                            Version compiled on
                               27 July 2001

Extremising the L_{p}–norm of a monic polynomial with roots in a given interval

Shayne Waldron

Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand e–mail: waldron@math.auckland.ac.nz (http:www.math.auckland.ac.nz/˜ waldron)

ABSTRACT

Let Θ be a multiset of n points in the interval [a, b], and

ω_{Θ} := ∏_{θ∈Θ}(· − θ), ∥ω_{Θ}∥_{p} := ∥ω_{Θ}∥_{L_{p}[a,b]}, 1 ≤ p ≤ ∞.

In this paper we investigate the extrema of Θ ↦ ∥ω_{Θ}∥_{p}. The main result is determining the maximum of ∥ω_{Θ}∥_{p} when some submultiset of Θ is prescribed. Applications given include L_{p}–error bounds for Hermite interpolation, error estimates for Gauss quadrature formulæ with multiple nodes, and certain quantitative statements about good and best approximation by polynomials of fixed degree.

Key Words: Hermite interpolation, (incomplete) beta function, Chebyshev polynomials, Legendre polynomials, good and best approximation by polynomials, Gauss quadrature formulæ with multiple nodes, s–orthogonal polynomials

AMS (MOS) Subject Classifications: primary 26D10, 41A44, 52A40; secondary 33C45, 41A05, 41A10, 41A50, 41A55, 41A80

1. Introduction

Let Θ be a multiset of n points in the interval [a, b], and

ω_{Θ} := ∏_{θ∈Θ}(· − θ),

which belongs to Π_{n} (the polynomials of degree ≤ n). In this paper we investigate the size of ∥ω_{Θ}∥_{p} := ∥ω_{Θ}∥_{L_{p}[a,b]} as a function of Θ. The constant ∥ω_{Θ}∥_{p} arises naturally in certain error bounds for Hermite interpolation. For example, if H_{Θ}f ∈ Π_{n−1} is the Hermite interpolant to f at the points Θ (counting multiplicities), then

∥f − H_{Θ}f∥_{p} ≤ \frac{∥ω_{Θ}∥_{p}}{n!} ∥D^{n}f∥_{∞},  ∀f ∈ W_{∞}^{n}[a, b],  (1.1)

with equality if and only if f ∈ Π_{n}. Here W_{p}^{n}[a, b] is the Sobolev space of functions f with D^{n−1}f absolutely continuous on [a, b] and D^{n}f ∈ L_{p} := L_{p}[a, b].

In Section 2, we give the main result: that if some of the points in Θ are prescribed, then ∥ω_{Θ}∥_{p} is maximised by an appropriate choice of the remaining points from {a, b}. In Waldron [W97] a corollary of this result was used to obtain L_{p}–error bounds for Hermite interpolation in cases where some of the points in Θ are known to be from {a, b}. We apply a variation of this result to recent work of Shadrin [S95] on the derivatives of the error in Hermite interpolation.

In Section 3, we show that ∥ω_{Θ}∥_{p} is minimised when Θ is the roots of the error in the unique best L_{p}–approximation from Π_{n−1} to any polynomial of (exact) degree n. This classical result is closely related to Gauss quadrature formulæ with multiple nodes (via s–orthogonal polynomials), for which we give new error bounds.

Section 4 gives further applications of these results on the maximum and minimum. These include error bounds for best L_{p}–approximation by polynomials of fixed degree, and certain quantitative statements about good and best approximation by polynomials of fixed degree.

2. Maximising ∥ω_{Θ}∥_{p}

Throughout,

∥ · ∥_{p} := ∥ · ∥_{L_{p}[a,b]}, 1 ≤ p ≤ ∞,

and Θ is used for a multiset of n points from [a, b].

Theorem 2.1 (Maximum of ∥ω_{Θ}∥_{p}). Let Ψ be a fixed multiset of ≤ n points from [a, b]. Then the maximum of

{∥ω_{Θ}∥_{p} : Θ ⊃ Ψ}

is attained when Θ \ Ψ is from {a, b}.

Proof: We seek the maximum of the convex function f ↦ ∥f∥_{p} over the compact set

W := {ω_{Θ} : Θ ⊃ Ψ} ⊂ Π_{n}.

For n ≥ 2 this set is not convex, e.g., ω_{{a,a}}, ω_{{b,b}} ∈ W while the convex combination

\frac{1}{2}ω_{\{a,a\}} + \frac{1}{2}ω_{\{b,b\}}

has complex roots, and so does not belong to W. Let C be the convex hull of W. Each point in C \ W can be written as a (nontrivial) convex combination of two points in C (otherwise it would be a point in W), and so the extreme points of C are contained in W. Since C → IR : f ↦ ∥f∥_{p} is a continuous convex function, it attains its maximum at an extreme point of C (which is a point in W).

Suppose ω_{Θ} ∈ W is an extreme point of C, with {ξ, Ψ} ⊂ Θ, for some ξ ∈ (a, b). Then for small ε

ω_{Θ} = \frac{1}{2}(· − (ξ − ε)) ω_{Θ\ξ} + \frac{1}{2}(· − (ξ + ε)) ω_{Θ\ξ},

a convex combination of points in W, contradicting the fact ω_{Θ} is an extreme point of C. Thus, the extreme points of C are given by ω_{Θ} where Θ consists of Ψ together with points from {a, b}.

We now use this result to find the maximum of Θ ↦ ∥ω_{Θ}∥_{p} over

A_{n}(i, j) := \{ \begin{matrix}the set of those Θ containing one endpoint at least i times \\ and the other endpoint at least j times, where i + j ≤ n \end{matrix} \} .

Here are some typical examples

A_{3}(0, 1) = {{a, θ_{1}, θ_{2}}, {b, θ_{1}, θ_{2}} : θ_{1}, θ_{2} ∈ [a, b]}, A_{3}(1, 2) = {{a, b, b}, {a, a, b}}.

Notice that A_{n}(i, j) is symmetric in i, j, that A_{n}(0, 0) consists of all possible Θ, and A_{n}(m, n − m) has at most two elements.

Let B be the beta function

B(x, y) := ∫_{0}^{1} t^{x−1}(1 − t)^{y−1} dt = \frac{Γ(x)Γ(y)}{Γ(x + y)}, ∀x, y > 0.

Recall that B is symmetric, and satisfies: 0 < B(x, y) ≤ min{1, 1/ max{x, y}}, ∀x, y > 0. The following result was used in [W97:p470].

Corollary 2.2. Let m := min{i, j}, and 0^{0} := 1. Then

max_{Θ∈A_{n}(i,j)} ∥ω_{Θ}∥_{p} = (b − a)^{n+ \frac{1}{p}} \{ \begin{matrix}B(pm + 1, p(n − m) + 1)^{\frac{1}{p}} , & 1 ≤ p < ∞ \\ m^{m}(n − m)^{n−m}/n^{n}, & p = ∞, \end{matrix}

with the maximum achieved if and only if Θ ∈ A_{n}(m, n − m).

Proof: By Theorem 2.1, the maximum occurs when all the points in Θ are from {a, b}. For Θ ∈ A_{n}(k, n − k), we compute

∥ω_{Θ}∥_{p} = (b − a)^{n+ \frac{1}{p}} \{ \begin{matrix}B(pk + 1, p(n − k) + 1)^{\frac{1}{p}} , & 1 ≤ p < ∞ \\ k^{k}(n − k)^{n−k}/n^{n}, & p = ∞, \end{matrix}

and then observe that the maximum of ∥ω_{Θ}∥_{p} over m ≤ k ≤ n − max{i, j} occurs when k = m.

This improves upon the weaker result of Agarwal [A91:p253], that

max_{Θ∈A_{n}(i,j)} ∥ω_{Θ}∥_{p} ≤ (b − a)^{n+ \frac{1}{p}} (2B_{1/2}(pm + 1, p(n − m) + 1))^{\frac{1}{p}} ,  1 ≤ p < ∞. (2.3)

where B_{1/2} is the incomplete beta function

B_{1/2}(x, y) := ∫_{0}^{1/2} t^{x−1}(1 − t)^{y−1} dt, ∀x, y > 0.

Observe that B_{1/2} is not symmetric, and satisfies B(x, y) ≤ 2B_{1/2}(x, y), ∀ 1 ≤ x ≤ y, with strict inequality unless x = y. Thus, Corollary 2.2 gives better bounds than (2.3) whenever m ≠ n − m, and the same bounds otherwise.

L_{p}–Error bounds for Hermite interpolation

Let 1 ≤ p, q ≤ ∞, and H_{Θ}f ∈ Π_{n−1} be the Hermite interpolant to f at Θ (counting multiplicities). Recently, the author has shown (see [W97:Th. 4.1]) that:

∥f − H_{Θ}f∥_{p} ≤ const_{n,p,q,Θ}(b − a)^{n+ \frac{1}{p} − \frac{1}{q}} ∥D^{n}f∥_{q},  ∀f ∈ W_{q}^{n},  (2.4)

where

const_{n,p,q,Θ} := \frac{n^{\frac{1}{q}}}{n!} ∥x ↦ \frac{ω_{Θ}(x)}{(diam\{x, Θ\})^{1/q}} ∥_{p}(b − a)^{−(n+ \frac{1}{p} − \frac{1}{q})}.

Here diam denotes the diameter of a (multi)set of points. Corollary 2.2 can be used to estimate the constants const_{n,p,q,Θ}.

Corollary 2.5 (Hermite error bounds). Let Θ ∈ A_{n}(i, j), with m := min{i, j} > 0. Then

const_{n,p,q,Θ} ≤ \frac{n^{\frac{1}{q}}}{n!} \{ \begin{matrix}B(pm + 1, p(n − m) + 1)^{\frac{1}{p}} , & 1 ≤ p < ∞ \\ m^{m}(n − m)^{n−m}/n^{n}, & p = ∞. \end{matrix}

Proof: Since m > 0, diam{x, Θ} = b − a, and we obtain

const_{n,p,q,Θ} = \frac{n^{\frac{1}{q}}}{n!} ∥ω_{Θ}∥_{p}(b − a)^{−(n+ \frac{1}{p})}.

Now apply Corollary 2.2.

This improves upon the bounds in [A91], which involve B_{1/2}. In the case m = 0, the above argument can be modified, by observing that

∥x ↦ \frac{ω_{Θ}(x)}{(diam\{x, Θ\})^{1/q}} ∥_{p} ≤ (∥ω_{Θ}∥_{p(1− \frac{1}{nq})})^{1− \frac{1}{nq}} ,  (2.6)

See [W97] for a full discussion, including the cases of equality in (2.4) and mention of some related inequalities of Brink [B72].

Application to the solution of ordinary differential equations

The Hermite interpolation error bounds of Corollary 2.5 can be applied to the analysis of the boundary value problem: D^{n}f = g, with Hermite multipoint conditions given by H_{Θ}f = 0 (see, e.g., Agarwal and Wong [AW93]).

L_{∞}–bounds for the error in Hermite interpolation

Shadrin [S95] proved the following sharp bound for the j–th derivative of the error in Hermite interpolation (where 0 ≤ j < n)

∥D^{j}(f − H_{Θ}f)∥_{∞} ≤ \frac{1}{n!}∥D^{j}ω_{Θ}∥_{∞}∥D^{n}f∥_{∞}, ∀f ∈ W_{∞}^{n}.

For j = 0 the result is well known and easily proved, but for j > 0 it is highly nontrivial. In light of this, one might seek to maximise ∥D^{j}ω_{Θ}∥_{∞} to obtain an universal bound. A simple modification of the proof of Theorem 2.1 gives the following extension.

Theorem 2.1^{∗}. Let Ψ be a fixed multiset of ≤ n points from [a, b] and 0 ≤ j < n. Then the maximum of

{∥D^{j}ω_{Θ}∥_{p} : Θ ⊃ Ψ}

is attained when Θ \ Ψ is from {a, b}.

Combining these two observations we have

Corollary 2.7 (Sharp Hermite error bounds). For 0 ≤ j < n, the error in Hermite interpolation satifies

∥D^{j}(f − H_{Θ}f)∥_{∞} ≤ \frac{(b − a)^{n−j}}{(n − j)!} ∥D^{n}f∥_{∞}, ∀f ∈ W_{∞}^{n}.

This inequality is best possible in the sense that equality is attained if and only if H_{Θ} is Taylor interpolation at one of the endpoints and f is a polynomial of degree n.

Proof: By Theorem 2.1^{∗}, the maximum of ∥D^{j}ω_{Θ}∥_{∞} occurs for some Θ consisting of endpoints. A simple calculation shows it is attained when one endpoint is repeated n times, giving

max_{Θ} ∥D^{j}ω_{Θ}∥_{∞} = ∥D^{j}((· − a)^{n})∥_{∞} = \frac{n!}{(n − j)!}(b − a)^{n−j}.

3. Minimising ∥ω_{Θ}∥_{p}

We now show that Θ ↦ ∥ω_{Θ}∥_{p} has a unique minimum, using the classical result that:

Theorem 3.1. If P ⊂ C[a, b] is an n–dimensional Haar space, then g^{∗}, the unique best L_{p}–approximation to f ∈ C[a, b] from P, interpolates f at n distinct points in (a, b).

For 1 ≤ p < ∞, the characterisation theorem for best L_{p}–approximation gives that g^{∗} is uniquely determined by

∫_{a}^{b} |f − g^{∗}|^{p−1}sign(f − g^{∗}) g = 0,  ∀g ∈ P,  (3.2)

where sign denotes the signum function. Taking P = Π_{n−1}, and f = (·)^{n}, we obtain:

Proposition 3.3 (Minimum of ∥ω_{Θ}∥_{p}). There is a unique choice of Θ which minimises ∥ω_{Θ}∥_{p}, namely the n distinct points in (a, b) that are the roots of M_{n,p} ∈ Π_{n}, which is defined to be the error in the unique best L_{p}–approximation to (·)^{n} from Π_{n−1}. We have

\frac{1}{4^{n}} (b − a)^{n+ \frac{1}{p}} ≤ min_{Θ} ∥ω_{Θ}∥_{p} = ∥M_{n,p}∥_{p} ≤ \frac{2}{4^{n}} (b − a)^{n+ \frac{1}{p}} ,

with equality only when p = 1, ∞, respectively. In addition

min_{Θ} ∥ω_{Θ}∥_{2} = ∥M_{n,2}∥_{2} = \frac{(n!)^{2}}{(2n)!\sqrt{2n + 1}}(b − a)^{n+ \frac{1}{2}} .

Proof: Taking P = Π_{n−1}, and f = (·)^{n} in Theorem 3.1 gives that M_{n,p} (the error in best approximation) is of the form M_{n,p} = ω_{Θ} for a certain Θ consisting of distinct points in (a, b). Thus, this choice of Θ uniquely minimises ∥ω_{Θ}∥_{p} (even when Θ is not restricted to lie within [a, b]).

From Hölder’s inequality, it follows that

p ↦ C_{p} := ∥M_{n,p}∥_{p}(b − a)^{−(n+ \frac{1}{p})} = min_{Θ} ∥ω_{Θ}∥_{p}(b − a)^{−(n+ \frac{1}{p})}

is strictly increasing.

For p = 1, M_{n,p} is, up to an affine change of variables equal to U_{n}, the Chebyshev polynomial of the second kind, and we calculate

C_{1} = ∥2^{−n}U_{n}∥_{1} 2^{−(n+ \frac{1}{1} )} = \frac{1}{4^{n}} .

Similarly M_{n,2}, M_{n,∞} are P_{n}, T_{n}. i.e., the Legendre, Chebyshev polynomials, respectively, and

\begin{matrix}C_{2} = ∥\frac{2^{n}(n!)^{2}}{(2n)!} P_{n}∥_{2} 2^{−(n+ \frac{1}{2} )} = \frac{(n!)^{2}}{(2n)!\sqrt{2n + 1}}, \\ C_{∞} = ∥2^{−(n−1)}T_{n}∥_{∞} 2^{−(n+ \frac{1}{∞})} = \frac{2}{4^{n}} . \end{matrix}

The facts about U_{n}, P_{n}, T_{n} that we have used above can be found in any standard book on orthogonal polynomials (see, e.g., Szegö [Sz59:p41]).

−1  1 −0.08 0.08

Fig 3.1 Graphs of the polynomials M_{7,1} (thick), M_{7,2} (medium), and M_{7,∞} (thin)

As mentioned in the proof, when p = 1, 2, ∞, the M_{n,p} are well known orthogonal polynomials. For other values of p, no recurrence relations are known for M_{n,p}. By (3.2), for 1 ≤ p < ∞, M_{n,p} is the unique m ∈ Π_{n} with leading term (·)^{n} and

∫_{a}^{b} |m|^{p−1}sign(m) g = 0,  ∀g ∈ Π_{n−1}.  (3.4)

It is possible to view (3.4) as a nonlinear system of equations in the roots of M_{n,p} (with a unique solution), and solve it numerically. For two different iterative schemes, together with sample results, see Burgoyne [Bu67], and Vincenti [V86].

Gauss quadrature formulæ with multiple nodes

The polynomials M_{n,p} have the following interesting connection with quadrature (see Turan [T50], also Ghizzetti and Ossicini [GO70:p74]).

If p = 2s + 2, s = 0, 1, 2, …, then (3.2) reduces to

∫_{a}^{b} m^{2s+1}g = 0, ∀g ∈ Π_{n−1}.

The corresponding m (= M_{n,2s+2}) is called s-orthogonal (with weight dx).

There is a quadrature formula of the form

Q(f) := ∑_{i=0}^{2s} ∑_{v∈Θ} w(i, v)D^{i}f(v),  (3.5)

for the integral I(f) := ∫_{a}^{b} f, of precision (2s + 2)n − 1, if and only if Θ is the zeros of M_{n,2s+2}. In keeping with the special case s = 0, such a Q is referred to as a Gauss formulæ with multiple nodes, or simply as a s–Gauss formula, and M_{n,2s+2} is called a Legendre s-polynomial.

The s–Gauss formulæ are interpolatory, i.e. Q(f) = I(H_{Θ^{∗}}f), where Θ^{∗} is any set of ≤ n(2s + 2) points, which contains each zero of M_{n,2s+2} with multiplicity at least 2s + 1. This allows us to estimate the error for these formulæ.

Corollary 3.6 (Error bound for s–Gauss formulæ). Let Θ be the zeros of M_{n,2s+2}. Then

|I(f) − Q(f)| ≤ \frac{1}{(n(2s + 2))!}(∥ω_{Θ}∥_{2s+2})^{2s+2}∥D^{n(2s+2)}f∥_{∞}, ∀f ∈ W_{∞}^{n(2s+2)},

with equality for all f ∈ Π_{n(2s+2)}. In addition

(∥ω_{Θ}∥_{2s+2})^{2s+2} < (\frac{2}{4^{n}} )^{2s+2}(b − a)^{n(2s+2)+1},

which differs from equality by a factor of < 2^{2s+2}.

Proof: With Θ^{∗} as above, by (1.1)

|I(f)−Q(f)| = |I(f −H_{Θ^{∗}}f)| ≤ ∥f −H_{Θ^{∗}}f∥_{1} ≤ \frac{1}{(n(2s + 2))!}∥ω_{Θ^{∗}}∥_{1}∥D^{n(2s+2)}f∥_{∞}. (3.7)

Let Θ^{∗} consist of the points Θ, each with multiplicity 2s + 2. For this choice,

∥ω_{Θ^{∗}}∥_{1} = (∥ω_{Θ}∥_{2s+2})^{2s+2}.

Further, if f ∈ Π_{n(2s+2)}, then f −H_{Θ^{∗}}f is a scalar multiple of ω_{Θ}^{2s+2}, which is nonnegative, and so equality holds in (3.7). Finally, by Proposition 3.3,

(∥ω_{Θ}∥_{2s+2})^{2s+2} < (\frac{2}{4^{n}} (b − a)^{n+ \frac{1}{2s+2}} )^{2s+2} = (\frac{2}{4^{n}} )^{2s+2}(b − a)^{n(2s+2)+1},

which differs from equality by a factor of < 2^{2s+2}.

Only when s = 0 is this result known (see, e.g., Davis and Rabinowitz [DR75:p98]). In this case ∥ω_{Θ}∥_{2} is the L_{2}–norm of a Legendre polynomial, and can be computed exactly. For a full account of s–Gauss formulæ, including other error estimates, see the survey article of Gautschi [G81].

By using (2.4) and (2.6), it is possible to run through the above argument, to get error bounds for s–Gauss formulæ in terms of ∥D^{m}f∥_{q}, where n(2s + 1) ≤ m ≤ n(2s + 2).

4. Further applications

Good approximation by polynomials

Combining Corollary 2.2 and Proposition 3.3, we obtain:

\frac{1}{2} \frac{4^{n}}{(np + 1)^{1/p}} ≤ \frac{max_{Θ} ∥ω_{Θ}∥_{p}}{min_{Θ} ∥ω_{Θ}∥_{p}} ≤ \frac{4^{n}}{(np + 1)^{1/p}} ,

where (np + 1)^{1/p} := 1, when p = ∞. Thus, a good choice of Θ can greatly improve the size of the constant ∥ω_{Θ}∥_{p} occurring in (1.1) over that for a poor choice.

For example, with Θ_{Eq} consisting of points with equal spacing h := (b − a)/(n − 1), and Θ_{Ch} the Chebyshev points, Isaacson and Keller [IK66:p267] provide the estimate:

\frac{∥ω_{Θ_{Eq}}∥_{∞}}{∥ω_{Θ_{Ch}}∥_{∞}} > \frac{\sqrt{2}}{n − 1}(\frac{4}{e})^{n−1},

for large n, in support of doing Lagrange interpolation at the Chebyshev points.

Best approximation by polynomials

By Theorem 3.1, the unique best L_{p}–approximation to f ∈ C[a, b] from Π_{n−1} is obtained by Lagrange interpolation at n points in (a, b). Thus, in view of (1.1), we expect some relation between min_{Θ} ∥ω_{Θ}∥_{p}, and the error

E_{n,p}(f) := inf_{g∈Π_{n−1}} ∥f − g∥_{p}

in best L_{p}–approximation. The main result in this direction is the following.

Theorem 4.1 (Phillips [P70]). If f ∈ C^{n}[a, b], then ∃ ξ ∈ [a, b] such that

E_{n,p}(f) = \frac{∥M_{n,p}∥_{p}}{n!} |D^{n}f(ξ)| ≤ \frac{∥M_{n,p}∥_{p}}{n!} ∥D^{n}f∥_{∞},

with equality if and only if f ∈ Π_{n}.

Along the same lines, Fink [F77] defines B(n, p, q) as the smallest constant such that

E_{n,p}(f) ≤ B(n, p, q)(b − a)^{n+ \frac{1}{p} − \frac{1}{q}} ∥D^{n}f∥_{q}, ∀f ∈ W_{q}^{n},

and gives some equivalent definitions.

Since best approximations are given by Lagrange interpolation, we might hope to estimate B(n, p, q) by interpolating f at some Θ, as does Phillips in Theorem 4.1, where he shows:

B(n, p, ∞) = \frac{∥M_{n,p}∥_{p}}{n!} (b − a)^{−(n+ \frac{1}{p})}.  (4.2)

Pursuing this idea, we are able to estimate B(n, p, q) to within a factor of 8n.

Corollary 4.3 (Estimate for Fink’s constant).

\frac{1}{n!} \frac{1}{4^{n}} ≤ B(n, p, q) ≤ \frac{n^{\frac{1}{q}}}{n!} (\frac{2}{4^{n}} )^{1− \frac{1}{nq}} ≤ 8n\frac{1}{n!} \frac{1}{4^{n}} .

Proof: Let b − a = 1. First the lower bound. Since M_{n,p} is the error in approximating f = (·)^{n}, which has D^{n}f = n!, we must have

B(n, p, q) ≥ \frac{∥M_{n,p}∥_{p}}{n!} ≥ \frac{1}{n!} \frac{1}{4^{n}} .

By (2.4) and (2.6):

B(n, p, q) ≤ \frac{n^{\frac{1}{q}}}{n!} ∥ω_{Θ}^{1− \frac{1}{nq}} ∥_{p} = \frac{n^{\frac{1}{q}}}{n!} (∥ω_{Θ}∥_{p(1− \frac{1}{nq})})^{1− \frac{1}{nq}} .  (4.4)

Choosing ω_{Θ} = M_{n,p(1−1/nq)}, then applying Proposition 3.3 to (4.4), we obtain the upper bound.

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