SOME INEQUALITIES BETWEEN THE BEST SIMULTANEOUS APPROXIMATION AND MODULUS OF CONTINUITY IN THE WEIGHTED BERGMAN SPACE Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 9, No. 2, 2023, pp. 165-174

DOI: 10.15826/umj.2023.2.0141

SOME INEQUALITIES BETWEEN THE BEST SIMULTANEOUS APPROXIMATION AND THE MODULUS OF CONTINUITY IN A WEIGHTED BERGMAN SPACE

Muqim S. Saidusainov

University of Central Asia, 155 Qimatsho Imatshoev, Khorog, GBAO, Republic of Tajikistan

[email protected]

Abstract: Some inequalities between the best simultaneous approximation of functions and their intermediate derivatives, and the modulus of continuity in a weighted Bergman space are obtained. When the weight function is y(p) = pa, a > 0, some sharp inequalities between the best simultaneous approximation and an mth order modulus of continuity averaged with the given weight are proved. For a specific class of functions, the upper bound of the best simultaneous approximation in the space ' Yl(p) = Pa, a > 0, is found. Exact

values of several n-widths are calculated for the classes of functions Wpr)(wm,q).

Keywords: The best simultaneous approximation, Modulus of continuity, Upper bound, n-widths.

1. Introduction

Extremal problems of polynomial approximation of functions in a Bergman space were studied, for example, in [8, 13-15]. Here, we will continue our research in this direction and study the simultaneous approximation of functions and their intermediate derivatives in a weighted Bergman space based on the works [4-6, 10]. Note that the problem of simultaneous approximation of periodic functions and their intermediate derivatives by trigonometric polynomials in the uniform metric was studied by Garkavi [1]. In the case of entire functions, this problem was studied by Timan [12].

To solve the problem, we first will prove an analog of Ligun's inequality [2].

Let us introduce the necessary definitions and notation to formulate our results. Let

U : = {z € C : |z| < 1}

be the unit disk in C, and let A(U) be the set of functions analytic in the disk U. Denote by B2>7 the weighted Bergman space of analytic functions f € A(U) such that [8]

2.7 := J J \f(z)\H\z\)day2 < oo, (1.1)

da is an area element, 7 := 7(|z|) is a nonnegative measurable function that is not identically zero, and the integral is understood in the Lebesgue sense. It is obvious, that the norm (1.1) can be written in the form

1 f1 f2n \ 1/2

1 ' ' ' ^ 1 Jt\ |2,

2'7=(i/ jj pl(p)\f(pelt)\2dpdt)

In the particular case of y = 1, Bq := Bq1 is the usual Bergman space. The mth order modulus of continuity in B2,Y is defined as

wm(/, t)2,Y = sup {|| Am(/, h) 1(2,7 : |h| < t} =

= SUP{(jLJ0 Jo :\h\<

where

m

Am(/; p,u,h) = £(-1)fc Cm f (pei(u+kh)).

Let Pn be the set of complex polynomials of order at most n. Consider the best approximation of functions f € B2)7:

En-i(/)2,y = inf {||/ - Pn-1(2,7 : Pn-i € Pn-1} Denote by Bf;) and B(r), r € N the class of functions f € A(U) whose rth order derivatives

f (r)(z) = dr f/dzr belong to the spaces B2,Y and B2, respectively. Define

an,r = n(n — 1) ■ ■ ■ (n — r + 1), n > r. It is well known [7, 8] that the best approximation of functions

f = E Ck(fG £2,

k=0

is equal to

En-i(f)2,Y = (£ |ck(f)|2 T P2k+1Y(p)dp)

1/2

2

x / œ r 1 \ 1/2

En-s-1 (f(s)) = (J2 |ck(f)|2 p2(k-s)+17(p)dp

, k=n 0

(1.2)

and the modulus of continuity of f G B2,Y is

œ 1 ^ 1/2

I f (r),t) = 2m/2 sup {y a2kr |ck (f )|2(1 - cos(k - r)h)m / p2(k-r)

2,Y w<* L k=r '

Denote by

x r œ f 1 ^ 1/2

(f(r),t) =2m/2 sup £ ak,r |ck (f)|2(1 - cos(k - r)h)m / p^+W^dp • (1.3) v 2,Y |h|<t I t^ ./0 J

Ps(7)= / Y (P)PS dp, s = 0,1, 2,... (1.4)

0

the moments of order s of the weight function y(p) on [0,1]. According to notation (1.4), we write equalities (1.2) and (1.3) in compact form:

/ œ \ 1/2

En-1(f )2,7 = (Y, lck (f )|2 ^2k+1(7n ,

^ k=n '

, x i ^ \ 1/2

En-s-1 (f(s)J = ^ |ck(f)|2 ak,s ^2(k-s)+1(Yn , (1.5)

' k=n

( œ N 1/2

Wrn (f(r),t) =2m/2 sup £ ak,r |Ck(f)|2(1 - cos(k - r)h)m ^(k-r)+1(Y» •

v 2,y ihi<iitr: J

œ

2. Analog of Ligun's inequality

For compact statement of the results, we introduce the following extremal characteristic:

, n 2m/2Ens-l (/<% Ji'm,n,r,s,p(Q> 7, h) = sup --'f < (f (r),i)2;7 q(t)dt)

1/p '

where m,n € N, r € Z+, n > r > s, 0 < p < 2, 0 < h < n/(n — r), and q(t) is a real, nonnegative, measurable weight function that is not identically zero on [0, h].

Theorem 1. Let k,m,n € N, r, s € Z+, k > n > r > s, 0 <p< 2, 0 <h < n/(n — r), and let q(t) be a nonnegative, measurable function that is not identically zero on [0, h]. Then

J?,wp(Qn,h) ~ 7'k) ~ inf J?k,r,sAQ,1,hV (2-1)

n<k<<x

where

^W?, 7, h) = ^ (At2(fc-r)+1|701/2 ( [h (1 - cos(fc - r)tr'2q{t)dt) 1/P -ak,s V^2(k-s)+1(7) / \J0 J

Proof. Consider the simplified variant of Minkowski's inequality [3, p. 104]:

ft- h f ^ \ p/2 \ 1/p f ^ ft- h \ 2/p\ 1/2

U(|>(i>|2) dt) >(£(i|gk(t)rdV ) • (2'2)

which is hold for all 0 < p < 2 and h € R+. Setting

gk = fkq1/p (0 <p < 2)

in (2.2), we get

rh / ^ \ p/2 \ 1/p / ^ / rh \ 2/p\ 1/2

œ xp/2 \ 1/P / œ / /-h \2/p\ 1/2

£|fk (t)lM q(t)d0 >(£(/l fk (t)lpq(t)dt) . (2.3)

k=n ' ' ^ k=n ^ ^ ^

From (1.3) with respect to (2.3), we get

fh \ 1/P ( r h } 1/P

yo ^mm(f(r),t)2i7q(t)d^ = |y (^m(f(r),t)2,7)p/2q(t)dtj

/»h ^^ /o N

X j (2^ ak,r|ck(f)|2(1 - cos(k - r)t)m^2(k-r)+1(7^ q(t)dt

0 k=n

œ r rh

> K] Î2mp/2^r|ck(f)|p / (1 - cos(k - r)t)mp/2 (^2(k-r)+1(7))P/2 q(t)dt

lk=nL 70

C œ r h

= 2m/2 E |ck(f)|2^2(k-r)+1(7) a?ktr (1 - cos(k - r)t)mp/2q(t)dt

2/P 1/2

k=n

<x

s uo

= 2™/2{ £ |Ck (f )|2«i,s ^2(k-s)+1(7),M2(k-r)+1(7)(^2(k-s)+1(7))-1

k=n

afc,r

P rh

ak,s / J0

(1 — cos(k — r)t)mp/2 q(t)dt

2/p ji/2

>2"V2 inf J \ 1/2 f fh{1_ cos(fc _ r)t)mp/2 {t)dS 1/P

/ ~ \ 1/2 X Elc* (f)|2ak,s ^2(k-s)+1(7n = 2m/2En-s-1(f(s))2,Y mf Lfc,w(9, Y, h),

y y n< k< OO

' k=n

and this yields the inequality

2m/2 En-s-1 (f(s))

2,7

< —-T7-TV (2-4)

/ rh \ 1/p " inf L r s p(q, Y, h)

(jf ^ (f (r),t)2,Y ?(t)dtj n<fc<^ fc,r,s,pW'7'

or

i(/-7'- inf (2-5)

n<fc<^

To estimate the value in (2.1) from below, consider the function

fo(z) = zn € . Simple calculation leads to the following relations:

/ f1 \ 1/2 /

En-s-^f0s^2,7 = p2(n-s)+1 Y(p)d^ = «n,s (fe(n-s)+1 (y)) 1/2 ,

f 1

^m (f0r), t) 2,7 = 2m<r (1 — cos(n — r)t)m ^ p2(n-r)+1 Y(p)dp

= 2m°n,r i1 — cos(n — r)t) mp2(n-r)+1(Y), using which, we get the lower estimate

2

r .>p (f (r) 0

Y,h) >

f ^m(f0r),t)2,Yq(t)dt 0

2m/2an,s (P2(n-s)+1(Y))

(2.6)

. /* h \ /1 ^^ n r s p (

2mp/2<r (^2(n-r)+1 (y))(1 — cos(n — r)t)mp/2q(t)dt »

0

1/p Ln,r,s,p(q, Y, h)'

Comparing the upper estimate (2.5) and the lower estimate (2.6), we obtain the required two-sided inequality (2.1). This completes the proof of Theorem 1. □

1

Corollary 1. The following two-sided inequality holds for Y1(p) = pa, a > 0, in Theorem 1:

< Jfm,n,r,s,p(l> Yi>fc)< r^1-7TTT' (2.7)

where

Gra,r,s,p,a(q, h) inf Gfc,r,s,p,a(q, h)

n<fc<^

iW«(?, = ^ f'if 'III!") 1/2 if t1 " cos(fc " r)i)mP/2 VP • (2.8)

a&,s V 2(k — r + 1) + a/ V./0 J

The following problem naturally arises from (2.7): to find an exact upper bound for the extremal characteristic

2™/2 . ,(f (s)

^m,n,r,s,p{Q> = SUP

^m,n,r,s,p

(r) / /.ft vl/p'

!>71

, ^(f(r) ,t)2,Yi q(t)dt 0

where m,n € N, r, s € Z+, n > r > s, 0 < p < 2, 0 < h < n/(n — r), Y1(p) = pa, and a > 0.

Theorem 2. Let a weight function q(t), t € [0, h], be continuous and differentiable on the interval. If the differential inequality

r-—1 p 2p(r — s) 1 \ 1 '

£ ~ \2(k — r + 1) + a](2(fc — s + 1) + a) " J 9(i) " ~k^tq'{t) " ° (2-9)

holds for all k € N, r, s € Z+, k>n>r > s, 0 <p < 2, and a > 0, then the following equality holds for all m, n € N and 0 < h < n/(n — r):

./,„„„„.(7.71, fc) = ^ 1/2 f [h (1 - cos(n - r)t)^/2 9(t)dA 1/P ■ (2.10)

an,r \2(n — s + 1) + a/ V./0 /

Proof. To prove equality (2.10), it suffices to show that the following equality holds in (2.7):

inf Gk (q, h) = Gn (q,h). (2.11)

n<k<^

We should note that a similar problem of finding a lower bound in (2.11) for some specific weights for p = 2 was considered in [2]. In the general case, this problem was studied in [9], where it was proved that, if the weight function q € C(1)[0, h] for 1/r < p < 2, r > 1, and 0 < t < h satisfies the differential equation

(rp — 1)q(t) — tq'(t) > 0,

then (2.11) holds.

Let us now show that, under all constrains on the parameters k, r, s, m, p, a, and h in Theorem 2, the function

m=M'/"e-«•<*-^r^m ^

\ak,sj \2(k — r + 1)+ a) J0 increases for n < k < to. Indeed, differentiating (2.12) and using the identity

d t d

- eos{k - r)t)mp/'2 = -—— -f (1 - cos(fc - r)t)mp/'\ dk k — r dt

we obtain

, r—1

k - I \2(k - r + l) + a/

s

p

+ Z^-s + l^yV2-! 4.-4T /2

1 - ' 2\2(k-r + l) + aJ [2(k-r + l)+a]2 J0 y y 11 qy '

aksj 2 \ 2(k - r + 1) + a

<W \2(k — r + 1) + a J J0 dk

_ /a^y_2p(r - s)_ Î2(k-s + l)+aXp/2 1

\ak J [2(k - r + 1) + a](2(fc - s + 1) + a) \2(k -r + l)+a) J

p /2(k - « + 1) + a\p/2

ak,AV 2(k - s + 1)+ a\p/2 i h

h fh

-— (1 - cos(k - r)h)mp/2q{h) + / (1 - cos(k - r)i)mp/2

k - r JO

ak,s/ \2(k - r + 1) + a/ [k - r

r-1

p 2p( r — «I I \ I

dt

x

___2p(r ~ s)___1_\ , . 1 ,, ,

^k-l [2(k — r + 1) + a](2(k — s + 1) + a) k-r)q() k-r'q()

l=s

This relation and condition (2.9) imply that ^(k) > 0, k > n>r > s, and we obtain equality (2.10). Theorem 2 is proved. □

Denote by Wp(r)(wm, q) (r € Z+, 0 < p < 2) the set of functions f € Bf^ whose rth derivatives

f(r) satisfy the following condition for all 0 < h < n/(n - r) and n > r:

/• h

I ^m(f(r),t)2,71 q(t)dt < 1.

Since, for f € , its intermediate derivatives f(s) (1 < s < r — 1) also belong to L2, the behavior

2,yi ,

of the value En-s-1 (f(s))2 for some classes M(r) C , n > r > s, n € N, and r, s € Z+, is of interest. More precisely, it is required to find the value

An,s(M(r)) := sup {En-s-1(f (s))2,7i : f € M(r)}.

Corollary 2. The following equality holds for all n € N, n > r > s, 0 < p < 2, and 0 < h < n/(n — r):

^(W^Wi)) := sup {Ens^fW)^ : / € l^Wq)} = 9m/2<y 1--^r- (2-13)

k J 2 Gn,r,s,p,a(q, h)

Moreover, there is a function g0 € Wp(r)(wm,q) on which the upper bound in (2.13) is attained.

Proof. Assuming that y = Y1(p) = pa in (2.4), with respect to (2.8), we can write

-h \ 1/P / t-h \ 1/P

\ 1/p / rh \

^m(f(r),t)2,Yiq(t)dtj ^m(f(r),t)2,Yiq(t)dtj

2m/2 inf Lfc,r,s,p(q,Y1,h) 2m/2 inf Gk,r,s,P,a(q, h)

Using equality (2.11) and the definition of the class Wp(r)(wm,q), we get

¿k—ii/'-W < /2 1 • (2.14)

2 г'n,r,s,p,a(q, h)

From (2.14), it follows the upper estimate of the value on the left-hand side of (2.13):

To obtain the lower estimate for this value, consider the function

2m/2an , r

and show that g0 belongs to Wpr)(wm,q). Differentiating this function r times, we obtain

si" W = V/2(""2^1) + ° (jf d - -<»- rW"» ■/('><*) "V-.

Using this equality and formulas (1.3), we get

,(p) N [1 - COS(» - r)t]m/2

Um [g° ' J 2,71 - 77-:

(1 - cos(n - r)t)mp/2 q(t)dt

0

Raising both sides of this inequality to a power p (0 < p < 2), multiplying them by the weight function q(t), and integrating with respect to t from 0 to h, we obtain

/ (¿m(g0r) ,t)2>7iq(t)dt = 1 0

or, equivalently,

¿mm (g0r) ,t)2,Yi q(t)d^ =1.

Thus, the inclusion g0 € Wpr)(wm,q) is proved. Since the relation

tf'M = + (jfo-™<» - -or'"2**)*)"""

holds for all 0 < s < r < n, n € N, and r, s € Z+, according to (1.5), we have

i Ui = 1 ^ ( fh{1_ cos{n _ r)f]mp ^ "1/P

raiiVyo;2,7l 2"V2 a„,r V 2(n - s + 1) + a \Jo /

2m/2 Gn,r,s,p,a(qi h)

Using this equality, we obtain the lower estimate

sup (/(i))2,7l : / € > = ,m/2<y 1-rTT. (2.16)

2 Gn,r,s,p,a(q, h)

Comparing the upper estimate (2.15) and the lower estimate (2.16), we obtain the required equality (2.13). □

1

3. Exact values of n-widths for the classes Wp(r)(^m,q) (r G Z+, 0 < p < 2)

Recall definitions and notation needed in what follows. Let X be a Banach space, let S be the unit ball in X, let An c X be an n-dimensional subspace, let An c X be a subspace of codimension n, let L : X ^ An be a continuous linear operator, let L±: X ^ An be a continuous linear projection operator, and let M be a convex centrally symmetric subset of X. The quantities

bn(M,X) = sup {sup (e > 0; eS n An+1 c M} : An+1 c X}, d„(M,X) = snf S sup (inf (||/ - g||x : g G A„} : / G M} : A„ C X}, 5„(M,X) =inf S inf (sup (||/ - L/||x : / G M} : LX C A„} : A„ C X}, d"(M, X) = inf S sup (||/ ||x : / G Mn An} : An C X}, n„(M,X) = inf S inf(sup{||/ - L±/||x : / G M} : LxX C A„} : A„ C X}

are called the Bernstein, Kolmogorov, linear, Gelfand, and projection n-widths of a subset M in the space X, respectively. These n-widths are monotone in n and related as follows in a Hilbert space X (see, e.g., [3, 11]):

6ra(M, X) < d"(M, X) < dra(M, X) = 5„(M, X) = nra(M, X). (3.1)

For an arbitrary subset M C X, we set

E„-i(M)x := sup SEra-i(/)2 : / G M}.

Theorem 3. The following equalities hold for all m,n G N, r G Z+, n > r, and 0 < h < n/(n — r):

Xn{W^\ujm,q),B2rn) = En_l{W^\ujm,q),B2rn)

1 12(n — r + 1) + a { [h. , ^mp MMY1/P (3"2)

f [1 - cos(n - r)t]mp q(t)dt Jo

2m/2 a„,ry 2(n + 1) + a where An(-) is any of the n-widths 6n(0, dn(^), dn(^), ¿n(-), and nn(^).

Proof. We obtain the upper estimates of all n-widths for the class Wp(r)(wm, q) with s = 0 from (2.14) since

<

En_i(,q))2>71 = sup{E„-i(/)2,7l : / G W;(r>(wm,q)}

/•h \ "VP

/ [1 - cos(n - r)t]mp q(t)dt

./0 y

' 2(n - r + 1) + a

—-4 /

2m/2 an,ry 2(n + 1) + a V./o Using relations (3.1) between the n-widths, we obtain the upper estimate in (3.2):

A„(Wr)(Wm,q)) < Era_i(Wr)(Wm,q))2Yl

< 1 l2(n-r + l)+a ( ¡hu (3"3)

2m/2 an,rV 2(n + 1) + a

f [1 - cos(n - r)t]mp q(t)dt 0

To obtain the lower estimate on the right-hand side of (3.2) for all n-widths in the (n + 1)-dimensional subspace of complex algebraic polynomials

n

Pn+i = {Pn(z) : Pn(z) = E afczk, afc G cj,

fc=0

we introduce the ball

B„+1: = {P„(.) € R, : IIP,,II < ^{^Ha (f H —(„.-,■)«]"■" ,i(i„«) },

where n > r, n € N, r € Z+, and show that Bn+1 C Wp(r)(wm,q). Indeed, for all pn(z) € Bn+1, from (1.3), we write

(p£\t) = 2m y <>fc(f )|2 (1 _ cos{k _ r)hy

/ 2,71 - r + 1)+ a v 77

(3.4)

m 2 m r- |ak (f )|2

< 2m max {a| r(l - cos(k - r)h)m} V /; |U'fcw;i-

r<k<n 1 k,rV v y y J ^ 2(k - r + 1)+ a

k=r

We have to prove that

max {a| r(1 — cos(k — r)h)m} = a^r(1 — cos(n — r)h)m, 0 < h < n/(n — r).

Consider the function

<^(k) = ak,r(1 — cos(k — r)h)m, r < k < n, 0 < h < n/(n — r).

We will show that the function <^(k) is monotone increasing for all accepted values k and h. To this end, it suffices to show that <^'(k) > 0. In fact

r—1 1

Lp'{k) = 2a| ,r E -—-(1 - cos(k - r)h)m + mha2k r sin(k - r)h( 1 - cos(k - r)h)m~l > 0. l=0

Hence, we can write (3.4) in the form

(p{r),t)9 < 2ma& r(l - cos(n - r)h)m Y |Qfc(/)'-

' ;2>yl - n,rV v ' ' 2(k - r + 1) + a

< 2ma£<r(l - cos (77, - r)/7,r E on + = 2m<r(l " cos (77 - r)/7)m|b„|||,71.

k=0 2(k r +1)+ a

k=r (3.5)

From (3.5), we have

wm(p(r), t)2m < 2m/2an,r(1 - cos(n - r)h)m/2||pnH2,Yi •

Raising both sides of this inequality to a power p (0 < p < 2), multiplying them by the weight function q(t), and integrating with respect to t from 0 to h, we obtain

rh rh

J ^(p(r),t)2>71 q(t)dt < 2mp/2an,r||pn||£>7l jf (1 — cos(n — r)h)mp/2q(t)dt < 1

for all pn € Bn+1. It follows that Bn+1 C Wpr)(w m, q). Then, according to the definition of the Bernstein n-width and (3.1), we can write the following lower estimate for all above listed n-widths:

An(Wp(r)(Wm,q),B2;7i) > 6n(Wp(r)(Wm,q),B2;7i) > 6n(Bn+1,B2)7i)

1 /2(77-r + 1) +a / fh r , , im„ , w \ ~1/p (3-6)

>-7--W-Ц-- / 1 - cos 77 - r)t]mpq(t)dt '

"2W2a„ V 2 77 + 1 +a vio

Comparing the upper estimate (3.3) and the lower estimate in (3.6), we obtain the required equality (3.2). Theorem 3 is proved. □

œ

4. Conclusion

Upper and lower estimates have been proven for extremal characteristics in a weighted Bergman space. In the case of a power function considered instead of a general weight, the values of n-widths have been calculated for a specific class of functions.

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