ψ, ζ m ), which is different from zero (since ζ m is not a frequency point), while Φ n0 (ψ Nk(ν), 0) tends to 1 by (4.2). It follows from (4.4) that lim ϕ n ν 0 (ψ Nk(ν),z N k(ν) m ) =, and hence by (2.3) that (4.3) is satisfied. The result of Theorem 4.1 is valid for the weights corresponding to zeros z N k(ν) m tending to ζ m for any convergent subsequence {B n (ψ Nk(ν),z):ν =1, 2,...}. Itisthusnotrequiredthatζ m is a simple zero of W n n0 ({N k(ν) },z) for any convergent subsequence, only that ζ m is not a frequency point. 5. Multiple zeros in the limit. We shall now consider a simple case of the situation where the frequency points (and in principle also other zeros ζ m ) occur as multiple zeros of B n ({N k(ν) },z). Without loss of generality we formulate the results in terms of convergence of the whole sequence {B n (ψ N,z):N =1, 2,...} and hence of the whole sequences {zm N : N =1, 2,...}, m =1, 2,...,n, to avoid troublesome formulations and notation in terms of subsequences. As before, let zm, N m =1, 2,...,n, be the zeros of B n (ψ N,z), and recall that they are all distinct. To simplify the notation we shall

12 640 L. DARUIS, O. NJÅSTAD, W. VAN ASSCHE suppress the index N in the following calculations. We introduce the polynomial T n as (5.1) T n (z) =(z z 3 ) (z z n ). Then the polynomials L 1 and L 2 in (2.2) may be written as (5.2) L 1 (z) = (z z 2)T n (z) (z 1 z 2 )T n (z 1 ), L 2(z) = (z z 1)T n (z) (z 2 z 1 )T n (z 2 ). We easily find that (5.3) L 1 (z)+l 2 (z) = T n(z)[(z z 2 )T n (z 2 ) (z z 1 )T n (z 1 )]. (z 1 z 2 )T n (z 1 )T n (z 2 ) In the following proposition we do not need to require that the values of the z k are distinct. Proposition 5.1. We may write (5.4) L 1 (z)+l 2 (z) = T n(z)p (z) T n (z 1 )T n (z 2 ), where P is a polynomial with the property (5.5) P (z 1 )=T n (z 1 ), when z 2 = z 1. Proof. From (5.3) we find P (z) = (z z 2)T n (z 2 ) (z z 1 )T n (z 1 ) z 1 z 2 = z(t n(z 2 ) T n (z 1 )) z 2T n (z 2 ) z 1 T n (z 1 ). z 1 z 2 z 1 z 2 Here T n (z 2 ) is a polynomial in z 2. The numerators are both divisible by z 2 z 1, hence (T n (z 2 ) T n (z 1 ))/(z 1 z 2 )and(z 2 T n (z 2 ) z 1 T n (z 1 ))/(z 1 z 2 ) both extend to polynomials in z 2, which are also defined for z 2 = z 1. From this expression it follows immediately that P (z 1 )=T n (z 2 ), and hence by continuity we have P (z 1 )=T n (z 1 )whenz 2 = z 1.

13 PARA-ORTHOGONAL POLYNOMIALS 641 We again consider a fixed degree n. We now assume that zm N ζ m as N for m =1, 2,...,n 0 and m = n 0 + p +1,...,n,andthat zn N 0 +j ζ j for j =1,...,p,where1 p n 0. Here it is assumed that ζ k ζ m for k = n 0 + p +1,...,n and m =1,...,n 0, and that the points ζ n0 +p+1,...,ζ n are distinct. Thus frequency points may have multiplicity at most 2 as zeros of B n (z), while zeros of B n (z) which are not frequency points are simple. In other words, it is assumed that W n n0 has only simple zeros. We introduce the notation (5.6) Tn(z) j =(z z1 N ) (z zj 1)(z N zj+1) N (z zn N 0 +j 1)(z zn N 0 +j+1) (z zn N ), for j =1,...,p. It follows from Proposition 5.1 that for j =1,...,p we may write [ (5.7) lim L N j (z)+l N n N 0 +j(z) ] = T n(z)p j j (z) Tn(ζ j, j ) 2 with z N m replaced by ζ m for m =1,...,n 0 and m = n 0 + p +1,...,n, and z N n 0 +j replaced by ζ j for j =1,...,p. Here P j (z) isapolynomial with the property P j (ζ j )=T j n(ζ j ). Furthermore (5.8) lim N LN m(z) =Λ m (z), for m = p +1,...,n 0 and m = n 0 + p +1,...,n,where (5.9) Λ m (z) = (z ζ 1 ) 2 (z ζ p ) 2 (z ζ p+1 ) (z ζ m 1 )(z ζ m+1 ) (z ζ n ) (ζ m ζ 1 ) 2 (ζ m ζ p ) 2 (ζ m ζ p+1 ) (ζ m ζ m 1 )(ζ m ζ m+1 ) (ζ m ζ n ). The ζ n0 +1,...,ζ n0 +p do not exist and do not occur in (5.9). We note that T j n(ζ j )P j (ζ j ) T n (ζ j ) 2 =1, for j =1,...,p, while T n (ζ k )P j (ζ k ) T n (ζ j ) 2 =0, for k j. Similarly Λ m (ζ m )=1form = p +1,...,n 0 and m = n 0 + p +1,...,n, while Λ m (ζ k )=0fork m.

14 642 L. DARUIS, O. NJÅSTAD, W. VAN ASSCHE Theorem 5.2. Let zm N ζ m as N for m =1,...,n 0 and m = n 0 + p +1,...,n,andzn N 0 +j ζ j for j =1,...,p. Assume that ζ 1,...,ζ n0,ζ n0 +p+1,...,ζ n are distinct points. Then (5.10) lim N (λn j + λ N T n 0 +j) = n(e j iθ )P j (e iθ ) Tn(ζ j dψ(θ), j ) 2 for j =1,...,p,and (5.11) lim N λn m = Λ(e iθ ) dψ(θ), for m = p +1,...,n 0 and m = n 0 + p +1,...,n. Proof. From the assumptions it follows that L N j (z) +LN n 0 +j (z) converges uniformly to Tn j(z)p j(z)/t n (ζ j ) 2 on T for j =1,...,p and that L N m(z) converges uniformly to Λ m (z) ont for m = p +1,...,n 0 and m = n 0 + p +1,...,n. The result then follows from Theorem 2.2 in the same way as Theorem 3.1 does. Corollary 5.3. Let the assumptions be as in Theorem 5.2. Then (5.12) (5.13) (5.14) ( lim λ N j + λ N ) n N 0 +j = λj, for j =1,...,p, lim N λn k = λ k, for k = p +1,...,n 0, lim N λn m =0, for m = n 0 + p +1,...,n. Proof. The result follows from Theorem 5.2 and the remarks preceding it, together with the definition of the measure ψ. Remark. The argument can be extended to allow double zeros among the zeros ζ n0 +p+1,...,ζ n. Furthermore in order to obtain the results (5.12) (5.13) it is not necessary to make any assumptions on ζ n0 +p+1,...,ζ n, except that they are all different from ζ 1,...,ζ n0. Example 5.1. Let the signal be as in Example 3.1, with n =3and τ = i. Then B 3 ( i, z) =(z i) 2 (z + i). Here ζ 1 = i, ζ 2 = i,

15 PARA-ORTHOGONAL POLYNOMIALS 643 ζ 3 = ζ 1 = i, with λ 1 = λ 2 = α 2. By numbering the zeros of B 3 (ψ N, i, z) such that z1 N ζ 1, z2 N ζ 2, ζ3 N ζ 1, we conclude from Corollary 5.3 that λ N 1 + λ N 3 α 2, λ N 2 α 2. Observations of the limiting behavior of zeros and weights will here indicate that n 0 = 2, that the frequency points are ±i, and that the amplitudes have modulus α. Example 5.2. Let the signal be as in Example 3.1, with n =5and τ = i. Accordingto(3.9)wehave B 5 (i, z) =(z 3 + i)(z i)(z + i) ( =(z i) 2 (z + i) z 1 )( 2 [ 3 i] z + 1 ) 2 [ 3+i]. Here we set ζ 1 = i, ζ 2 = i, ζ 3 = [ 3 i]/2, ζ 4 = [ 3+i]/2, ζ 5 = i. We number the zeros of B 5 (ψ N,i,z) such that zk N ζ k for k =1, 2, 3, 4, and z5 N ζ 1. It follows from Corollary 5.3 that ( lim λ N 1 + λ N ) 5 = α 2, lim N N λn 2 = α 2, lim N λn 3 = lim N λn 4 =0. Observations of the limiting behavior of zk N and λn k will indicate that n 0 = 2, that the frequencies are ±π/2, and that the amplitudes have modulus α. ENDNOTES 1. Available at REFERENCES 1. L. Daruis and P. González-Vera, Szegő polynomials and quadrature formulas on the unit circle, Appl. Numer. Math. 36 (1999), L.Daruis,P.González-Vera and O. Njåstad, Szegő quadrature formulas for certain Jacobi-type weight functions, Math.Comp.71 (2002),

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17 PARA-ORTHOGONAL POLYNOMIALS 645 Universidad de La Laguna, Tenerife, Department of Mathematical Analysis, Tenerife, Canary Islands, Spain address: Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway address: Katholieke Universiteit Leuven, Department of Mathematics, Leuven, Belgium address: