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Ukrainian Mathematical Journal, Vol. 59, No. 8, 2007

BEST APPROXIMATION BY HOLOMORPHIC FUNCTIONS. APPLICATION TO THE BEST POLYNOMIAL APPROXIMATION OF CLASSES OF HOLOMORPHIC FUNCTIONS V. V. Savchuk

UDC 517.9

We establish necessary and sufficient conditions under which a real-valued function from Lp (T ), 1 ≤ p < ∞, is badly approximable by the Hardy subspace Hp0 := {f ∈ Hp : f (0) = 0}. In a number of cases, we obtain the exact values of the best approximations in the mean of functions holomorphic in the unit disk by functions holomorphic outside this disk. We use the obtained results for finding the exact values of the best polynomial approximations and n-widths of some classes of holomorphic functions. We establish necessary and sufficient conditions under which the generalized Bernstein inequality for algebraic polynomials on the unit circle is true.

1. Introduction. Preliminary Information Assume that D := z ∈ C : |z| < 1 is the unit disk in the complex plane C, T := z ∈ C : |z| = 1 is the unit circle in the complex plane, σ is the normalized Lebesgue measure on T, Lp (T), 1 ≤ p ≤ ∞, is the space of functions f summable on T with respect to σ with the norm  1/p

     |f |p dσ  , 1 ≤ p < ∞,   

T f L (T) := p       ess sup |f (z)|, z∈T

p = ∞,

and Hp is the Hardy space of functions f holomorphic in D with the norm

f Hp :=

 1/p 

 

 p   sup  f ( ·) dσ  , 1 ≤ p < ∞,  0 0. Since the function f is real-valued, it follows from (3) that 0 ≤ F f, i.e., Im F = 0 almost everywhere on T. Therefore, according to the Schwartz formula, we get

1 + wz F (z) = i Im F (w) dσ(w) + Re F (0) = Re F (0) ∀z ∈ D. (5) 1 − wz T

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V. V. S AVCHUK

Thus, F ≡ c, c ∈ R. Further, on the set Sf− , we have 0 ≤ cf. Thus, c ≤ 0. However, we also have 0 ≤ cf on the set Sf+ . Therefore, c ≥ 0. Thus, 0 = c ≡ F and, by virtue of (3), f ≡ 0, which contradicts the conditions of the theorem. The contradiction obtained proves the relation 0 = min σ(Sf− ), σ(Sf+ ) < max(σ(Sf− ), σ(Sf+ ) ≤ 1. Thus, the function f is sign-preserving on T and F ≡ c, c ∈ R, |c| = 1. If 1 < p < ∞, then these facts, together with relation (4), imply that f = cf p = const almost everywhere on T. (ii) ⇒ (i). Assume, for definiteness, that f ≥ 0 almost everywhere on T if p = 1 and f = const > 0 if 1 < p < ∞. On the one hand, min f + gp ≤ f p .

g∈Hp0

On the other hand, according to relation (2),

min f + gp = max F f dσ

≥ 1 · f dσ = f p . F ∈Hq g∈Hp0

F q ≤1 T

T

Thus, min f + gp = f p

g∈Hp0

and, according to Remark 1, the function g ≡ 0 is the unique element of Hp0 that best approximates the function f. Remark 2. In the case where a real-valued function f ∈ Lp (T), 1 < p < ∞, is approximated by the set Hp , one can prove the following statement: min f + gp = f p ⇔ f ≡ 0.

g∈Hp

Indeed, if this equality is true, then f p = min f + gp ≤ min f + gp ≤ f p . g∈Hp

g∈Hp0

Thus, by virtue of Theorem 1, the function f is constant. According to the duality relation, equality (3) in which the dual extremal function F belongs to Hq0 holds almost everywhere on T. Thus, it follows from the Schwartz formula (5) that F ≡ 0. However, relation (4) can be satisfied only if f = 0 almost everywhere on T. A characterization of a subset of functions continuous on T that are badly approximable by the space H∞ was obtained in [3, 4].

B EST A PPROXIMATION BY H OLOMORPHIC F UNCTIONS

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3. Approximation in the Mean of Functions Holomorphic in D by Functions Holomorphic in D∞ In this section, we use the following notation: H+ and H− are the sets of functions holomorphic in D and D∞ , respectively; 0 is the set of functions f holomorphic in D H− ∞ and such that f (∞) = 0;

Pn is the set of algebraic polynomials of degree not higher than n, n ∈ N; S + and S − are the sets of Cauchy–Stieltjes-type integrals in D and D∞ , respectively; R+ is the set of functions holomorphic in D for which Re f ≥ 0; Hp,− is the Hardy space in D∞ . Recall that a function f ∈ H+

0 f ∈ H− of the form

f (z) = T

dµ(w) 1 − wz

∀z ∈ D

(z ∈ D∞ ),

(6)

where µ is a complex charge on T, is called a Cauchy–Lebesgue-type integral in D respectively, in D∞ . 0 , and Let f ∈ H+ , g ∈ H− f +

g∗1

1 := lim f ( ·) + g · . →1−

1

∗ The value inf g∈H− 0 f + g1 is called the best approximation in the mean of the function f ∈ H+ by the set 0 . If this value is finite, then we say that the function f can be approximated by the set H0 . H− − The main result of [5] yields a characterization of the set of functions from H+ that can be approximated by 0 . We can reformulate the main result of [5] as follows: the set H− 0 , it is necessary and Proposition 1. In order that a function f ∈ H+ can be approximated by the set H− + 0 − sufficient that f ∈ S ; in this case, the approximating subset of H− is S . 0 as well as S + and S − . Proposition 1 remains true if one formally interchanges H+ and H− Let us show the relationship between the set S + and some sets of holomorphic functions used in what follows. First of all, note that, by virtue of theorems of F. Riesz, M. Riesz, and V. Smirnov (see, e.g., [1, pp. 39, 40; 2, p. 100]), we have H1 ⊂ S + ⊂ Hp . 0 0 be approximated in the mean by the set H− − f f 0 , where in the disk D. For every fixed ∈ [0, 1), we have f ( ·) ∈ H1 and gf (1/ ·) ∈ H1,− 0 0 := f ∈ H− : z → f (1/z) ∈ H1 , z ∈ D . H1,− Therefore, by virtue of Theorem 1, we get min f ( ·) + g(·)1 = min f ( ·) + gf (1/ ·) + h(·)1 = f ( ·) + gf (1/ ·)1 = f0 .

0 g∈H1,−

0 h∈H1,−

(9)

According to the duality theorem, gf (1/ ·) is the unique function for which the lower bound in (9) is realized. 0 , one has This implies that, for any function h ∈ H− f ( ·) + gf (1/ ·)1 ≤ f ( ·) + h(1/ ·)1

∀ ∈ [1, 0),


1169

whence f + gf ∗1 ≤ f + h∗1 . 0 for which f + g ∗ = inf ∗ If g1 is another function from H− 0 f + g1 , then, by virtue of relation (9) 1 1 g∈H− and the fact that the function z → f (z) + g1 (1/z) in the disk D is a function from the harmonic Hardy space h1 (see, e.g., [1, p. 2]), we have

f0 = f ( ·) + gf (1/ ·)1 ≤ f ( ·) + g1 (1/ ·)1 ≤ f0

∀ ∈ [0, 1).

Thus, according to the duality theorem, we get g1 = gf . (ii) ⇒ (i). For every fixed ∈ [0, 1), by virtue of the duality theorem (see Remark 1), there exists a unique 0 for which function g∗ ∈ H1,−

min f ( ·) + g(·)1 =

0 g∈H1,−

|f ( w) + g∗ (w)|dσ(w) = f0 .

T

0 , we also have Since g∗ ∈ H1,−

(f ( w) + g∗ (w))dσ(w) = f0 .

T

Thus, the following relation holds almost everywhere on T: 0 ≤ f ( w) + g∗ (w) = |f ( w) + g∗ (w)|.

(10)

Further, we consider the function ϕ (z) := g∗ (1/z) − f ( z) + f0 , z ∈ D. The function ϕ belongs to and, according to (10), we have Im ϕ = 0 almost everywhere on T. Therefore, by virtue of the Schwartz formula [see (5)], we get ϕ (z) = c, c ∈ R, for all z ∈ D. Since ϕ ∈ H10 , we have 0 = ϕ (0) = c. Thus, ∗ Re g (w) = Re f ( w) − f0 almost everywhere on T, and relation (10) takes the form 0 ≤ 2 Re f ( w) − f0 .

H10

Remark 3. If the function f belongs to the Hardy space H1 ⊂ H+ , then, throughout the proof of Theorem 2, we can assume that = 1, and the functions f and gf can be understood as their limit values on T. Now consider the best approximation of a function f ∈ H+ by the set 0 0 := f : f = p + g, p ∈ Pn−1 , g ∈ H− Pn−1 + H− for fixed n ∈ N. It is clear that the possibility of this approximation is is completely characterized by Proposition 1. 0 , then, by virtue of the Cauchy formula, we get If f ∈ H+ and g ∈ Pn−1 + H− 1 fn = n

w T

n

f ( w) + g

1 w

dσ(w) ∀ ∈ [0, 1),

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V. V. S AVCHUK

whence inf

0 g∈Pn−1 +H−

f + g∗1 ≥ |fn |.

(11)

Our immediate aim is to find conditions for the function f under which relation (11) turns into equality. Let Sn−1 (f )(z) :=

n−1

fk z k

k=0

be a partial sum of the Taylor series of a function f ∈ H+ . The following statement is a direct corollary of Theorem 2: Theorem 3. Suppose that n ∈ N, f ∈ H1 , and |fn | > 0. The following assertions are equivalent: (i) 2 Re

f (z) − Sn−1 (f )(z) ≥ 1 in the disk D; z n fn

(i)

inf

0 g∈Pn−1 +H1,−

f + g1 = f + gf 1 = |fn |, where

gf (z) =

n−1

f2n−k ei2 arg fn − fk z k + ei2 arg fn z 2n f (1/z) − S2n (f )(1/z)

k=0 0 that best approximates the function f. is the unique element from Pn−1 + H1,−

Proof. Consider the functions ϕ(z) :=

1 z n fn

(f (z) − Sn−1 (f )(z)) ,

z ∈ D,

and 0 , gϕ (z) := ϕ(1/z) − ϕ

z ∈ D∞ .

The equivalence of assertions (i) and (ii) of the theorem is proved by the application of Theorem 2 to the functions ϕ and gϕ with regard for Remark 3 and the equality ϕ(w) + gϕ (w) =

=

=

1 fn wn

1 wn fn 1 wn fn

(f (w) − Sn−1 (f )(w)) + f (w) +

n−1

f2n−k e

k=0

(f (w) + gf (w)),

which holds at almost every point w ∈ T.

1 fn wn

i2 arg fn

f (w) − Sn−1 (f )(w) − 1

− fk wk + e

i2 arg fn

w2n f (w) − S2n (f )(w)


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We now give some other corollaries of Theorem 2. Corollary 1. If f belongs to H∞ , f = const, and |f | ≤ K in the disk D, then inf f + g1 ≤ min K, 2(K − |f0 |) .

g∈H1,−

Proof. It is obviously sufficient to prove the inequality inf f + g1 ≤ 2(K − |f0 |).

g∈H−

To this end, we consider the function ϕ := 2K − |f0 | − e−i arg f0 f.

It is clear that ϕ 0 = 2 K − |f0 | and 0 > 0. 2 Re ϕ = 2 2K − |f0 | − Re(e−i arg f0 f ) ≥ 2 K − |f0 | = ϕ Thus, by virtue of Theorem 2, we get inf f + g1 =

g∈H1,−

inf e−i arg f0 f + g 1 ≤ min ϕ + g1 = ϕ 0 = 2 K − |f0 | . 0 g∈H1,−

g∈H1,−

A function f, f (0) = 0, holomorphic in D is called convex in D if it is univalent and the image f (D) is a convex domain. Corollary 2. If f is a convex function in D, then the function gf (z) := z 2 f (1/z) − f1 z is the unique element from H− that best approximates the function f and min f + g∗1 = f + gf ∗1 = |f1 |.

g∈H−

Proof. Since the function f is univalent, we have f1 = f (0) = 0. Consider the function F (z) :=

f (z) . f1 z

Since F (0) = 1 and Re F > 1/2 (see, e.g., [6, p. 45]), by virtue of Theorem 2 we get 1 = min F + g∗1 = F + GF ∗1 = 0 g∈H−

where GF (z) := F (1/z) − 1.

1 f + gf ∗1 , |f1 |

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V. V. S AVCHUK

This yields the relation min f + g∗1 ≤ f + gf ∗1 = |f1 |,

g∈H−

which, with regard for (11), is, in fact, an equality. Since, according to Theorem 2, the function GF (the element of the best approximation for F ) is unique, this implies the uniqueness of the function gf . Note that relation (11) and Corollaries 1 and 2 directly yield the following estimates for the Taylor coefficients of holomorphic functions: |fn | ≤ min K, 2(K − |f0 |)

∀n ∈ N

if

|f | ≤ K

and |fn | ≤ |f1 | ∀n ∈ N

if f is convex.

Each of these estimates is unimprovable on the corresponding class of functions. The first estimate is known as the Carathéodory inequality and the second is the Löwner–Privalov inequality (see, e.g., [6, p. 46]). Corollary 3. Suppose that f belongs to H+ and

f (z) = 1 +

∞

fk z k ,

z ∈ D.

k=1

If ∞ k=1

1 |fk | ≤ , 2

0 that best approximates the then the function gf (z) = f (1/z) − 1, z ∈ D∞, is the unique element from H1,− function f and

min f + g1 = f + gf 1 = 1.

(12)

0 g∈H1,−

Proof. It is easy to see that 2 Re f (eit ) − 1 = 1 + 2

∞

|fk | cos(kt − arg fk ) ≥ 1 − 2

k=1

∞

|fk | ≥ 0,

k=1

i.e., 2 Re f ≥ 1 everywhere on T. Thus, by virtue of Theorem 2, equality (12) is true.

A function h : D → C is called inner if h is holomorphic in D, h(z) ≤ 1 for all z ∈ D, and |h(z)| = 1 for almost all z ∈ T.


1173

A function holomorphic in D and having the form  Q(z) = c exp 

T

 w+z log ψ(w)dσ(w) , w−z

where c ∈ C, |c| = 1, and ψ is a nonnegative function from Lp (T) such that log ψ ∈ L1 (T), is called outer in the space Hp , p > 0. It is convenient to introduce the notions of inner and outer functions in the space Hp,− , p > 1 as follows: A function f holomorphic in D∞ is called inner in the space Hp,− if the function z → f (1/z), z ∈ D, is outer. A function f holomorphic in D∞ is called outer in the space Hp,− , p > 0, if the function z → f (1/z), z ∈ D, is outer in the space Hp . It is known that every function f ∈ Hp , p > 0, can be uniquely represented, up to a factor of unit absolute value, in the form f = hQ, where h is an inner function and Q is an outer function. Denote (13) Hpn := f ∈ Hp : fj = 0, j = 0, n , n ∈ Z+ . Below, we give an elementary statement necessary for what follows. Proposition 2. Suppose that n ∈ N, 1 ≤ p ≤ ∞, f : T → C is a function from Lp (T), and h is an inner function in D. Then min hf + gp ≤ minn f + gp ≤ minn hf + gp .

g∈Hpn

g∈Hp

g∈Hp

(14)

Proof. If g ∈ Hpn , then hg ∈ Hpn . Thus, min hf + gp = minn h(f + hg)p = minn f + hgp ≥ minn f + gp ,

g∈Hpn

g∈Hp

g∈Hp

g∈Hp

(15)

and the right inequality in (14) is proved. Continuing relation (15), we get min f + gp = minn h(f + g)p ≥ minn hf + gp .

g∈Hpn

g∈Hp

g∈Hp

Remark 4. If f belongs to H2,− , then relations (14) turn into equalities for p = 2 because, in this case, the function g ≡ 0 is the unique element that best approximates the functions hf, f, and hf. Using Proposition 2 and Theorem 1, we obtain the following corollaries: Corollary 4. Suppose that 1 ≤ p ≤ ∞, f ∈ Hp , and f = hQ is the factorization of f into the inner and outer multipliers. Then min Q + gp ≤ min f + gp .

0 g∈Hp,−

(16)

0 g∈Hp,−

Proof. Indeed, the function h1 (z) := h(1/z), z ∈ D∞ , is inner in Hp,− . Therefore, by virtue of the second inequality in (14), we get min Q + gp ≤ min h1 · Q + gp = min h · Q + gp = min f + gp .

0 g∈Hp,−

0 g∈Hp,−

0 g∈Hp,−

0 g∈Hp,−

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V. V. S AVCHUK

Corollary 5. Let 1 ≤ p ≤ ∞ and let h be an inner function in D. Then min h + gp = hp = 1.

g∈Hp0

Proof. According to the first inequality in (14), we have min 1 + gp ≤ min h + gp ≤ hp = 1.

g∈Hp0

g∈Hp0

However, by virtue of Theorem 1, the left-hand side of the last relation is equal to 1 for 1 ≤ p < ∞. 0 that best approximates 1, one has |1+g | ≤ 1 almost everywhere If p = ∞, then, for the function g1 ∈ H∞ 1 on T, which implies that Re(−g1 ) ≥ 0 in the disk D, i.e., −g1 ∈ R+ . Thus, according to the Riesz–Herglotz theorem [see (7)], we have

dµ(w) + g1 (0) ∀z ∈ D, −g1 (z) = 1 − wz T

where µ is the Borel measure on T. However, g1 (0) = 0. Therefore,

dµ = |µ|(T).

0= T

Since µ ≥ 0, this implies that µ ≡ 0 and g1 = 0. 4. Best Approximation of Classes of Holomorphic Functions by Algebraic Polynomials For a sequence of complex numbers ψ := {ψk }∞ k=0 such that lim

k

k→∞

|ψk | ≤ 1,

we construct the function Ψ(z) :=

∞

ψk z k ,

z ∈ D.

k=0

If a function f belongs to H+ , then the sum of the series (f ∗ Ψ)(z) :=

∞

fk ψk z k

k=0

defines a function holomorphic in D and is called the Hadamard convolution of the functions f and Ψ. If, for a given sequence ψ, a function f ∈ H+ can be represented in the form of the convolution g ∗ Ψ with a certain function g ∈ H+ , then one says [7] that f is the ψ-integral of the function g. In turn, the function g is called the ψ-derivative of the function f and is denoted by f ψ . If X is a certain normed linear space, then U X denotes the unit ball in it.


1175

We define a class Hpψ , 1 ≤ p ≤ ∞, as follows: Hpψ := f ∈ H+ : f ψ ∈ U Hp . ∞ ψk z k ∈ H+ is the generating kernel of the class Hpψ if it satisfies the We say that a function Ψ(z) = k=0 conditions presented above and |ψk | > 0 for all k ∈ Z+ . It is clear that if Ψ is the generating kernel and f ∈ Hpψ , then f ψ (z) =

∞ 1 k fk z ∈ U Hp , ψk

z ∈ D.

k=0

Consider the sequence of linear operators {Un }∞ 0 defined on H+ and acting according to the rule

Un (f )(z) = Un,Λ (f )(z) =

n−1

λnk fk z k ,

(17)

k=0

where λnk are elements of the infinite lower triangular matrix Λ := {λnk } , n = 1, ∞, k = 0, n − 1, over the field of complex numbers. Thus, any lower triangular matrix Λ generates, according to rule (17), a certain linear polynomial method of approximation of functions from H+ . Let X be a normed linear space of functions holomorphic in D and let A be a subset of H+ . The value Ln (A; X) := inf sup f − Un (f )X , Λ f ∈A

n ∈ N,

where the lower bound is taken over the set of all lower triangular numerical matrices Λ, is called the best linear approximation of the class A in the space X. If there exists a matrix Λ∗ that generates a sequence of operators {Un∗ }∞ 0 such that sup f − Un∗ (f ) = Ln (A; X), n = 0, ∞, X

f ∈A

then one says that the matrix Λ∗ generates the best linear method of approximation of the class A in the space X. The value En (A; X) := sup

inf

f ∈A Pn−1 ∈Pn−1

f − Pn−1 X

is called the best polynomial approximation of order n of the class A in the space X. Let C(T ) be the Banach space of functions f continuous on T := {z ∈ C : |z| = }, > 0, with the norm f C(T ) := max |f (z)|.

Theorem 4. Let a function Ψ(z) := Then the following assertions are true:

∞ k=0

z∈T

ψk z k be the generating kernel of the class Hpψ , 1 ≤ p ≤ ∞.

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V. V. S AVCHUK

(i) for every fixed ∈ [0, 1] and natural n, the equality ψ ψ ; C(T ) = Ln H∞ ; C(T ) = |ψn | n En H ∞

(18)

holds if and only if 2 Re

∞ 1 ψk+n z k ψn

≥1

∀z ∈ D;

(19)

k=0

in this case, there exists a unique best linear method of approximation Λ∗ = {λ∗n k } of the form λ∗n k =1−

ψ2n−k 2i arg ψn 2(n−k) e

, ψk

k = 0, n − 1;

(20)

(ii) if inequality (19) is true, then, for every fixed ∈ [0, 1] and natural n, the following relation holds for all p ∈ [1, ∞) : En Hpψ ; Lp (T ) = Ln Hpψ ; Lp (T ) = sup f − Un,Λ∗ (f )L (T ) = |ψn | n . p ψ

(21)

f ∈Hp

Remark 5. If the numbers ψk , k = n, ∞, are real, then condition (19) is equivalent to the following condition: The harmonic function ∞

1 ψk+n z k ψn + Re 2 k=1

is sign-preserving in the disk D.

Under these restrictions imposed on the function Ψ, the implication (19) ⇒ Ln Hpψ ; Lp (T ) = |ψn | n was proved in [8], where the best linear method of approximationin the form (20) was also constructed. Under these restrictions on Ψ, the implication (19) ⇒ En Hpψ ; Lp (T ) = |ψn | n follows from the results of [9, 10]. ψ , the following relation is true: Proof. (i) For every function f ∈ H∞

f ( z) +

n−1 k=0

αk k k fk z = ψk

f ψ (wz) Ψ( w) +

n−1

αk k wk + g

k=0

T

1 w

dσ(w),

0. where αk are arbitrary complex numbers, z ∈ T, ∈ [0, 1), and g is an arbitrary function from H− By virtue of the duality theorem [formula (2)], this relation yields

n−1 n−1

1 αk k

k −k

fk = sup h(w) Ψ( w) + sup f ( ) + αk w + g w dσ(w)

h∈U H∞ ψk

f ∈H ψ ∞

k=0

k=0

T

n−1 = inf Ψ( ·) + αk ( ·)k + g (·) . 0 g∈H1,− k=0

1

(22)


1177

ψ Since the class H∞ is invariant under the translation of the argument, we have

ψ Ln H∞ ; C(T ) = inf sup |f ( ) − Un,Λ (f )( )| Λ

ψ f ∈H∞

n−1

αk

fk k = inf sup f ( ) + αk

ψk f ∈H ψ ∞

k=0

n−1 αk ( ·)k + g (·) = inf inf Ψ( ·) + αk g∈H 0 1,− k=0

=

1

Ψ( ·) + g (·)1 .

(23)

Ψ( ·) + g (·)1 = n |ψn |

(24)

inf

0 g∈Pn−1 +H1,−

Thus, if equality (18) is true, then inf

0 g∈Pn−1 +H1,−

and, according to Theorem 3, the function Ψ( ·) and, hence, the function Ψ satisfies condition (19). Conversely, if relation (19) is true, then equality (24) holds by virtue of Theorem 3. Thus, according to (23), equality (18) is true. (ii) Estimating the integral on the right-hand side of (22) and using the Minkowski integral inequality, we obtain

n−1 n−1 αk 1 k ψ k k fk ( ·) ≤ f (w ·) Ψ( w) + αk w + g w dσ(w) f ( ·) + ψk

k=0 k=0 p T

p

n−1 1 ≤ f ψ p Ψ( ·) + αk ( ·)k + g · ,

k=0

1

whence En Hpψ ; Lp (T ) ≤ Ln Hpψ ; Lp (T ) ≤

inf

0 g∈Pn−1 +H1,−

Ψ( ·) + g(·)1 .

Since relation (24) is satisfied under condition (19), we have En Hpψ ; Lp (T ) ≤ Ln Hpψ ; Lp (T ) ≤ n |ψn |.

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V. V. S AVCHUK

On the other hand, by virtue of the duality theorem [formula (1)], the function f∗ (z) = ψn z n satisfies the following relation:

min

Pn−1 ∈Pn−1

where

f∗ ( ·) − Pn−1 (·)

Hp

= sup f∗ ( w)g(w)dσ(w)

, g∈Lq,n (T)

T

 

  g(w)wk dσ(w) = 0, k = 0, n − 1 . Lq,n (T) := g ∈ U Lq (T) :   T

Thus, En

Hpψ ; Lp (T ) ≥

min

Pn−1 ∈Pn−1

f∗ ( ·) − Pn−1 (·)Hp

n

≥ f∗ ( w)w dσ(w)

= n |ψn |,

T

which completes the proof of the second part of the theorem. Let us find conditions for the generating function Ψ under which the space of algebraic polynomials is the optimal approximating n-dimensional space for the classes Hpψ in the sense of n-widths (see definitions below). For this purpose, we use the statement presented below, which gives necessary and sufficient conditions for the function Ψ under which the generalized Bernstein inequality for algebraic polynomials on the circle T in terms of ψ-derivatives is true. Theorem 5. Let ψ := {ψ0 , . . . , ψn }, n ∈ N, be a collection of nonzero complex numbers. Then the following assertions are true: (i) in order that, for any algebraic polynomial Pn ∈ Pn , the generalized Bernstein inequality ψ Pn

C(T)

≤

1 Pn C(T) |ψn |

(25)

be true, it is necessary and sufficient that there exist a sequence of numbers {ck }∞ k=0 such that n ∞ ψn k n+1 k z +z ck z 2 Re ≥1 ψn−k k=0

∀z ∈ D;

(26)

k=0

(ii) if inequality (26) is true, then, for any algebraic polynomial Pn ∈ Pn , the following relation holds for every p ∈ [1, ∞) : ψ Pn

Lp (T)

The equality in (27) is realized for Pn (z) = z n .

≤

1 Pn L (T) . p |ψn |

(27)


1179

Remark 6. Condition (26) is equivalent to each of the following conditions: (i) there exists a Borel measure µ on T such that

w−k dµ(w) =

T

ψn , ψn−k

k = 0, . . . , n;

(ii) for every natural m and arbitrary complex numbers λk , k = 0, 1, . . . , the following relation is true: m m

γk−l λk λl ≥ 0,

(28)

k=0 l=0

where  ψn   , k = 0, . . . , n,  ψn−k γk =   c , k = n + 1, . . . , k and γ−k = γk , k ≥ 1. Indeed, denoting F (z) := 1 + 2

∞ n ψn k z + 2z n+1 ck z k , ψn−k k=1

k=0

we conclude that condition (26) is equivalent to the statement that Re F ≥ 0 in the disk D, i.e., F belongs to R+ , which, by virtue of the Riesz–Herglotz theorem, is equivalent to the equality

F (z) = T

∞

dλ(w) dλk z k − 1 = (dλ0 − 1) + 1 − wz

∀z ∈ D,

k=1

where λ is a positive Borel measure on T. On the other hand, by virtue of the Carathéodory theorem (see, e.g., [6, p. 40]), F belongs to R+ if and only if relation (28) is true. In the case where the numbers ψk are such that condition (26) is satisfied for ck = 0, k = 0, 1, . . . , the implication (26) ⇒ (25) was proved by Szegö (see the references in [11, p. 173]) and proved again in [12]. Proof. Assume that relation (25) holds for any polynomial Pn ∈ Pn . It obviously turns into equality for the polynomial Pn∗ (z) = z n . Thus, sup Pnψ C(T) =

Pn ∈U Pn

where U Pn := Pn ∈ Pn : Pn C(T) ≤ 1 .

1 , |ψn |

(29)

1180

V. V. S AVCHUK

Since Pnψ (z)

zn = ψn

T

Pn (w) wn

n ψn w k dσ(w) ψn−k z

∀z ∈ C

k=0

and the set U Pn is invariant under the translation of the argument, i.e., (Pn ∈ U Pn ) ⇒ (Pn (eiθ ·) ∈ U Pn ∀θ ∈ [0, 2π]), it follows from the duality theorem [formula (1)] and relation (29) that 1 = |ψn | sup Pnψ Pn ∈U Pn

C(T)

= |ψn | sup Pnψ (1) Pn ∈U Pn

n

Pn (w)

ψn k

= sup w dσ(w)

= inf |µ|(T), n w ψn−k Pn ∈U Pn

µ∈Mn (ψ) k=0

(30)

T

where  

  !k := w−k dµ(w) = ψn , k = 0, n Mn (ψ) := µ ∈ M (T) : dµ   ψn−k T

and M (T) is the Banach space of complex charges µ on T with the total variation |µ|(T) in the role of the norm. Let {µν }∞ ν=1 be a sequence of charges from Mn (ψ) such that lim |µν |(T) =

ν→∞

inf

µ∈Mn (ψ)

|µ|(T) = 1.

Consider the Poisson–Stieltjes integral of the charge µν :

P (dµν )(z) := T

1 − |z|2 dµν (ζ), |1 − ζz|2

z ∈ D.

Since, for every ν ∈ N, one has |P (dµν )(z)| ≤

1 + |z| sup |µν |(T) 1 − |z| ν∈N

on every compact set in D, we conclude that the sequence of functions P (dµν ), ν = 1, 2, . . . , is uniformly bounded in D. Therefore, according to the accumulation principle for harmonic functions (see, e.g., [13, p. 25]), one can choose a subsequence of the sequence {P (dµν )}∞ ν=1 that converges uniformly

a certain harmonic function,

in D to

say, u. In this case, the absolute value |u| is the limit of the subsequence P (dµνj ) on every compact set K in D, i.e.,

|u(z)| = lim P (dµνj )(z) j→∞

∀z ∈ K ⊂ D.


1181

Using the Fatou lemma and equality (30), we obtain the following relation for any ∈ [0, 1):

|u( w)|dσ(w) ≤ lim inf ν→∞

T

|P (dµν )( w)|dσ(w) ≤ lim |µν |(T) = ν→∞

T

inf

µ∈Mn (ψ)

|µ|(T) = 1.

(31)

It follows from the uniform convergence of the subsequence P (dµνj ) to u that (dµ k as j → ∞ for νj )k → u each k ∈ Z. Thus, the harmonic function u in the disk D can be represented in the form n ψn k k

w + u( w) = ψn−k k=0

ck |k| wk

k =0,1,...,n

n ψn k k 1 =

w + h( w) + g w ψn−k

∀ ∈ [0, 1) ∀w ∈ T,

k=0

where ck are certain numbers such that the functions h(z) := z

n+1

∞

z ∈ D,

ck z k ,

k=0

and g(z) :=

∞

c−k

k=1 n and H0 , respectively. belong to H+ −

Now consider the best approximation of the function

1 , zk

z ∈ D∞ ,

n k=0

According to (8) and (31), we have

ψn k 0. z + h(z) by the set H− ψn−k

∗ ∗ n n ψ ψ n n (·)k + h(·) + f (·) ≤ (·)k + h(·) + g(·) = u∗1 ≤ 1. 1 ≤ inf 0 ψn−k ψn−k f ∈H− k=0

1

k=0

1

Thus, the equality is realized everywhere in the last relations. By virtue of Theorem 2, we get n ψn 2 Re z k + h(z) ≥ 1 ψn−k

∀z ∈ D,

k=0

which completes the proof of the necessity of the conditions of the theorem. Assume that condition (26) is satisfied. To prove the sufficiency of this condition, note that, for any function 0 and any ∈ [0, 1), the following relation holds at every point z ∈ T: g ∈ H− 1 Pnψ (z) = ψn n

T

Pn ( zw) wn

∞ n ψn 1 ( w)k + ( w)n+1 ck ( w)k + g w dσ(w). ψn−k

k=0

k=0

1182

V. V. S AVCHUK

Properly choosing the function g and using the Hausdorff–Young inequality, we get ψ Pn ≤ p

n ∞ ψn 1 k k+n+1 P (

·) + c (

·) 2 Re − 1 n p k n |ψn |

ψn−k k=0

=

1 Pn p , |ψn | n

k=0

1

1 ≤ p ≤ ∞.

Letting tend to 1, we obtain relations (25) and (27). Assume that X is a complex normed linear space, A is a subset of X, Xn is an n-dimensional subspace of X, X n is a subspace of X of codimension n, and Ln : X → Xn is a continuous linear operator of rank n. The values bn (A; X) := sup sup {r : rU Xn+1 A} , Xn+1

dn (A; X) := inf sup inf f + gX , Xn f ∈A g∈Xn

dn (A; X) := infn X

sup

f ∈A∩X n

f X ,

δn (A; X) := inf sup f + Ln (f )X Ln

f ∈A

are called, respectively, Bernstein, Kolmogorov, Gel’fand, and linear n-widths. For the first three widths presented above, the subspaces for which the upper bound for bn and the lower bounds for dn and dn are realized are called optimal subspaces. By analogy, an operator Ln is called optimal for δn if the lower bound is realized for it. The n-widths introduced above relate to one another as follows (see, e.g., [14], Chap. 2): bn (A; X) ≤

dn (A; X) dn (A; X)

≤ δn (A; X).

(32)

∞ ψk z k is the generating kernel of the class Hpψ , Theorem 6. Suppose that 1 ≤ p ≤ ∞, Ψ(z) := k=0 n ∈ N, and Dn is one of the n-widths bn , dn , dn , and δn . If conditions (19) and (26) are simultaneously satisfied for Ψ, then the following relation holds for any ∈ [0, 1] : Dn (Hpψ ; Lp (T )) = |ψn | n . In this case, the following assertions are true: (i) the space Pn is an optimal subspace for bn ; (ii) the space Pn−1 is an optimal subspace for dn ;

(33)


1183

(iii) the space Hpn−1 [see (13)] is an optimal subspace for dn ; (iv) the operator Ln := Un,Λ∗ , where the elements of the matrix Λ∗ are defined by (20), is optimal for δn . In the case where the sequence {ψk }∞ k=0 satisfies the conditions of Remark 5 and condition (26) with ck = 0, k = 0, ∞, equality (33) was proved in [12], and its special cases for specific values of the sequence ψk were proved in [15, 16]. Proof. According to relation (32), it suffices to prove the estimates bn Hpψ ; Lp (T ) ≥ |ψn | n and δn Hpψ ; Lp (T ) ≤ |ψn | n . The first estimate follows from the definition of width bn and Theorem 5, which guarantees that the set

n |ψn |U Pn := P ∈ Pn : P ( ·)

Lp (T)

≤ n |ψn |

is contained in Hpψ . The second estimate is a corollary of Theorem 4 because δn Hpψ ; Lp (T ) ≤ Ln Hpψ ; Lp (T ) . The optimality of the space Pn for bn follows from Theorem 5. The optimality of the space Pn−1 for dn is a corollary of Theorem 4 and equality (33). The optimality of the space Hpn−1 for dn is a corollary of equality (33) and the fact that the function f (z) = ψn z n belongs to Hpψ ∩ Hpn−1 . The optimality of the operator Ln is a corollary of equalities (33) and (21). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

P. Duren, Theory of Hp Spaces, Academic Press, New York (1970). J. B. Garnett, Bounded Analytic Functions [Russian translation], Mir, Moscow (1984). S. J. Poreda, “A characterization of badly approximable functions,” Trans. Amer. Math. Soc., 169, 249–256 (1972). T. W. Gamelin, J. B. Garnett, L. A. Rubel, and A. L. Shields, “On badly approximable functions,” J. Approxim. Theory, 17, No. 3, 280–296 (1976). G. Ts. Tumarkin, “On Cauchy–Stieltjes-type integrals,” Usp. Mat. Nauk, 11, No. 4, 163–166 (1956). C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen (1975). A. I. Stepanets and V. V. Savchuk, “Approximation of Cauchy-type integrals,” Ukr. Mat. Zh., 54, No. 5, 706–740 (2002). V. I. Belyi and M. Z. Dveirin, “On the best linear methods of approximation on classes of functions defined by associated kernels,” in: Metric Problems in the Theory of Functions and Mappings [in Russian], Vol. 5, Naukova Dumka, Kiev, (1971), pp. 37–54. K. I. Babenko, “Best approximations of classes of analytic functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 222, No. 5, 631–640 (1958). J. T. Scheick, “Polynomial approximation of functions analytic in a disk,” Proc. Amer. Math. Soc., 17, 1238–1243 (1966). S. N. Bernstein, Collection of Works [in Russian], Vol. 2, Academy of Sciences of the USSR, Moscow (1954). L. V. Taikov, “Widths of some classes of analytic functions,” Mat. Zametki, 22, No. 2, 285–295 (1977). G. M. Goluzin, Geometric Theory of Functions of Complex Variables [in Russian], Nauka, Moscow (1966). A. Pinkus, n-Widths in Approximation Theory, Springer, Berlin (1985). V. M. Tikhomirov, “Widths of sets in a functional space and theory of best approximations,” Usp. Mat. Nauk, 15, No. 3, 81–120 (1960). L. V. Taikov, “On the best approximation in the mean of some classes of analytic functions,” Mat. Zametki, 1, No. 2, 155–162 (1967).