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A3

An Interpolation Theorem

Overview

Appendix 3 presents an interpolation theorem for linear functions that is used in Section 7.1. From this Haar’s characterization of Chebyshev spaces follows, as do alternate proofs of many of the basic inequalities.

An Interpolation Theorem

The main result of this section, Theorem A.3.3, is an interpolation theorem that plays an important role in Section 7.1. Further applications are given in the exercises.

Throughout this appendix we use the following notation. Let Q be a compact Hausdorff space. Let C(Q) be the space of all real- or complexvalued continuous functions defined on Q. Let P be a (usually finitedimensional) linear subspace of C(Q) over ℝ if C(Q) is real or over ℂ if C(Q) is complex. The function f ∈ C(Q) is said to be orthogonal to P, written as f ⊥ P, if

∥f∥_{Q} ≤ ∥f + p∥_{Q} for all p ∈ P .

This is an L_{∞} analog of the more usual L_{2} notion of orthogonality, as in Section 2.2. The following two lemmas give necessary and sufficient conditions for the relation f ⊥ P.

An Interpolation Theorem 383

Lemma A.3.1. Let 0 ≠ f ∈ C(Q). The function f is orthogonal to P if and only if there exists no p ∈ P such that

(A.3.1)  Re(\overline{f(x)}p(x)) > 0

holds on

(A.3.2)  E := E(f) := {x ∈ Q : |f(x)| = ∥f∥_{Q}} .

Proof. Assuming there exists p ∈ P satisfying (A.3.1) on E defined by (A.3.2), we show that

∥f − ϵp∥_{Q} < ∥f∥_{Q}

for some ϵ > 0. Since the set E defined by (A.3.2) is compact, Re(\overline{f(x)}p(x)) attains its positive minimum, say, 2δ > 0, on E, and there exists an open set G containing E such that

Re(\overline{f(x)}p(x)) > δ > 0 , x ∈ G .

Since G^{c} := Q \ G is also compact, there exists an α > 0 such that

|f(x)| < (1 − α)∥f∥_{Q} , x ∈ G^{c} .

Thus, with a sufficiently small ϵ > 0,

|f(x) − ϵp(x)|^{2} ≤ ∥f∥_{Q}^{2} − 2ϵδ + ϵ^{2}∥p∥_{Q}^{2} < ∥f∥_{Q}^{2} , x ∈ G ,

while

|f(x) − ϵp(x)| ≤ (1 − α)∥f∥_{Q} + ϵ∥p∥_{Q} < ∥f∥_{Q} , x ∈ G^{c} .

Therefore

∥f − ϵp∥_{Q} < ∥f∥_{Q}

if ϵ > 0 is small enough.

Conversely, if f is not orthogonal to P, then there exists p ∈ P such that ∥f − p∥_{Q}^{2} < ∥f∥_{Q}^{2}, so

2 Re(\overline{f(x)}p(x)) > ∥p∥_{Q}^{2} ≥ 0 , x ∈ E ,

384 A3. An Interpolation Theorem

and the proof is finished. ⊓ ⊔ Lemma A.3.2. Let 0 ≠ f ∈ C(Q) and let P be an n-dimensional linear subspace of C(Q) over ℝ if C(Q) is real or over ℂ if C(Q) is complex. Then the function f is orthogonal to P if and only if there exist points x_{1}, x_{2}, …, x_{r} in E(f) defined by (A.3.2) and positive real numbers c_{1}, c_{2}, …, c_{r}, where 1 ≤ r ≤ n+1 when C(Q) is real, and 1 ≤ r ≤ 2n+1 when C(Q) is complex, such that

(A.3.3)  ∑_{i=1}^{r} c_{i}\overline{f(x_{i})}p(x_{i}) = 0 ,  p ∈ P .

Proof. Suppose (A.3.3) holds with some positive real c_{1}, c_{2}, …, c_{r} satisfying ∑_{i=1}^{r} c_{i} = 1. As |f(x_{i})| = ∥f∥_{Q}, we have

∥f∥_{Q}^{2} = ∑_{i=1}^{r} c_{i}\overline{f(x_{i})}f(x_{i}) = ∑_{i=1}^{r} c_{i}\overline{f(x_{i})}(f(x_{i}) − p(x_{i}))

≤ ∥f∥_{Q} ∑_{i=1}^{r} c_{i} max_{1≤j≤r} |f(x_{j}) − p(x_{j})| ≤ ∥f∥_{Q}∥f − p∥_{Q}

for every p ∈ P, so f ⊥ P. (Note that r ≤ n + 1 or r ≤ 2n + 1, respectively, was not needed for this part of the proof, so the sufficiency of (A.3.3) is valid with no hypothesis about r.)

Conversely, suppose f ⊥ P. Let {φ_{1}, φ_{2}, …, φ_{n}} be a basis for P over ℝ (or ℂ, respectively), and consider the map T : Q → ℝ^{n} (or ℂ^{n}, respectively) defined by

T (x) = \overline{f(x)}(φ_{1}(x), φ_{2}(x), …, φ_{n}(x)) .

Observe that the origin is in the convex hull of

T (E) := {T (x) : x ∈ E} ,

otherwise by the principle of separating hyperplanes (a corollary of the Hahn-Banach theorem; see Rudin [73]), there would exist complex numbers a_{1}, a_{2}, …, a_{n} such that

Re (∑_{i=1}^{n} a_{i}\overline{f(x)}φ_{i}(x)) > 0 , x ∈ E .

Hence, with p := ∑_{i=1}^{n} a_{i}φ_{i} ∈ P,

Re(\overline{f(x)}p(x)) > 0 , x ∈ E

and f is not orthogonal to P by Lemma A.3.1, which contradicts our assumption.

Now it follows from Caratheodory’s lemma (see E.1) that there exist points x_{1}, x_{2}, …, x_{r} in E and positive real numbers c_{1}, c_{2}, …, c_{r}, where 1 ≤ r ≤ n+1 when C(Q) is real and 1 ≤ r ≤ 2n+1 when C(Q) is complex, such that

∑_{i=1}^{r} c_{i}\overline{f(x_{i})}φ_{k}(x_{i}) = 0, k = 1, 2, …, n.

Therefore

∑_{i=1}^{r} c_{i}\overline{f(x_{i})}p(x_{i}) = 0, p ∈ P ,

and the lemma is proved. ⊓ ⊔

Theorem A.3.3 (Interpolation of Linear Functionals). Let C(Q) be the set of real- (complex-) valued continuous functions on the compact Hausdorff space Q. Let P be an n-dimensional linear subspace of C(Q) over ℝ (ℂ). Let L ≠ 0 be a real- (complex-) valued linear functional on P. Then there exist points x_{1}, x_{2}, …, x_{r} in Q, and nonzero real (complex) numbers a_{1}, a_{2}, …, a_{r}, where 1 ≤ r ≤ n in the real case and 1 ≤ r ≤ 2n − 1 in the complex case, such that

(A.3.4)  L(p) = ∑_{i=1}^{r} a_{i}p(x_{i}) ,  p ∈ P

and

(A.3.5)  ∥L∥ = ∑_{i=1}^{r} |a_{i}| ,

where

∥L∥ := sup{|L(p)| : p ∈ P , ∥p∥_{Q} ≤ 1} .

Proof. Because of the finite dimensionality of P, there exists an element p^{∗} ∈ P (called an extremal element for L) such that ∥p^{∗}∥_{Q} = 1 and L(P^{∗}) = ∥L∥. Let P_{0} denote the null-space of L, so

P_{0} := {p ∈ P : L(p) = 0} .

Now p^{∗} is orthogonal to P_{0} because if

∥p^{∗} + p_{0}∥_{Q} < ∥p^{∗}∥_{Q} = 1

for some p_{0} ∈ P, then g := p^{∗} + p_{0} satisfies ∥g∥_{Q} < 1 and L(g) = ∥L∥, which is impossible. Note that the dimension of P_{0} over ℝ is n − 1 in the real case and 2n − 2 in the complex case. So by Lemma A.3.2 there exist points x_{1}, x_{2}, …, x_{r} in

E := {x ∈ Q : |p^{∗}(x)| = 1}

and positive real numbers c_{1}, c_{2}, …, c_{r}, where 1 ≤ r ≤ n in the real case and 1 ≤ r ≤ 2n − 1 in the complex case, such that

∑_{i=1}^{r} c_{i}\overline{p^{∗}(x_{i})}p_{0}(x_{i}) = 0 , p_{0} ∈ P_{0} .

Since L(p)p^{∗} − L(p^{∗})p ∈ P_{0} for all p ∈ P, we have

L(p) ∑_{i=1}^{r} c_{i}|p^{∗}(x_{i})|^{2} = L(p^{∗}) ∑_{i=1}^{r} c_{i}\overline{p^{∗}(x_{i})}p(x_{i}) , p ∈ P .

Since L(p^{∗}) = ∥L∥ and |p^{∗}(x_{i})| = 1 for each i, we obtain (A.3.4) by taking

a_{i} = \frac{c_{i}\overline{p^{∗}(x_{i})}}{∑_{j=1}^{r} c_{j}} ∥L∥ .

Using the fact that |p^{∗}(x_{i})| = 1 for each i in the above formula for a_{i}, we get (A.3.5). ⊓ ⊔

Comments, Exercises, and Examples.

We have followed Shapiro [71], which gives a long discussion of questions related to the best uniform approximation of a function f ∈ C(Q) from a (usually finite-dimensional) linear subspace P ⊂ C(Q). Some of these, together with other applications, are discussed in the exercises.

E.1 Caratheodory’s Lemma. If A ⊂ ℝ^{n}, then every point from the convex hull co(A) of A can be written as a convex linear combination of at most n + 1 points of A.

Proof. Let x ∈ A. After a translation if necessary, we may assume that x = 0. Suppose

(A.3.6)  0 = ∑_{i=1}^{r} α_{i}x_{i} ,  x_{i} ∈ A , α_{i} > 0 , r > n + 1 .

Since r > n+1, the elements x_{2}, x_{3}, …, x_{r} are linearly dependent, so there exist real numbers β_{i}, i = 2, 3, …, r, not all zero, such that

∑_{i=2}^{r} β_{i}x_{i} = 0 .

Let β_{1} := 0. For all λ ∈ ℝ, we have

0 = ∑_{i=1}^{r} α_{i}x_{i} + λ ∑_{i=1}^{r} β_{i}x_{i} = ∑_{i=1}^{r} (α_{i} + λβ_{i})x_{i} .

When λ = 0, each term in the last sum is positive. We now define c := min |α_{i}/β_{i}|, where the minimum is taken for all indices i for which β_{i} ≠ 0. If the index j is chosen so that |α_{j}/β_{j}| = c, and if λ := −α_{j}/β_{j}, then at least one of the numbers α_{i} + λβ_{i} is zero, and all are nonnegative. Also α_{1} + λβ_{1} = α_{1} > 0. We have thus obtained a representation of the same form as (A.3.6), but with s terms, where 1 ≤ s ≤ r − 1. If s > n + 1, then the process can be repeated, and after a finite number of steps we obtain the desired representation. ⊓ ⊔

E.2 Reformation in Terms of Integrals. Lemma A.3.2 can be reformulated as follows. Under the assumptions of Lemma A.3.2, f ∈ C(Q) is orthogonal to P if and only if there exists a nonzero nonnegative Borel measure ρ on Q whose support consists of r points of E(f) defined by (A.3.2), where 1 ≤ r ≤ n + 1 in the real case and 1 ≤ r ≤ 2n + 1 in the complex case, such that \overline{f(x)} dρ(x) annihilates P, that is,

(A.3.7)  ∫_{Q} \overline{f(x)}p(x) dρ(x) = 0 ,  p ∈ P .

This reformation is not only a notational convenience, but it is essential in generalizations where P is no longer finite-dimensional. Moreover, (A.3.7) with any nonzero nonnegative Borel measure (not necessarily discrete) is sufficient for f ⊥ P. This is often useful, even when P is finite-dimensional; see Shapiro [71].

E.3 Haar’s Characterization of Chebyshev Spaces.

a] Let f_{0}, f_{1}, …, f_{n} be real- or complex-valued continuous functions defined on a (not necessarily compact) Hausdorff space Q. Show that P := span{f_{0}, …, f_{n}}, where the span is taken over ℝ (or ℂ), is a Chebyshev space if and only if there exists no real (or complex) measure on Q annihilating P whose support consists of less than n + 1 points.

Hint: Use Proposition 3.1.2. ⊓ ⊔

b] Let P be an n-dimensional linear subspace of C(Q), the space of real- (or complex-) valued continuous functions defined on a compact Hausdorff space Q containing at least n points. The space P is a Chebyshev space if and only if for each f ∈ C(Q), there is a unique best uniform approximation to f from P.

Proof. First suppose P is a Chebyshev space of dimension n, and p_{1} and p_{2} are best uniform approximations to some f ∈ C(Q) from P. Then p_{3} := \frac{1}{2}(p_{1} + p_{2}) is also a best uniform approximation to f from P. As f − p_{3} ⊥ P, Lemma A.3.2 yields that |f(x) − p_{3}(x)| attains its maximum on Q at r points, x_{1}, x_{2}, …, x_{r}, that support an annihilating measure for P, where r ≥ n + 1 by part a]. Note that

p_{1}(x_{j}) − f(x_{j}) = p_{2}(x_{j}) − f(x_{j}) , j = 1, 2, …, n + 1 ,

and hence, since P is a Chebyshev space, p_{1} − p_{2} = 0.

Conversely, if P is not a Chebyshev space, then there exist n distinct points x_{1}, x_{2}, …, x_{n} in Q such that the system of homogeneous linear equations

( \begin{matrix}g_{1}(x_{1}) & … & g_{1}(x_{n}) \\ ⋮ & ... & ⋮ \\ g_{n}(x_{1}) & … & g_{n}(x_{n}) \end{matrix} ) ( a_{1} ; ⋮ ; a_{n} ) ,

where {g_{1}, …, g_{n}} is a basis for P, has a nontrivial solution. Then also the homogeneous system formed with the transposed matrix has a nontrivial solution, so there exist constants b_{i}, not all zero, so that

∑_{i=1}^{n} b_{i}g_{i}(x_{j}) = 0 , j = 1, 2, …, n .

Thus, with g := ∑_{i=1}^{n} b_{i}g_{i}, we have

g(x_{j}) = 0 , j = 1, 2, …, n .

We may assume, without loss, that ∥g∥_{Q} = 1. Some of the constants a_{1}, a_{2}, …, a_{n} may be zero; however, the set Γ of indices j for which a_{j} ≠ 0 is not empty. By Tietze’s theorem there exists an f ∈ C(Q) such that ∥f∥_{Q} = 1 and

f(x_{j}) = \frac{\overline{a}_{j}}{|a_{j}|} , j ∈ Γ .

Setting h(x) := f(x)(1 − |g(x)|), we have

h(x_{j}) = \frac{\overline{a}_{j}}{|a_{j}|} , j ∈ Γ .

We claim that ∥h − p∥_{Q} ≥ 1 for every p ∈ P. Indeed, if ∥h − p∥_{Q} < 1 for some p ∈ P, then

|f(x_{j}) − p(x_{j})|^{2} = |f(x_{j})|^{2} − 2 Re(p(x_{j})\overline{f(x_{j})}) + |p(x_{j})|^{2} < 1

for every j ∈ Γ; hence

Re(a_{j}p(x_{j})) > 0 , j ∈ Γ .

Since a_{j} = 0 if j ∉ Γ, we have Re(a_{j}p(x_{j})) = 0 for each j ∉ Γ. Thus

Re (∑_{j=1}^{n} a_{j}p(x_{j})) > 0 .

However, if p := ∑_{i=1}^{n} c_{i}g_{i}, then

∑_{j=1}^{n} a_{j}p(x_{j}) = ∑_{j=1}^{n} a_{j} ∑_{i=1}^{n} c_{i}g_{i}(x_{j}) = ∑_{i=1}^{n} b_{i} ∑_{j=1}^{n} a_{i}g_{i}(x_{j}) = 0 ,

which contradicts the previous inequality and shows that ∥h − p∥_{Q} ≥ 1 for every p ∈ P.

Finally, for all λ ∈ [0, 1], λg is a best uniform approximation to h from P because

|h(x) − λg(x)| ≤ |f(x)|(1 − |g(x)|) + λ|g(x)|
≤ 1 + (λ − 1)|g(x)| ≤ 1

for all x ∈ Q, so the best uniform approximation to h ∈ C(Q) from P is not unique. ⊓ ⊔

E.4 Unicity of the Extremal Function. Assume the notation of Theorem A.3.3. Show that if P ⊂ C(Q) is an n-dimensional real Chebyshev space and r = n, then the extremal element p^{∗} ∈ P satisfying ∥p^{∗}∥_{Q} = 1 and L(p^{∗}) = ∥L∥ is unique.

The interesting relations of Theorem A.3.3 to the Riesz representation theorem, the Krein-Milman theorem, and the Hahn-Banach theorem are discussed in Shapiro [71].

E.5 Applications of the Interpolation Theorem. As before, let

D := {z ∈ ℂ : |z| < 1}  and  K := ℝ (mod2π) .

Prove the following statements. Each of them may be proven by characterizing the extremal polynomial for the given inequality with the help of Theorem A.3.3. A detailed hint is given only to part a].

a] Bernstein’s Inequality.

|t′(θ)| ≤ n∥t∥_{K} , t ∈ T_{n} , θ ∈ ℝ .

Hint: Let θ_{0} ∈ ℝ be fixed, and study the linear functional L(t) := t′(θ_{0}), t ∈ T_{n}. Observe that an extremal p in Lemma A.3.3 must satisfy |p(x_{i})| = 1 for each i = 1, 2, …, r and r must equal 2n. Note that r ≤ 2n holds by Theorem A.3.3, while the argument for r ≥ 2n is similar to the corresponding step in the proof of Theorem 7.1.7. Finally, show that the extremal element t satisfying L(t) = ∥L∥ is of the form

t(θ) = cos(nθ − α)

for some α ∈ K. ⊓ ⊔

b] Markov’s Inequality.

|p′(1)| ≤ n^{2}∥p∥_{[−1,1]} , p ∈ P_{n} .

c] Chebyshev’s Inequality.

|p(x)| ≤ |T_{n}(x)| ∥p∥_{[−1,1]} , p ∈ P_{n} , x ∈ ℝ \ [−1, 1] ,

where T_{n} is the Chebyshev polynomial of degree n defined by (2.1.1).

d] Bernstein’s Inequality.

|p(z)| ≤ |z|^{n}∥p∥_{D} , p ∈ P_{n}^{c} , z ∈ ℂ \ \overline{D} .

e] Bernstein’s Inequality.

|p′(z)| ≤ n|z|^{n−1}∥p∥_{D} , p ∈ P_{n}^{c} , z ∈ ℂ \ \overline{D} .

Hint: Use Theorem 1.3.1 (Lucas’ theorem). ⊓ ⊔

f] Riesz’ Identity. There are real numbers a_{i} with ∑_{i=1}^{2n} |a_{i}| = n such that

t′(θ) = ∑_{i=1}^{2n} a_{i}t(θ + θ_{i}) , t ∈ T_{n} , θ ∈ ℝ ,

where

θ_{i} := \frac{2i−1}{2n} π , i = 1, 2, …, 2n .

(This is, apart from the explicit determination of the number a_{i}, an identity discovered by M. Riesz [14].)

g] Show that in part f],

a_{i} = (−1)^{i+1} \frac{1}{4n sin^{2}(θ_{i}/2)} , i = 1, 2, …, 2n .

h] Bernstein’s Inequality in L_{p}.

∫_{0}^{2π} |t′(θ)|^{p} dθ ≤ n^{p} ∫_{0}^{2π} |t(θ)|^{p} dθ , t ∈ T_{n} , p ≥ 1 .

Hint: Use part f] and Jensen’s inequality (see E.20 of Appendix 4). ⊓ ⊔

Arestov [81] shows that the inequality of part h] is valid for all p > 0. Golitschek and Lorentz [89] gives a simpler proof of this.

i] Find all extremal polynomials in parts a] to e] and h].

An Interpolation Theorem 391

E.6 An Inequality of Szegő. If p ∈ P_{n}^{c} and z_{1}, z_{2}, …, z_{2n} are any equally spaced points on the unit circle ∂D, then

∥p′∥_{D} ≤ n max_{1≤k≤2n} |p(z_{k})| .

Proof. See Frappier, Rahman, and Ruscheweyh [85]. ⊓ ⊔