K-functionals and exact values of n-widths in the Bergman space Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 3, No. 2, 2017

K-FUNCTIONALS AND EXACT VALUES OF n-WIDTHS IN THE BERGMAN SPACE

Mukim S. Saidusaynov

Tajik National University, Dushanbe, Tajikistan [email protected]

Abstract: In this paper, we consider the problem of mean-square approximation of complex variables functions which are regular in the unit disk of the complex plane. We obtain sharp estimates of the value of the best approximation by algebraic polynomials in terms of k-functionals. Exact values of some widths of the specified class of functions are calculated.

Key words: Bergman space, Best mean-square approximation, k-functional, n-width.

Introduction and preliminary facts

We consider the problem of mean-square approximation by Fourier sums of complex functions f which are regular in a simply connected domain D c C and belong to the space L2 := L2(D) with the finite norm

11/11 := II/IIl2(p) = ^ ff^ \f(z)\2da^j ,

where the integral is understood in the Lebesgue sense and da is an element of area.

The study of the mean-square approximation of functions in the domain D c C is closely related to the theory of orthogonal functions. A sequence of complex functions (z)} (k = 0,1,2,...) is called an orthogonal system on the domain D if

- ff <pk(z)<pi(z)da = 0, k^l.

n J J(D)

d orthonormal system if

// <pk(z)<pi(z)da = ökj, J J (D)

1

n J J(d)

Such a sequence of functions is called orthonormal system if

1

n J ad)

where ök,l = 0, k = l, and ök,k = 1, k € N. If f € L2, then the numbers

auU) = ~ ff f(z)^(z)da (1)

n J J(D)

are called the Fourier coefficients of the function f with respect to the orthonormal system (z)} (k = 0,1,2,...). We associate with a given function f its Fourier series with respect to the specified orthogonal system:

<x

f(z) ak(fVk(z). (2)

k=0

Let

n—1

Sn-1(f,z) = ak(f Vk(z)

k=0

be the partial sum of order n of the series (2). We form a linear combination of the first n functions of the system {fk(z)}:

n— 1

Pn-i(z) = ^ dk fk (z), k=0

where dk € C are arbitrary complex coefficients. We call this linear combination a generalized polynomial. It is well known (see, for example, [1], p.263) that

En-i(f) = inf {||f - pn-i|| : dk € C} f

= If - Sn-i(f)|| = ]T |<ч(f )l

Л1/2 (3)

„ k=n

where ak (f) are the Fourier coefficients of the function f defined by (1).

In the case of the mean approximation of complex functions in a simply connected domain D C C by Fourier series with respect to an orthogonal system of functions (z)}£=0 on D, the problem of finding the exact constant in the Jackson-Stechkin inequality was studied in [2]. Recall that Jackson-Stechkin inequalities are inequalities in which the value of the best approximation of a function by a finite dimensional subspace of a given normed space is estimated by the modulus of smoothness of the function itself or some its derivative. In this paper, we use the same methods as in [2, 3, 5, 15].

We study in more detail the case where D is the unit disk U := {z € C : |z| < 1}. In this case, it is clear that the system of functions (z) = zk(k = 0,1,2,...) is orthogonal in the disk U:

-[[ Vk{z)^{z)d(T = - f1 r rk+l+lei{-k~l)tdrdt = 0, k^l. n J J(U) n J0 J0

However, this system is not orthonormal, since

iff 1 f1 f2n

1 I' ■ - m2 * 1

if \Mz)\2da = - [ [\2k+ldrdt = J J (U) n J0 J0

n J J(U) n Jo Jo k + 1

Therefore, the system of functions (p*k(z) = л/к + 1zk (k = 0,1,2,...) is orthonormal. We denote by A(U) the set of all functions f analytic in U. The Maclaurin series of such a function has the form

f (z) = Ë Ck (f )zk, (4)

-kU )zk k=0

where Ck(f ) are the Maclaurin coefficients of f. We note that

il/il2 = £^ff = (5)

k=0 k=n

It was proved in the monograph [1] that the Fourier series of a function / with respect to the orthonormal system <p*k{z) = \/k + Izk, k = 0,1,2,..., coincides with the series (4) for / € A(U); i.e.,

<x <x

f (z)^ ak (f )fk (z)^ Ck (f )zk. (6)

k=0 k=0

Therefore, the series (6) can be differentiated term by term any number of times and, according to the Weierstrass theorem [6, p.107], for any r € N, we get

f (r)(z) ^ Ck (f )k(k - 1) ■ ■ ■ (k - r + 1)zk—r ak,r Ck (f )zk—r, (7)

k=r k=r

where

akr := k(k - 1) ••• (k - r + 1), k € N, r € Z+, k > r.

We denote by L2r) := L2r)(U) (l20) := L2(U)) the class of all functions f € L2 such that f(r) € L2 (r € Z+, f20) = f).

1. Sharp estimates of the value of the best approximation by means of

K-functionals

In this section, we prove some sharp inequalities relating the value En-1(f) of the best approx-

(r)

imation of functions in the class L2 ) and Peetre K-functionals. The definition and some properties of Peetre K-functionals are given in [7]. The direct and inverse theorems of the theory of approximation by means of K-functionals were proved in [8, 9]. We define the K-functional constructed by the spaces L2 and L2m) as follows:

Km(f, tm)2 := K (f, tm; L2; L2m)) = inf {||f - + tm ■ ||g(m) M2 : g € L^} , (8)

where m € N and 0 < t < 1. We note that a weak equivalence of the K-functional defined by (8) and a special generalized modulus of continuity of order m was established in [8].

Theorem 1. Let n, m € N and r € Z+ be arbitrary numbers such that n > r + m. Then the following equality holds:

gup V(n + l)/(n-r + l)-QV.Era_i(/) = i

K

t)

/eLM I „^ /n — r — m + 1

//pr m I ' y n — r + 1

an—r

1

Proof. Using (7), we easily find that

¿£-r-i(/(r)) = E ^r^TT^. r G z+- (10)

k=n

Taking into account equality (10), we obtain

?2 m_y !<*(/) fc-r + 1 O M/)l

- > , , - > 77——TT-Ö- • ak

J k — r + 1 I ^ 2 |ck(f)| (11)

< max < —-——5— > • > atr----(J-J-J

- fc€N ) (fc + lW [ ^fc-r+l

k k + 1 kT (k + 1)«kr 'r k — r + 1

k=n k=n k>'

k—r+1 / _

n 1 (k + 1)ak,r

n — r + 1 1

k>n

k,r J k=n

n I 1 Gi^n r

E^r-l f(r)

(r)

Now, for an arbitrary function f € L2 , we write

*.-,</) < ^ + 1. J-En-r^ (/<->) < JE^L. _L||/M _ S„_„_I(9)||, (12)

2

where Sn-r-1(g) is the partial sum of order n — r of the Fourier series of an arbitrary function g € L2m). In view of (2) and (11), we get

- Sn-r-i(g)\\ = K-r-i(g) < J-—r ---—K-r-m-i (g{m)

n r + 1 an—r,m ^

¡n — r — m + 1 1

(13)

n - r + 1 an—r It follows from inequalities (12) and (13) that

En-^f) < ^H+I. _L {||/W _+ II, _Sn_r_l{g)||}

V n + 1 ara,r I 11J y" V n-r + 1 a.n-r

g(m)

(14)

Now, we note that the left-hand side of inequality (14) does not depend on g € L2m). Therefore, passing to the infimum over all functions g € L2m) on the right-hand side of (14) and using the definition (8) of K, we get

n — r + 1 1 _ / _(r) n - r — m + 1

£n-i(/) < \i" ; ; ^ • —( /M,

, -, • -m i t/ j . -,

n + 1 an r \ V n — r + 1 an—r

This implies the following upper bound:

qnn + l)/(w - r + 1) • a^rEn-ijf)

sup " ; , -I ■ ^ XJ

/e4r) £ ffM <,/n~r~m + 1 ■ 1 ^

№ mV V n-r + 1 an-r,mJ

where Pr is the subspace of complex algebraic polynomials of degree at most r.

To obtain a lower bound of the extremal characteristic on the left-hand side of (15), in (8), we put f (z) := (z), where (z) is an arbitrary complex algebraic polynomial in Since the function g(z) = 0 belongs to the class L2m), we obtain from (8) the upper bound

Km(pra; tm)2 < ||pn||.

Since the function g(z) := (z) also belongs to the class L2m), we find from (8) that

Km(Pn; )2 < ||pnm) ||. Thus, the last two relations imply that, for any element (z) € ,

Km(Pn; tm)2 < min { ||pra||; ||} . (16)

We consider the function f0(z) = zn. Since

f0r+m) = n(n — 1) ■ ■ ■ (n — r + 1) ■ ■ ■ (n — r — m + 1)zn-r-m = an,r ■ an-r,mzn-r-m, according to (16), we have

n - r + 1 an—r,m / V n - r + 1 an—

xn—r,m

n r m* + 1 1 an,r ' an—r,m an,r

n-r + 1 tt„-r,m - r - m + 1 V™ - r + 1 '

1

Using the obtained inequality and the second equality in (5), we establish that

y/(n + 1 )/{n - r +1) • On,rEn-l(f)

sup /

/eL(r) ^ I In — r — m + 1 1

f^Pr n-r + 1 an_

,__" 7 (17)

> v(n + l)/(n-r + l) • anirEn_i(fo) >

y0\ n - r + 1 ara_r,m J

We obtain equality (9) by comparing the upper bound (15) with the lower bound (17). The theorem is proved.

2. Exact values of n-widths of a class of functions

We assume that S is the unit ball in the space L2, An c L2 is an n-dimensional subspace, and An c L2 is a subspace of codimension n. Let L : L2 ^ An be a continuous linear operator, let L± : L2 ^ An be a continuous linear projection operator, and let M be a convex centrally symmetric subset of L2. The quantities

bn(M, L2) = sup {sup (e > 0; eS n Ara+1 c M} : Ara+1 c L2} ,

d„(M, L2) = inf (sup (inf (Mf - g|| : g € Ara} : f € M} : Ara C L2} , 5n(M, L2) = inf (inf (sup (Mf - Lf || : f € M} : LL2 C Ara} : Ara C L2} , dn(M, L2) = inf (sup (||f 112,y : f € M n An} : An C L2} , n„(M, L2) = inf {inf {sup {||f - Lxf || : f € M} : L^ C Ara} : Ara C L2}

are called, respectively, the Bernstein, Kolmogorov, linear, Gelfand, and projection n-widths of the subset M in the space L2. These widths are monotone with respect to n, and the following relation holds (see, for example, [10, 11]):

bn(M, L2) < dn(M, L2) < dn(M, L2) = 5n(M, L2) = nra(M, L2). (18)

We recall (see, for example, [12, p. 25]) that a nondecreasing function ^ on R+ is called a k-majorant if the function t_k^(t) is nonincreasing in R+, ^(0) = 0, and ^(t) ^ 0 as t ^ 0. For k = 1, the function ^ is simply called a majorant.

Let W2(r)(Km, r € Z+,m € N, be the class of all functions f € L2r) whose derivatives f(r) satisfy the condition

Km(f (r),tm) < ^(tm), 0 <t< 1.

In this definition, ^ is a majorant, L20) = L2, and W2(0)(Km, = W2(Km, For any subset M C L2, we define

Era_1(M)L2 := sup (Era_1(f) : f € M} .

We note that, in the Bergman space, values of widths of some classes of analytic functions in a disk were calculated, for example, in [13-19].

/Cm

Theorem 2. Let tf be the majorant defining the class W2(r)(Km, tf), m € N, and r € R+. Then, for any natural number n > m + r, we have

A„ (w2(r)(Km, tf), L2) = Era-i (wir)(Km, tf))

n-r + 1 1 / ln-r - m +1 1 \ n + 1 an;i. \ V n r + 1 an—r,m /

where An(-) is any of the n-widths bn(-), dn(■), dn(-), ¿n(-), and nn(-).

Proof. Let n be a natural number such that n > m + r. In view of the definition of the class W2(r)(Km, tf), relations (15) and (18) imply that

An (w2(r)(Km, *), L2) < dn (w2(r)(Km, *), L2)

. „ t\A , ln-r +1 1 / [n — r — m + 1 1

— a/-—:---* . , 1

n + 1 an r \ V n - r + 1 an—r

(20)

To find the corresponding lower bound, in view of (18), it suffices to estimate the Bernstein n-width of the class W2(r) (K m, tf). On the set Pn n L 2, we define the ball

1 ^ un n — r + 1 1 / /n — r — m + 1 1 ■Mn+i ■= {Vn^Vn: \\pn\\ < 4/-—---tf

n ■ WFti 1 _ 1, -1 — 1 » / . -1

n + 1 an r \ V n — r + 1 an—r

Now, we note that, in view of formula (7) and the identity ak , r+m = ak , r ak-r> m, for an arbitrary pn(z) = ^fc=0 ak(pn)zk € Pn, the following equality holds:

PT+m)(z)= £ ak(Pn)ak,r+mzk r m := £ ak(pn)ak ,r ■ ak—r,mz

k=r+m k=r+m

Hence, using the Parseval equality and the inequality ak>r < an,r, k < n, we obtain the Bernstein type inequality

( n \ 1/2

||p(r+m)|| = J ^ |afc(pn)|2ak>r ■ ak- < «n,r ■ an- r,m||Pn||. (21)

l k=r+m J

By definition, for the majorant tf and for any 0 < ti < t2 < 1, we have the inequality r1tf(r2) < r2tf(r1). Therefore, for any 0 < t1 < t2 < 1, setting t1 = i™ and t2 = i™, we obtain

i-mtf(im) > t-mtf(im). (22)

We now show that Mn+1 C W2(r)(Km, tf). Thus, we need to prove that, for any polynomial Pn C Mn+1,

Km(pnr),im) < tf(tm), 0 <i < 1.

Since, by assumption, m, n € N, r € Z+, and n > m + r, we consider two cases:

---\ l/rri

, n — r — m + 1 1

0 < i <

n - r + 1 an—r

n

n

and

-t 1 \ l!m

n — r — m + l l \ <¿<1

n r + 1 an_r,m /

First, assume that

1/m

. /n — r — m + 1 1 0 < i < ' 1

n r + 1 an_r,m /

In this case, using inequality (22) with

1/m

, In — r — m + 1 1

tl = t, t2

n r + 1 an_r,m /

and applying (12) and (21), for any € Mn+1, we obtain

Km(pir), tm)2 < tm ■ |pir+m) || < tm ■ a„,r ■ an_r,m||p

" n 1 1 ^n.r

n — r + 1 1 T / /n — r — m + 1 1

n + 1 an,r \ V n - r + 1 ara_r,m / (23)

n_r + l ^ M-r-m + 1 1

xn—r,m n , — 11/ , ^

n - r - m + 1 \V n - r + 1 an_

Now, let

-r \ 1/m

n — r — m + 1 1 \

-1-1---< i < 1-

n r + 1 an_r,m y

Then using (16) and the Bernstein type inequality

||pir)|| < an,r ■ ||pn|

and taking into account that the majorant ^ is nondecreasing, we find that

Km(pir),tm)2 < ||pir)|2 < an,r ||Pn|2

n - r + 1 1 T / n - r - m + 1 1

< an,r\l-—:---^

n + 1 an,r \ V n - r + 1 an_r

In — r +1 / In — r — m +1 1 \ (24)

< \ -— • tf

n + 1 \V n — r + 1 an_

n._r,m

<\[/l n — r — m + 1 1 \ ^ xj/^ni^

n r + 1 an_r,m I

The definition of the class W2(r)(Km, along with (23) and (24) implies that Mn+1 C W2(r)(Km, Then, taking into account the definition of the Bernstein n-width and (18), we obtain

(r) (r)

A„ (W2(r)(Km, *), L2) > bn (W2(r)(Km, L2)

, , . , r N ln-r + 1 1 T / ln-r -m +1 1 \ (25)

> bn(Mn+1;L2) > J-—---^ J-—---.

V n + 1 an,r y V n - r + 1 a„_r,m J

Comparing the upper bound (20) and the lower bound (25), we get the required equality (19). The theorem is proved.

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