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Bounds for Vandermonde type determinants of orthogonal polynomials.

Abstract. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a system of monic orthogonal polynomials. We establish upper and lower estimates for determinants of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the proofs, we have to study the monic orthogonal system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] obtained by inserting the polynomial w(x) := [[product].sup.k.sub.v=1](x - [z.sub.v]) as a weight into the inner product defining [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We also express the recurrence formula for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in terms of Vandermonde type determinants.

Key words. Vandermonde type determinants, orthogonal systems, polynomial weights, inequalities.

AMS subject classifications. 42C05, 15A15, 15A45, 30A10.

1. Introduction and statement of results. First we want to introduce some terminology for orthogonal polynomials, referring to [1, 2, 7] for standard results.

We denote by [sigma] an m-distribution, that is, a non-decreasing bounded function [sigma] : R [right arrow] R which attains infinitely many distinct values and is such that the moments

[[mu].sub.n] := [[integral].sup.[infinity].sub.-[infinity]][x.sup.n]d[sigma](x) (n [member of] [N.sub.0])

exist. Then there exists a uniquely determined sequence of polynomials

[P.sub.0](z), [P.sub.1](z), ..., [P.sub.n](z), ...,

called the sequence of monic orthogonal polynomials with respect to d[sigma](x); with the following properties:

(i) each [P.sub.n] is a monic polynomial of degree n;

(ii) <[P.sub.n], [P.sub.m]> := [[integral].sup.[infinity].sub.-[infinity]] [P.sub.n](x)[P.sub.m](x)d[sigma](x) = 0 for m [not equal to] n.

For any polynomial f, we define the norm

(1.1) [parallel]f[parallel] := [([[integral].sup.[infinity].sub.- [infinity]] [[absolute value of f(x)].sup.2]d[sigma](x)).sup.1/2]

and introduce the numbers

(1.2) [[gamma].sub.n] := [parallel] [P.sub.n][[parallel].sup.2] (n [member of] [N.sub.0]).

The system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies a recurrence formula

(1.3) [P.sub.n](x) = (x - [[alpha].sub.n])[P.sub.n-1](x) - [[beta].sub.n-1][P.sub.n-2](x) (n [member of] N),

where [P.sub.-1](x) [equivalent to] 0, [P.sub.0](x) [equivalent to] 1, [[beta].sub.0] = 1, and

(1.4) [[alpha].sub.n] = 1/[[gamma].sub.n-1] [[integral].sup.[infinity].sub.-[infinity]] x[P.sup.2.sub.n-1](x)d[sigma](x) and [[beta].sub.n] = [[gamma].sub.n]/[[gamma].sub.n-1] (n [member of] N),

see [7, [section] 3.2].

It is known that the polynomials [P.sub.n] (n [member of] N) have only real zeros. Denoting by [J.sub.n] the smallest interval containing the zeros of [P.sub.n], we introduce the n-th distance function

(1.5) [d.sub.n](z) := min {[absolute value of z - [zeta]] : [zeta] [member of] [J.sub.n]} (z [member of] C).

Since the zeros of consecutive orthogonal polynomials interlace, we have

[d.sub.1](z) [greater than or equal to] [d.sub.2](z) [greater than or equal to] ... [greater than or equal to] [d.sub.n](z) [greater than or equal to] ... [greater than or equal to] [absolute value of [[??].sub.z]].

In this paper, we want to estimate the following generalized Vandermonde type determinants:

(1.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easily seen that [V.sub.0]([z.sub.1], ..., [z.sub.k]) is equal to the classical Vandermonde determinant of [z.sub.1], ..., [z.sub.k]. In fact, each polynomial [P.sub.m] may be written as

[P.sub.m](z) = [z.sup.m] + [m-1.summation over ([mu]=0)][c.sub.m[mu]] [P.sub.[mu]](z),

with certain constants [c.sub.m[mu]]. Hence, if we add to each column in (1.6) an appropriate linear combination of its predecessors, and do it first for the last column, then for the last but one and so on, we find that

(1.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

There is no simple explicit formula for [V.sub.n]([z.sub.1], ..., [z.sub.k]) when n [greater than or equal to] 1, and therefore we are interested in bounds for these determinants.

It is usually not a big problem to find some upper bound for a determinant. For sake of completeness, we present the following result.

PROPOSITION 1.1. Let [z.sub.1], ..., [z.sub.k] [member of] C. Then, with the preceding notations,

(1.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover, for any z [member of] C,

(1.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [x.sub.1], ..., [x.sub.m] are the zeros of [P.sub.m].

Lower estimates for determinants in general and for our Vandermonde type determinants in particular are much more delicate. We shall establish a lower bound for the modulus of

(1.10) [V.sub.n+1]([z.sub.1], ..., [z.sub.k])/[V.sub.n]([z.sub.1], ..., [z.sub.k]).

Used repeatedly for n, n - 1, n - 2, ... and combined with (1.7), it allows us to estimate [absolute value of [V.sub.n+1]([z.sub.1], ..., [z.sub.k])] for n [member of] [N.sub.0] from below. Here we admit that some (or all) of the points [z.sub.1], ..., [z.sub.k] may coalesce. In this case, we define the quotient (1.10) by continuous continuation. More precisely, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then we replace the polynomials in the [j.sub.1]-st, [j.sub.2]-nd, ..., [j.sub.l]-th row in (1.6) by their first, second, ..., l-th derivative.

As usual, we shall denote by [??]x[??] the largest integer not exceeding x. We are now ready for presenting the main result.

THEOREM 1.2. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a sequence of monic orthogonal polynomials with associated intervals [J.sub.n], constants [[gamma].sub.n], and distance functions [d.sub.n] (n [member of] N) as specified in (1.2)-(1.5). Let w(x) = [[product].sup.k.sub.j=1](x - [z.sub.j]) be a real polynomial which has no zero in [J.sub.n]+[??]k/2[??]+1]. Denote by [m.sub.1] and [m.sub.2] the number of zeros (counted according to their multiplicities) in the left and the right component of R \ [J.sub.n+[??]k/2[??]+1], respectively, Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and l := (k - m)/2. Suppose that

[d.sub.n+[??]k/2[??]+1]([z.sub.j]) [greater than or equal to] r (j = 1, ..., k),

with r > 0. Then, for the determinants (1.6),

(1.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 1. Note that m = 0 if w(x) does not change sign on R. Furthermore, when m = 0, then the right-hand side of (1.11) remains positive even if r [right arrow] 0. Therefore (1.11) holds with a positive lower bound even if w(x) has zeros on [J.sub.n+[??]k/2[??]+1] provided that their multiplicities are even and m = 0. However, if w(x) changes sign on [J.sub.n+[??]k/2[??]+1], then the left-hand side of (1.11) may vanish, and so we cannot have a non-trivial lower bound.

The proof of Theorem 1.2 will show that (1.11) can be refined by working with individual bounds [r.sub.j] instead of r such that [d.sub.n+[??]k/2[??]+1]([z.sub.j]) [greater than or equal to] [r.sub.j] for j = 1, ..., k.

Remark 2. At the conference in Inzell (3rd Workshop 'Orthogonal Polynomials, Approximation, and Harmonic Analysis', April 2000), Michael Skrzipek gave a lecture on the inversion of Vandermonde type matrices of orthogonal polynomials. Theorem 1.2 includes a sufficient condition for invertibility.

Remark 3. The proof of Theorem 1.2 rests on repeated application of Lemmas 2.2 and 2.3 given in [section] 2 below. In the proofs of these lemmas, all considerations are based on equations. It is only at the end that a lower bound is deduced from a mean value. We can as well deduce an upper bound from that mean value and establish an inequality analogous to (1.11), but in the opposite direction. More precisely, we can proceed as follows. Analogously to (1.5), we define

[D.sub.n](z) := max {[absolute value of z - [zeta]] : [zeta] [member of] [J.sub.n]} (z [member of] C).

Then

[D.sub.1](z) [less than or equal to] [D.sub.2](z) [less than or equal to] ... [less than or equal to] [D.sub.n](z) [less than or equal to] ....

Now suppose that in the situation of Theorem 1.2, we have

[D.sub.n+[??]k/2[??]+1]([z.sub.j]) [less than or equal to] R (j = 1, ..., k).

Then, for the determinants (1.6),

(1.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 4. Except for trivial cases, inequality (1.11) is not sharp. In view of Remark 3 and an analysis of the proofs given below, we find that the precision of (1.11) depends on the length of [J.sub.n+[??]k/2[??]+1]. If this interval is relatively small, then the estimate (1.11) is quite accurate. If [J.sub.n+[??]k/2[??]+1] is unbounded as n [right arrow] [infinity], then (1.11) will be less accurate when n is large, but it will be non-trivial nevertheless.

The proof of Theorem 1.2 will show that the points [z.sub.1], ..., [z.sub.k] can be involved successively as real singles and pairs of conjugates. Therefore the location of these points, relative to one another, is not crucial for the accuracy of (1.11). This may be surprising since, on the left-hand side of (1.11), the numerator and the denominator tend to zero as two of the points [z.sub.1], ..., [z.sub.k] approach each other.

If in Theorem 1.2 the hypothesis on w(x) holds for some n [member of] N, then it automatically holds for all smaller indices n, and m and l do not change when n is reduced. This allows us to deduce the following lower estimate for [absolute value of [V.sub.n+1]([z.sub.1], ..., [z.sub.k])] by iterating (1.11) and employing (1.7).

COROLLARY 1.3. Suppose that in the statement of Theorem 1.2 the hypothesis on w(x) holds for some n [member of] N. Then, introducing the polynomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the proof of Theorem 1.2, we shall employ another orthogonal system which reveals why the determinants [V.sub.n]([z.sub.1], ..., [z.sub.k]) are of interest.

Let w(x) = [[product].sup.k.sub.j=1](x - [z.sub.v]) be a real polynomial which is non-negative on the real line. Then there exists a uniquely determined sequence of monic orthogonal polynomials

(1.13) [P.sup.[w].sub.0](z), [P.sup.[w].sub.1](z), ..., [P.sup.[w].sub.n] (z), ...

with respect to w(x)d[sigma](x): We want to distinguish all the quantities associated with this system from those associated with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by attaching a superscript [w]. Thus

[[gamma].sup.[w].sub.n] := [[integral].sup.[infinity].sub.-[infinity]] [([P.sup.[w].sub.n](x)).sup.2] w(x)d[sigma](x),

[J.sup.[w].sub.n] is the smallest interval containing the zeros of [P.sup.[w].sub.n], and [d.sup.[w].sub.n](z) is the distance of z from [J.sup.[w].sub.n].

In 1858 already, Christoffel had observed (in the case where [sigma] (x) = x) that

(1.14) w(x)[P.sup.[w].sub.n](x) = [(-1).sup.k] [V.sub.n](x, [z.sub.1], ..., [z.sub.k])/[V.sub.n]([z.sub.1], ..., [z.sub.k]) (n [member of] [N.sub.0]),

see [7, [section] 2.5], where the result is given for general [sigma].

When w(x) changes sign on the real line, then w(x)d[sigma](x) may not be an admissible differential for defining an inner product in the space of polynomials. But if w(x) is nonnegative on [J.sub.n+[??]k/2[??]+1]; then w(x)d[sigma](x) is admissible for the subspace [P.sub.n] consisting of all polynomials of degree at most n. In fact, for any f, g [member of] [P.sub.n], the integral

(1.15) [[integral].sup.[infinity].sub.-[infinity]]f(x)[bar.g(x)]w(x)d [sigma](x)

can be calculated by means of the Gaussian quadrature formula [7, [section] 3.4] whose nodes are the zeros of [P.sub.n+[??]k/2[??]+1], and so we need only the restriction of w to [J.sub.n+[??]k/2[??]+1]. Thus we find that (1.15) defines an inner product on [P.sub.n] and that the polynomials

(1.16) [P.sup.[w].sub.0](z), [P.sup.[w].sub.1](z), ..., [P.sup.[w].sub.n](z),

as given by (1.14), form an orthogonal basis for [P.sub.n]. Moreover, if the support of d[sigma](x) is contained in an interval J (such an interval is called an interval of orthogonality) and w(x) is non-negative on J, then (1.14) defines an infinite sequence of orthogonal polynomials.

If, in the previous paragraph, w(x) is non-positive on [J.sub.n+[??]k/2[??]+1] (respectively, on J), then the polynomials (1.16) (respectively, those in (1.13) with unrestricted n), exactly as defined by (1.14), form a sequence of monic orthogonal polynomials with respect to -w(x)d[sigma](x).

In order to establish the recurrence formula for the system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we need a modification of the determinants [V.sub.n]([z.sub.1], ..., [z.sub.k]). We denote by [V.sup.*.sub.n]([z.sub.1], ..., [z.sub.k]) the determinant obtained from [V.sub.n]([z.sub.1], ..., [z.sub.k]) by replacing the index n of the polynomials in the first column by n - 1, that is,

(1.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

THEOREM 1.4. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a monic orthogonal system satisfying a recurrence formula

[P.sub.n](x) = (x - [[alpha].sub.n])[P.sub.n-1](x) - [[beta].sub.n-1] [P.sub.n-2](x) (n [member of] N)

with [P.sub.-1](x) [equivalent to] 0 and [P.sub.0](x) [equivalent to] 1. Let J be an interval of orthogonality, and suppose that w(x) = [[product].sup.k.sub.j=1](x - [z.sub.j]) is a real polynomial that does not change sign on J. Then

(1.18) [P.sup.[w].sub.n](x) = (x - [[alpha].sup.[w].sub.n]) [P.sup.[w].sub.n-1](x) - [[beta].sup.[w].sub.n-1] [P.sup.[w].sub.n-2](x) (n [member of] N)

with [P.sup.[w].sub.-1](x) [equivalent to] 0, [P.sup.[w].sub.0] (x) [equivalent to] 1, and

(1.19) [[alpha].sup.[w].sub.n] = [[alpha].sub.n] + [[beta].sub.n-1] [V.sup.*.sub.n-1]([z.sub.1], ..., [z.sub.k])/[V.sub.n-1]([z.sub.1], ..., [z.sub.k]) - [[beta].sub.n] [V.sup.*.sub.n]([z.sub.1], ..., [z.sub.k])/[V.sub.n]([z.sub.1], ..., [z.sub.k]),

(1.20) [[beta].sup.[w].sub.n] = [[beta].sub.n] [V.sub.n+1]([z.sub.1], ..., [z.sub.k])[V.sub.n-1] ([z.sub.1], ..., [z.sub.k])/[([V.sub.n]([z.sub.1], ..., [z.sub.k])).sup.2] (n [member of] N).

Note that [[beta].sup.[w].sub.0] is not needed and may therefore be arbitrarily defined.

While (1.19) and (1.20) give explicit representations for [[alpha].sup.[w].sub.n] and [[beta].sup.[w].sub.n], Gautschi [3] proposed an algorithm for a recursive computation of these quantities. However, as far as the computation of the polynomials [P.sup.[w].sub.n](x) is concerned, Skrzipek [6] pointed out that the use of the recurrence formula (1.18) may have disadvantages. He proposed an alternative approach.

In [5], we have proved several inequalities for [[gamma].sup.[w].sub.n] and [parallel][P.sup.[w].sub.n][parallel]; see [5, Lemmas 3-5]. They imply further inequalities for [V.sub.n]([z.sub.1], ..., [z.sub.k]) and its modifications. Some of these inequalities are sharp.

2. Lemmas. Continuing in using the notations of [section] 1, we shall prove the following auxiliary results.

LEMMA 2.1. Let [P.sub.n] and [P.sub.n+1] be consecutive monic orthogonal polynomials, and denote by [x.sub.1], ..., [x.sub.n+1] the zeros of [P.sub.n+1]. Then

(2.1) [P.sub.n](z)/[P.sub.n+1](z) = [n+1.summation over (v=1)] [[lambda].sub.v]/z-[x.sub.v] where [[lambda].sub.v] > 0 (v = 1, ..., n + 1)

and

(2.2) [n+1.summation over (v=1)] [[lambda].sub.v] = 1.

Proof. For (2.1), see [7, p. 47, Theorem 3.3.5]). Multiplying both sides of the equation in (2.1) by z and letting z [right arrow] [infinity], we readily conclude that (2.2) holds.

LEMMA 2.2. Let N [member of] N and w(x) = x - [xi] with [xi] [member of] R \ [J.sub.N+1]. Then

(2.3) [J.sup.[w].sub.n] [subset] [J.sub.n+1]

and

(2.4) [[gamma].sup.[w].sub.n] [greater than or equal to] [[gamma].sub.n] [d.sub.n+1]([xi])

for n = 0, ..., N.

Proof. By (1.14),

(2.5) w(x)[P.sup.[w].sub.n](x) = [P.sub.n+1](x) - [P.sub.n+1]([xi])/ [P.sub.n]([xi]) [P.sub.n](x) (n = 0, ..., N).

Now let [x.sub.1], ..., [x.sub.n+1] be the zeros of [P.sub.n+1] in increasing order. Then

w([x.sub.v])[P.sup.[w].sub.n]([x.sub.v]) = - [P.sub.n+1]([xi])/ [P.sub.n]([xi]) [P.sub.n]([x.sub.v]) (v = 1, ..., n + 1).

Since the polynomials [P.sub.n] and [P.sub.n+1] are monic and their zeros interlace, we have

(2.6) sgn [P.sub.n]([x.sub.v]) = [(-1).sup.n+1-v] (v = 1, ..., n + 1).

Taking into account that w(x) does not change sign on [J.sub.N+1], we find that

sgn [P.sup.[w].sub.n]([x.sub.v]) = [(-1).sup.n-v] sgn [P.sub.n+1]([xi])/w ([x.sub.1])[P.sub.n]([xi]) (v = 1, ..., n + 1).

Hence the zeros of [P.sup.[w].sub.n] and [P.sub.n+1] interlace, and this implies (2.3).

The polynomials [P.sup.[w].sub.0], ..., [P.sup.[w].sub.N] are orthogonal with respect to [+ or -]w(x)d[sigma](x), the sign depending on the sign of w(x) on [J.sub.N+1]. In any case,

[[gamma].sup.[w].sub.n] = [absolute value of [[integral].sup.[infinity].sub.-[infinity]] [([P.sup.[w].sub.n](x)).sup.2]w(x)d[sigma](x)] = [absolute value of [[integral].sup.[infinity].sub.-[infinity]]w(x)[P.sup.[w].sub.n](x) [P.sub.n](x)d[sigma](x)].

Substituting (2.5) into the right-hand side, we readily find that

(2.7) [[gamma].sup.[w].sub.n] = [[gamma].sub.n] [absolute value of [P.sub.n+1]([xi])/[P.sub.n]([xi])].

Now, by Lemma 2.1,

[absolute value of [P.sub.n]([xi])/[P.sub.n+1]([xi])] [less than or equal to] 1/[min.sub.1[less than or equal to]v[less than or equal to]n+1] [absolute value of [xi] - [x.sub.v]] [less than or equal to] 1/[d.sub.n+1]([xi]),

and so (2.4) follows from (2.7).

While in Lemma 2.2 w(x) was linear, we now establish a corresponding result for a quadratic w(x).

LEMMA 2.3. Let N [member of] N and w(x) = (x - [[zeta].sub.1]) (x - [[zeta].sub.2]), where either [[zeta].sub.2] = [[bar.[zeta]].sub.1] or [[zeta].sub.1], [[zeta].sub.2] [member of] R \ [J.sub.N+2]. Then

(2.8) [J.sup.[w].sub.n] [subset] [J.sub.n+1] (n = 0, ..., N).

If, in addition, [[zeta].sub.1] and [[zeta].sub.2] do not lie in different components of R \ [J.sub.N+2], then

(2.9) [[gamma].sup.[w].sub.n] [greater than or equal to] [[gamma].sub.n+1] + [[gamma].sub.n][d.sub.n+1]([[zeta].sub.1])[d.sub.n+1]([[zeta].sub.2]) (n = 0, ..., N).

Proof. We shall prove the lemma under the additional hypothesis that [[zeta].sub.1] [not equal to] [[zeta].sub.2]. An extension to [[zeta].sub.1] = [[zeta].sub.2] will be achieved by continuous continuation, as we have explained in the paragraph following (1.10).

Using the notation (1.17), we deduce from (1.14) by Laplace expansion (with respect to the first row) of the determinant in the numerator that

(2.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now let [x.sub.1], ..., [x.sub.n+1] be again the zeros of [P.sub.n+1] in increasing order. It follows from the recurrence formula (1.3) and from (1.4) that

[P.sub.n+2]([x.sub.v]) = - [[gamma].sub.n+1]/[[gamma].sub.n] [P.sub.n] ([x.sub.v]) (v = 1, ..., n + 1).

Hence (2.10) gives

w([x.sub.v])[P.sup.[w].sub.n]([x.sub.v]) = [P.sub.n]([x.sub.v]) ([V.sub.n+1]([[zeta].sub.1], [[zeta].sub.2])/[V.sub.n]([[zeta].sub.1], [[zeta].sub.2]) - [[gamma].sub.n+1]/[[gamma].sub.n]) (v = 1, ..., n + 1).

The term in parentheses must be different from zero since [P.sup.[w].sub.n] would have n + 1 zeros otherwise. Recalling (2.6), we easily conclude that the zeros of [P.sup.[w].sub.n] and [P.sub.n+1] interlace. This shows that (2.8) holds.

Now we want to estimate [[gamma].sup.[w].sub.n] from below. Clearly

[[gamma].sup.[w].sub.n] = [[integral].sup.[infinity].sub.-[infinity]] [([P.sup.[w].sub.n](x)).sup.2]w(x)d[sigma](x) = [[integral].sup.[infinity].sub.-[infinity]]w(x)[P.sup.[w].sub.n](x) [P.sub.n](x)d[sigma](x).

Substituting (2.10) into the right-hand side, we readily find that

(2.11) [[gamma].sup.[w].sub.n] = [[gamma].sub.n][V.sub.n+1] ([[zeta].sub.1], [[zeta].sub.2])/[V.sub.n]([[zeta].sub.1], [[zeta].sub.2]).

Employing the recurrence formula (1.3), we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence (2.11) may be rewritten as

(2.12) [[gamma].sup.[w].sub.n] = [[gamma].sub.n+1] + [[gamma].sub.n] ([[zeta].sub.2] - [[zeta].sub.1])[P.sub.n+1]([[zeta].sub.1])[P.sub.n+1] ([[zeta].sub.2])/[V.sub.n]([[zeta].sub.1], [[zeta].sub.2]).

Using Lemma 2.1 for a partial fraction decomposition of [P.sub.n]/[P.sub.n+1], we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [[zeta].sub.1], and [[zeta].sub.2] are either a pair of conjugate zeros or a pair of real zeros lying in the same component of R\[J.sub.N+2], we see that the last sum is a mean value of positive terms. Therefore

[V.sub.n]([[zeta].sub.1], [[zeta].sub.2])/([[zeta].sub.2] - [[zeta].sub.1])[P[[zeta].sub.n+1]([[zeta].sub.1])[P[[zeta].sub.n+1] ([[zeta].sub.2]) [less than or equal to]1/[d.sub.n+1]([[zeta].sub.1]) [d.sub.n+1]([[zeta].sub.2]).

Combining this estimate with (2.12), we obviously obtain (2.9).

3. Proofs of the results in [section] 1.

Proof of Proposition 1.1. Let A = ([a.sub.[mu]v]) be a matrix in [C.sup.n x n]. Then, by an inequality of Hadamard [4, p. 418, Theorem 13.5.3] applied to the transpose of A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This estimate can be generalized. If D [member of] [C.sup.n x n] is a non-singular diagonal matrix with diagonal entries [d.sub.1], ..., [d.sub.n], then det([D.sup.-1]AD) = det A, and therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This shows that [V.sub.n]([z.sub.1], ..., [z.sub.k]) may be estimated as

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the Christoffel-Darboux formula (see [7, p. 43, (3.2.3) and (3.2.4)])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

But this is the quantity [[DELTA].sub.m](z), defined in Proposition 1.1. Thus (3.1) gives (1.8).

Employing Lemma 2.1, we can avoid the distinction between real and non-real z in the definition of [[DELTA].sub.m](z). In fact, let [x.sub.1], ..., [x.sub.m] be the zeros of [P.sub.m], and let [[lambda].sub.1], ..., [[lambda].sub.m] be the coefficients in the partial fraction decomposition of [P.sub.m-1]/[P.sub.m] according to Lemma 2.1.

If z [member of] R, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and if z [member of] C \ R, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence for any z [member of] C,

[[DELTA].sub.m](z) = 1/[[gamma].sub.m-1] [m.summation over ([mu]=1)] [[absolute value of [P.sub.m](z)/z-[x.sub.[mu]]].sup.2],

which gives (1.9) at once.

Proof of Theorem 1.2. First we note that

(3.2) [absolute value of [V.sub.n+1]([z.sub.1], ..., [z.sub.k])/ [V.sub.n]([z.sub.1], ..., [z.sub.k])] = [[gamma].sup.[w].sub.n]/ [[gamma].sub.n].

In fact, using (1.14), we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, expanding the determinant inside the integral with respect to the first row and paying attention to the orthogonality of the system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we readily obtain (3.2).

In view of (3.2), we have to estimate [[gamma].sup.[w].sub.n] from below. For this we can use Lemmas 2.2 and 2.3 repeatedly, taking advantage of the obvious fact that the operation of attaching a superscript [w] is multiplicative in the following sense. If w = uv, then [P.sup.[w].sub.n] = [([P.sup.[u].sub.n]).sup.[v]] and, consequently, [[gamma].sup.[w].sub.n] = [([[gamma].sup.[u].sub.n]).sup.[v]].

Obviously, we may factor the polynomial w as

w(x) = p(x)[q.sub.1](x) ... [q.sub.l](x),

where p is a monic real polynomial of degree m such that, if m = 2, then the zeros of p lie in different components of R [J.sub.n+[??]k/2[??]+1], and where [q.sub.[lambda]] ([lambda] = 1, ..., l) are monic real polynomials of degree two, each having either a pair of conjugate zeros or a pair of real zeros lying in the same component of R \ [J.sub.n+[??]k/2[??]+1]. In particular, each [q.sub.[lambda]](x) is positive for x [member of] [J.sub.n+[??]k/2[??]+1].

Applying Lemma 2.2 m times, we readily see that

[[gamma].sup.[p].sub.v] [greater than or equal to] [[gamma].sub.v] [r.sup.m] (v = 0, ..., n + l).

Now we define [w.sub.0](x) := p(x) and

[w.sub.[lambda]](x) := p(x)[q.sub.1](x) ... [q.sub.[lambda]](x) ([lambda] = 1, ..., l).

We claim that

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The inequality for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an easy consequence of Lemmas 2.2 and 2.3. The inequality for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] may be proved by induction on [lambda] as follows.

Let [[zeta].sub.1] and [[zeta].sub.2] be the zeros of [q.sub.[lambda]+1]. Using again Lemma 2.3, we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now the induction hypothesis applies and gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This completes the proof of (3.3).

Finally, noting that w(x) = [w.sub.l](x), and combining (3.2) and (3.3), we readily obtain (1.11).

Proof of Theorem 1.4. Let sgnw(x) =: [epsilon] for x [member of] J. It is clear, from the general theory of orthogonal polynomials, that a recurrence formula of the form (1.18) holds, where, according to (1.4),

(3.4) [[alpha].sup.[w].sub.n] = [epsilon]/[[gamma].sup.[w].sub.n-1] [[integral].sup.[infinity].sub.-[infinity]] x[([P.sup.[w].sub.n-1](x)).sup.2]w(x)d[sigma](x)

and

(3.5) [[beta].sup.[w].sub.n] = [[gamma].sup.[w].sub.n]/ [[gamma].sup.[w].sub.n-1]

for n [member of] N.

?From (3.2) and the discussion of the influence of the sign of w (see [section] 1), we know that

(3.6) [[gamma].sup.[w].sub.n] = [epsilon][(-1).sup.k][[gamma].sub.n] [V.sub.n+1]([z.sub.1], ..., [z.sub.k])/[V.sub.n]([z.sub.1], ..., [z.sub.k]) (n [member of] [N.sub.0]),

which gives (1.20) at once.

A verification of (1.19) is more sophisticated. In view of (1.14), we may write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the orthogonality of the system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, when we expand the determinant with respect to the first row and calculate [[alpha].sup.[w].sub.n] according to (3.4), we find that

(3.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next, we have to calculate the integral on the right-hand side. By the recurrence formula for the system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have

x[P.sub.n-1](x) = [P.sub.n](x) + [[alpha].sub.n][P.sub.n-1](x) + [[beta].sub.n-1][P.sub.n-2](x).

This implies that

(3.8) [[integral].sup.[infinity].sub.-[infinity]] x[P.sub.n-1](x) [P.sup.[w].sub.n-1](x)d[sigma](x) = [[alpha].sub.n][[gamma].sub.n-1] + [[beta].sub.n-1] [[integral].sup.[infinity].sub.-[infinity]] [P.sub.n-2](x)[P.sup.[w].sub.n-1](x)d[sigma](x).

It remains to calculate the integral on the right-hand side. For this, we proceed as follows. By (1.14),

w(x)[P.sup.[w].sub.n-2](x) = [(-1).sup.k] [V.sub.n-2](x, [z.sub.1], ..., [z.sub.k])/[V.sub.n-2]([z.sub.1], ..., [z.sub.k]).

Multiplying both sides by [P.sup.[w].sub.n-1](x), expanding the Vandermonde type determinant in the numerator, with respect to the first row, and integrating with respect to d[sigma](x); we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and so

(3.9) [[integral].sup.[infinity].sub.-[infinity]][P.sub.n-2](x) [P.sup.[w].sub.n-1](x)d[sigma](x) = [[gamma].sub.n-1][V.sup.*.sub.n-1] ([z.sub.1], ..., [z.sub.k])/[V.sub.n-1]([z.sub.1], ..., [z.sub.k]).

Finally, combining (3.6)-(3.9), we arrive at (1.19).

Acknowledgment. The author wants to thank the referee for his careful reading of the manuscript. Parts of Remarks 1-4 and Corollary 1.3 were incorporated in response to questions raised by him. He also drew attention to the paper of Gautschi, whose Oberwolfach lecture had escaped the author's memory.

REFERENCES

[1] T. S. CHIHARA, An Introduction to Orthogonal Polynomials, Gordon & Breach, New York, 1978.

[2] G. FREUD, Orthogonal Polynomials, Pergamon Press, Oxford, 1971.

[3] W. GAUTSCHI, An algorithmic implementation of the generalized Christoffel theorem, in Numerical Integration, G. Hammerlin, ed., ISNM 57, Birkhauser, Basel, 1982, pp. 89-106.

[4] L. MIRSKY, An Introduction to Linear Algebra, Clarendon Press, Oxford, 1955.

[5] G. SCHMEISSER, Orthogonal expansion of real polynomials, location of zeros, and an L2 inequality, J. Approximation Theory 109 (2001), pp. 126-147.

[6] M.-R. SKRZIPEK, Orthogonal polynomials for modified weight functions, J. Comput. Appl. Math. 41 (1992), pp. 331-346.

[7] G. SZEGO, Orthogonal Polynomials, Fourth ed., American Mathematical Society, Providence, RI, 1975.

* Received October 30, 2000. Accepted for publication March 7, 2001. Communicated by Sven Ehrich.

GERHARD SCHMEISSER ([dagger])

([dagger]) Mathematical Institute, University of Erlangen-Nuremberg, D-91054 Erlangen, Germany (schmeisser@mi.uni-erlangen.de).
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Date:Jul 1, 2002
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