Double cosine-sine series and Nikol’skii classes in uniform metric Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 8 (26), No3, 2019, pp. 187-203

DOI: 10.15393/j3.art.2019.6650

187

UDC 517.518.476

S. S. VOLOSIVETS

DOUBLE COSINE-SINE SERIES AND NIKOL'SKII CLASSES IN UNIFORM METRIC

Abstract. In the this paper, we give neccessary and sufficient conditions for a function even with respect to the first argument but odd with respect to the second one to belong to the Nikol'skii classes defined by a mixed modulus of smoothness of a mixed derivative (both have arbitrary integer orders). These conditions involve the growth of partial sum of Fourier cosine-sine coefficients with power weights or the rate of decreasing to zero of these coefficients. A similar problem for generalized "small" Nikol'skii classes is also treated.

Key words: double cosine-sine series, mixed modulus of smoothness, Nikol'skii classes

2010 Mathematical Subject Classification: 42B05, 42B35, 42A32

1. Introduction. Let {ak}(= be a sequence of real numbers, such

x

that |ak | < x. Then the functions k=1

x

f (x) = ^^ ak cos kx (1)

k= 1

and

X

g(x) = ak sin kx (2)

k= 1

are continuous and 2n-periodic (i.e., f,g G C2n) and (1) is the Fourier

series of f, correspondingly, (2) is the Fourier series of g. Lorentz [13]

(

established that the condition |ak| = O(n-a), n G N, for 0 < a < 1

k=n

© Petrozavodsk State University, 2019

implies f E Lip(a) or g E Lip(a). For ak ^ 0, Boas [2] proved the following

Theorem 1. (i) Let ak ^ 0, k E N, 0 < a < l and {ak be the sequence of Fourier sine or cosine coefficients of p. Then p E Lip(a) if

<x n

and only if ak = O(n-a), n E N, or, equivalently, kak = O(n1-a),

k=n k=1

n E N.

(ii) If ak ^ 0, k E N, {ak}^=1 is the sequence of Fourier sine coefficients

<x

of g, then g E Lip(l) if and only if kak < x>.

k= 1

Similar result to (i) was earlier obtained by Rubinstein [19] (see also Theorem A in [26]).

Nemeth [15] established several generalizations of Theorem A and gave a sharp version of Theorem 3 from [2]. Dyachenko [6] studied trigonometric series with coefficients of fractional order monotonicity, and obtained conditions for sums of such series to belong Lipschitz classes.

If u is increasing and continuous on [0; 2n], u(0) = 0, then u E $. A

<x

function u E $ belongs to the Bary class B, if Y1 k-1u(k-1) = O(u(n-1)),

k=n

n E N; respectively, it belongs to the Bary-Stechkin class Ba, a > 0, if

n

E ka-1u(k-1) = O(nau(n-1)), n E N (see [1]). k= 1

In the paper [3] by Butzer et al, several properties of fractional modulus of smoothness up(f,S), ft > 0, and its applications to the approximation theory were studied. Tikhonov [21], [22] proved a generalization of the Boas results in the case of fractional modulus of smoothness. In [23], the same author obtained the Boas type results for the Nikol'skii spaces WaH^ of functions. Let us note, that the previous results of the Boas type connected with Nikol'skii classes belong to Chan [4] and Nemeth [16].

For multiple complex Fourier series one can note the papers by Moricz and Fulop: [10] and [14]. Their results were generalized by the author of this paper in [26]. For the double cosine-sine series, Tevzadze [20] proved the following

Theorem 2. Let m,n E N, aik ^ 0 for all i,k E N, aik < <x,

i,k= 1

<x

h(x, y) = Y1 aik cos ix sin ky and u(t, t) be an increasing in each variable

i, k=1

function on [0,1]2, such that u(0, 0) = 0 and i i

u(u, v)u-m-iv-n-i du dv ^ Ciu(t, t)t-mT-n, t,T e (0,1], (3)

t T t 1

u(u,v)u-iv-n-i dudv ^ C2u(t,T)t-n, t,T e (0,1], (4)

0 t

1 T

u(u,v)u-m-iv-i dudv ^ C3u(t,T)t-m, t,T e (0,1], (5)

t0 t T

u(u,v)u-iv-i dudv ^ C4u(t,T), t,T e (0,1]. (6)

00

Then h e Hm'n(u) (see the next section) if and only if p q

^ ^ atk ^ Cu(1/p, 1/q), p,q e N n [2, +&>).

i=[p/2] k=[q/2]

Similar results were obtained in [20] for double sine and cosine series. Fiilop (see [8] and [9]) gave the necessary and sufficient conditions for sums of sine, cosine, and mixed double series to belong the space A*(2) = H2'2(^i;i) (see the next section), where ua,p(u,v) = uav@, 0 < a, ft ^ 1. Donskikh [5] proved some multidimesional analogues of Fulop results. Results from [8] and [9] were generalized by Yu [28] for classes HH = H1' l(u) (see the next section), where u satisfies the conditions similar to (3)-(6) in the case m = n =1. Han, Li, and Yu [11] considered mixed modulus of smoothness of natural orders and obtained

Theorem 3. Let u(u,v) be a conitinuous on [0, 2n]2, increasing and subadditive in each variable function, such that u(0, 0) = 0, ajk ^ 0 for all

x x

j,k e N, J2 ajk < x>, r,s e N. j=i k=i

(i) If the following relations

m n

jrksajk = O(mrnsu(1/m, 1/n)), m,n e N; (7)

j=i k=i

m ro

£ £

j=1 k=n+1

jrajk = O(mru(1/m, 1/n)), m,n G N;

ro

y^ s^2/ksajk = O(nsu(1/m, 1/n)), m, n G N;

j=m+1 k=1

oo oo

£ £ ajk = O(u(1/m, 1/n)), m,n G N; (10)

j=m+1 k=n+1

ro ro

are valid, then g(x,y) = ^ ^ ajk sin jx sin ky belongs to Hm'n(u).

k=1k=1 (ii) If g G Hm>n(u), then

mn

££ f kS ajk = O(mr* ns* u(1/m, 1/n)), j=1 k=1

where m,n G N, r* = r + 1 for even r and r* = r for odd r.

Using several conditions on u, the authors of [11] obtained some criteria for g G Hm,n(u) in terms of Fourier coefficients of g, but the confusion in formula numeration in [11, Theorem C] makes understanding of statements hard. The conditions (7)-(10) are not independent (see Lemmas 2 and 4 below). Using these facts, results of Theorem 3 were rewritten in a new form and extended to the Nikol'skii classes in [27].

In the present study, we extend Theorem 3 and its counterpart from [27] to the case of differentiable even with respect to the first argument and odd with respect to the second one functions, using the mixed modulus of smoothness and derivatives of arbitrary natural orders (Theorem 4). In the case of weak monotone Fourier coefficients with a given rate of decreasing, we give the sharp conditions for h from Theorem 2 to belong to the Nikol'skii classes Wr'sHm'n(u) (Theorem 5). Finally, we obtain an o-analogue of Theorem 4 (Theorem 6). The results are dependent on the evenness of m + r and n + s.

2. Definitions. Let r,s G Z+ = {0,1,...}, {ajk}j)keN C R and

roro

ksj | < rc. (11)

j=1 k=1

It follows From (11) that the series

roro

ajk cos jx sin ky (12)

j=1 k=1

and

x x

^2^2 jr ksajk cos(jx + nr/22) sin(ky + ns/2) (13)

j=i k=i

converge absolutely and uniformly to functions h(x, y) and ^>(x, y), respectively. By the classical theorem on differentiabilty of function series, we have

h<"S'(x.y):= ^^ = ^

everywhere. Let

m n / \/ \

Km,'nf(x,y) = {k)f(x +(m—2j)t/2,y +(n-2k)T/2)

be the mixed difference of orders m, n with steps t, t. For AmTnh(r's) (x,y) see Lemma 5. Let us consider the class $(2) of positive on [0, 2n]2 \ {(0, 0)} functions u, for which u(0, 0) = 0, u(xi,yi) ^ u(x2, yi), u(xi,yi) ^ u(xi,y2) if x-2 ^ xi, y2 ^ yi, xi,yi e [0, 2n], i = 1, 2. If u e $(2) is such that

t Bij)-u (^ f) = O (u f)) , m, n e N,

i=m j=n

then u belongs to the class BB.

If m,n > 0 and for u e $(2) the inequality

11m'k"'u (T'T) = o (jm'"u (j f)) ■ j-' e n,

is valid, then u belongs to the class BmBn. One-dimensional analogues of these classes were introduced by Bary and Stechkin [1]; for the two-dimensional case see, for example, [28]. For m,n e N and u e $(2), we will write f e Hm,n(u), if for all Si,S2 e [0,2 n] the inequality

umn(f,Si,S2) := sup{|A mff (x,y)| : 0 ^ t ^ Si, 0 ^ t ^ S2} ^ Cu(Si,S2)

holds and f e Wr'sHm'n(u), r,s e Z+, if f(r's exists everywhere and belongs to Hm'n(u). Learn more about these classes in Lp setting, e.g., in [18]. We will also consider

hm'n(u) = {f e Hm'n(u) : umn(f,Si,S2) = o(u(Si,S2))},

where 81,82 ^ 0+, and Wr'shm'n(u) are defined similarly to Wr'sHm'n(u). In the case r = s = 0, m = n =1 and ua,ß(u, v) = uavß, 0 < a, ß ^ 1, we denote Wr'sHm'n(uaß) by Lip(a,ß).

We shall write u e A2, if u(2t, t) ^ C1u(t, t) for all 2t, t e [0, 2n] and u(t, 2t) ^ Ciu(t, t) for all t, 2t e [0, 2n].

3. Auxiliary propositions. Lemmas 1-4 are proved in [26].

Lemma 1. Let m,n > 0, u e $(2).

(i) If u e BmBn, then u e A2.

(ii) If u e BB n A2, then u(^,t) e B for any fixed t e [0, 2n].

Lemma 2. Let {ajk}j,keN C R+ = [0, u e $(2), m,n > 0.

(i) If u e BmBn, then the condition

t t j = O (u (£, j)) , M,N e N, (14)

j=M k=N V V 7 7

implies

M N ( (2n 2n))

£ £ jmknajk = O MmNnu(— ,- , M,N e N. (15) j=1 k=1 ^ ^ ' '

(ii) If u e BB n A2, then (14) follows from (15).

Lemma 3. Let {ajk}jtk^N C R+, u e $(2), m,n> 0.

(i) If u e BmBnj {ajk} j,keN satisfies (14) and

^ ™ ( (2n 2n))

£5>k = 0[u{ % j^jj , M,N ^ (16)

j=M k=N M N

then

M N ( (2n 2n))

£ £ jmkna3k = o MmNnu — , - , M,N ^. (17) j=1 k=1 ^ ^ ' '

(ii) If u e BB n A2, {ajk}j,keN satisfies (17), then (16) is valid.

Lemma 4. (i) Let {ajk}j)keN C R+, u e $(2), u(-,t) e B for all t e [0, 2n], m,n > 0 and the relation (15) is valid. Then

± £ kno,jk = O (nnu (|,!)) , M,N e N.

j=M k=1 V V 7 7

(ii) If, instead of (15) in (i) we have (17), then x N ( ( 2n 2n W

EEknj = o(N^(--)), M,n

KM N

j=M k=l v v

Lemma 5. Ifm,k G N, Aff (x) = E(-1)j{m)f (x + (m-2j)t/2), then

H J (x) = z^(-1)" j j=o j '

we have

mn

Am cos kx = (2 sin kt/2)m cos(kx + —),

Am sin kx = (2 sin kt/2)m sin(kx + —).

In particular, for h defined as the sum of (12) and ^ defined as the sum of (13), respectively, under condition (11), we obtain

x x

A mnh(x, y) = E E ajk cos jx + "2r)sin (ky + T^x j=i k=i

x (2 sin jt/2)m(2 sin kr/2)n. (18)

x x

,-r 7„S ,

A mT>(x,y) = EE j r ks ajk x j=i k=i

/ (m + r)n\ ,, (n + s)n. ,m. , . .„ X cos jx + --sin (ky + -—(2 sin jt/2)m(2 sin kT/2)n.

(19)

Proof. Let zimf (x) be as above. It is known that Ajf (x) = Ai(Am-if (x)) and that Aieikx = (2i sin kt/2)eikx; therefore,

Am cos kx = Re(Ameikx) = Re[(2i sin kt/2)meikx].

The last expression equals to (— 1)m/2 cos kx(2 sin kt/2)m for even m and to (—1)(m+i)/2 sin kx(2 sin kt/2)m for odd m. Thus, Am cos kx = (2 sin kt/2)mx x cos(kx+mn/2). The second formula is proved in a similar manner. Since Am,nh(x,y) is the composition of Am with respect to x and Anf with respect to y, we obtain (18). But differentiation and the m-th difference commute, also the equalities cos(r)(x) = cos(x + rn/22), sin(r)(x) = sin(x + rn/2) hold, hence (19) is valid. □

Lemma 6. Let r,s G Z+, f be 2n-periodic in each variable continuous function. If f (r's) exists everywhere and is continuous, then for any m,n G N one has

Um+r,n+s(f ,81,82) ^ 8[8^Umn(f ^'^,81,82), 81,82 G [0, 2n].

Lemma 6 may be proved in the same way as similar one-dimensional result (see [25, Ch.3, §3.3, (1)]) using representation of higher order difference by means of derivative (see [25, Ch.3, §3.3, (4)]) and the equality of type Am+r = Ah(Am) with respect to both variables. For this lemma in the case of mixed Lp moduli of smoothness see [17] or [18].

4. Main results

Theorem 4. (i) Assume that r,s G Z+, {ajk jj'fceN satisfies the condition (11) and h(x,y) is the sum of (12). If m,n G N, u G BB f A2 and the condition

M N f f 2n 2n W

Y.Y.1 m+r kn+s\ajk | = O ( MmNnu( —, — jj, M, N G N, (20)

j=1 k=1 ^ ^ ' '

holds, then h G Wr'sHm'n(u).

(ii) Let {ajk}j,keN C R+ = [0, ro) satisfy the condition (11), h(x,y) be the sum of (12), m, n, r, s be as in the part (i). If u G BB f A2 and m + r is even, n + s is odd, then from h G Wr'sHm'n(u) it follows that the condition (20) is valid. If m+r is odd or n+s is even and u G BmBn f BB, then h G Wr'sHm'n(u) also implies (20).

Proof. (i) Let the condition (20) be valid. By (19) from Lemma 5, we have for t, t > 0 an upper estimate for \Amj.nh(r's (x,y)\ of the type

(M N ro N M ro \

EE + E E+E E + EE jrksM*

j=1 k=1 j=M +1 k=1 j=1 k=N+1 j>M k>N J

x

• jt sm-

kT

sinT

-• I(1) +1(2) +1(3) +1(4) (21)

-• IMN + IMN + IMN + IMN, (21)

where M = [2n/t], N = [2n/T]. By virtue of the inequality \ sinx\ ^ \x\, x G R, and (20), we find that for t,T G (0, 2n]

MN

I(mN ^ tmTn EE j m+r kn+s\a3k \ = O(u(t,T)). j=i k=i

m

n

Using Lemma 5, inequalities | sinx| ^ 1 and | sinx| ^ |x|, x e R, we have

N

iMN ^ 2mTn ^ r kn+sj l

j=M+i k=i

One may rewrite (20) in the form

(22)

MN

EE j mkn(jr ks\ajk |) = O ( M mNnu ( j=1 k=l ^ ^

M, f) >' M,N e N (23)

and by Lemma 1(ii) and Lemma 4(i) we obtain

oo N

E Ekn(jrks\ajk\) = O(N

j=M+1 k=1

M f)), M,N e N. (24)

From (22) and (24) we deduce that lMN = O(u(t,T)), t,T e [0, 2n]. Similarly, we estimate IM'N. Finally, by the condition u e BB n A2 and part (ii) of Lemma 2 (we again write (20) in the form (23))

xx

ImN ^ t t 2m+njr ksj | = O(u(t,T)), t,T e [0, 2n].

j=M +i k=N+i

Combining the obtained estimates, we see that

|Amтnh(r>s)(x,y)| = O(u(t,T)), t,T e [0, 2n],

whence the statement h e Wr'sHm'n(u) follows.

(ii) In Steps I-IV, we assume that r = s = 0 and set t = M-i, t = N-i for M,N e N.

Step I. Let m be even, n be odd. Then, by Lemma 5, we have

x x

Ciu(t,T) ^ |Am;nh(0,v)| = | 11 ajk(2 sin jt/2)m(2sin kT/2)n cos kv

j=i k=i

(25)

Since the series in the right-hand side of (25) converges uniformly in v, it may be integrated term by term over v e [—t/2,t/2], and we obtain

xx

CiTu(t,T) ^ 11 k-i(2 sin jt/2)m(2 sin kT/2)n+iajk ^ j=i k=i

MN

> C2 EE(jt)m(kT )n+1k-1ajk

j=i k=i

and

MN

EE j mknajk < C3M mNnu(t, t ) ^ C3M mN nu(2n/M, 2n/N ). j=i k=i

Step II. Let m, n be even. Then, by Lemma 5, we have

x x

Ciu(t,T ) ^ \A mra;nh(0, v)\ = I EE ajk (2 sin jt/2)m(2sin kT/2)n sin kv

j=i k=i

The series in the right-hand side may be integrated term by term over v e [0, t], and we obtain

xx

CiTu(t,T) ^ EE k-i(2 sin jt/2)m(2 sin kT/2)na]k(1 - cos kT) = j=i k=i

xx

= 2— EE k-i(2 sin jt/2)m(2 sin kT/2)n+2a3k > j=i k=i

MN

> C4 EE(jt)m(kT )n+2k-iajk

j=i k=i

and

MN

EE j mkn+iajk ^ C5M mN n+iu(2n/M, 2n/N ), M,N e N. j=i k=i

By the condition u e BB n A2 (see Lemma 1) and Lemma 2 (ii), we obtain (14), while the condition u e BmBn, (14) and Lemma 2 (i) imply (20).

Step III. Let m , n be odd. Then, by Lemma 5, we have Ciu(t,T) ^ \Amfh(u,v)\ = I x x I

= | EE ajk (2 sin jt/2)m(2 sin kT/2)n sin ju cos kv (26)

j=i k=i

for u,v G R. The series in the right-hand side of (26) uniformly converges in v and may be integrated term by term over v G [—t/2,t/2]. Therefore,

œ œ

„-1.

u e R. (27)

Ci Tu(t,T) ^ k-1ajk (2 sin jt/2)m(2 sin kT/2)n+1 sin ju

j=1 k=1

Substituting u = t/2 into (27), we obtain

ro ro

Ci Tu(t, t ) ^ 2-1 EE k-1ajk (2 sin jt/2)m+1(2 sin kT/2)n+1 ^ j=i k=i

MN

^ C6 EE(jt)m+1(kT)n+1k-1ajk j=i k=i

and

MN

EE j m+1kn ajk ^ C7 M m+1 N nu (2n/M, 2n/N), M,N G N. j=i k=i

By Lemmas 2 and 1 and the condition u G BB f BmBn, we deduce, similarly to Step II, that (20) holds.

Step IV. Let m be odd, n be even. By Lemma 5, we have

u,v e R. (28)

C\u(t,T) ^ | EE ajk (2 sin jt/2)m(2 sin kT/2)n sin ju sin kv j=i k=i

Integrating (28) over v G [0,t], we obtain

| ro ro

CiTu(t,T) ^ | E E k-1ajk(2 sin jt/2)m(2 sin kT/2)n sin ju(1 - cos kT) j=i k=i

(29

where u G R. Substituting u = t/2 into (29), we obtain

roro

CiTu(t,T) ^ 4-1 EE k-1ajk (2 sin jt/2)m+1(2 sin kT/2)n+2 > j=i k=i

MN

^ C8 EE(jt)m+1(kT)n+2k-1 ajk j=i k=i

and

M N

EE j m+1kn+1ajk ^ C7Mm+1Nn+1u(2n/M, 2n/N ), M,N G N. j=i k=1

Using Lemmas 2 and 1 and the condition u G BB f BmBn, we deduce, again, that (20) holds.

Step V. Secondly, we consider the general case r, s G Z+. If h G Wr'sHm'n(u), then, by Lemma 6,

um+r, n+s (h, 81,82) ^ 8[ 8Sum'n(h(r'S ,81,82 ) ^ C5 8{ 8^(81,82),

i.e., Wr'sHm'n(u) c Hm+r'n+s(Qr,s), where ^(81,82) = 8[8^(81,82). If u G BB f A2, then Qr>s also belongs to BB f A2, while if u G BmBn, then, by definition, belongs to Bm+r^n+s. Applying the results obtained in Steps I-IV in all cases, we have

m n 2 2

EE j m+r kn+sajk = 0^Mm+r Nn+s 0,s( M, N)) = j=1 k=1

= o(MmNnj2n, ^

V \M N J

Thus, (20) holds under conditions of part (ii) of Theorem 1. □

Corollary 1. If m = n =1, u G BB f B1B1, {ajk}xk=i c R+ satisfies (11) and h(x,y) is the sum of (12), then the conditions

MN

EE jkajk = 0(MNu(2n/M, 2n/N )), M,N G N, j=1 k=1

and h G H 1'1(u) are equivalent.

Corollary 2. If m = 2, n =1, u G BB f A2, {ajk}xk=l c R+ satisfies (11) and h(x,y) is the sum of (12), then the conditions

MN

EE j 2kajk = 0(M 2Nu(2n/M, 2n/N )), M,N G N,

2

j=1 k=1 2'1

and h G H2'1(u) are equivalent. In particular, u2'1(f, 81, 82) = 0 (8282),

x x

2

81, 82 G [0, 2n], if and only if the series E E j2kajk converges.

j=1k=1

Corollary 3. Let {ajk}Cjkk=1 and h be as in Corollary 2. Then, h G A* (2) (see Introduction) if and only if the condition

roro

EE ajk = O(M-1N-1), M,N G N,

j=M k=N

holds.

Remark. Corollary 3 is obtained by Fulop [9] together with its "small" analogue that may be derived from Theorem 6. Let us note that a particular case u(81,82) = 8282 in Corollary 2 corresponds to the exclusive case (ii) in Theorem 1.

5. Concluding remarks. We say that {ajk}ro°k=1 is weak monotone if ajk ^ 0 for all j,k and aj ^ Caki for all i G [k, 2k — 1], j G [l, 2l — 1]. The famous Lorentz theorem [13] states that if {an}ro=1 decreases to zero and 0 < a < 1, then the assertions (i) an = O(n-a-i), n G N,

roro

(ii) f (x) = an cos nx G Lip(a) and (iii) g(x) = ^ an sin nx G Lip(a)

n=1 n=1

are equivalent. Several one-dimensional generalizations of the Lorentz theorem to generalized Lipschitz or Nikol'skii spaces and classes of general monotone sequences may be found in papers of Tikhonov [23] and [24]. The following theorem is an extension of the Lorentz result and results from [23] to the two-dimensional mixed case. Applications of one-dimensional weak monotonicty can be found in [12]. Also, we note the paper by Dyachenko and Tikhonov [7], where the estimates of Fourier coefficients satisfying another definition of weak monotonicity are given.

Theorem 5. Let {ajk}Cjkk=1 be weak monotone, m,n G N, r, s G Z+, u G BB f BmBn. If {ajk}Cjkk=1 satisfies (11), h(x,y) is the sum of (12), then the conditions

(i) h G Wr'sHm'n(u); and

(ii) ajk = O(j-r-1k-s-1u(2n/j, 2n/k)), j, k G N, are equivalent.

The proof of Theorem 5 is similar to the proof of Theorem 2 from [27]. The last theorem is the o-analog of Theorem 4.

Theorem 6. (i) Let m,n G N, r,s G Z+, u g BB f A2. If {ajk}rok=l satisfies (11), h is the sum of (12) and the conditions (20) and

MN

EE jm+rkn+s\a3k\ = o(MmNnu(2n/M, 2n/N)), M,N ^ ro, (30) j=i k=i

are valid, then h G Wr,shm'n(u).

(ii) If m,n G N, r, s G Z+, aj,k ^ 0 for all j, k G N, {ajk}Cjkk=1 satisfies (11), h is the sum of (12), u G BB f A2, m + r is even, n + s is odd, and h G Wr'shm'n(u), then (30) is valid. If, in addition to the previous conditions of (ii), we have u G BmBn, then also (30) is valid.

Proof. Similarly to the proof of Theorem 4, we use the estimate (21) for t,T G (0, 2n] and M = [2n/t], N = [2-k/t]. By (30) and the condition u G A2, we find that

M N

4)N ^ EE jr ks\ajk \(jt)m(kT )n ^ j=l k=l

^ £tmTnMmNnu(2n/M, 2n/N) ^ Ci£u(t, t), 0 <t,T< 8i(e). (31)

By Lemma 4 (ii) applied to {jrksajk}j°k=1 and (30) for M, N > n0(£), we obtain

ro N / 2n 2n \ IMN < C2T^ rkn+sajk <£TnNnu(-, - ^ C3£u(t,T),

j=M+1 k=1 ^ '

(32)

where 0 <t,T < 82(e). The quantity iM^N is estimated in a similar manner for 0 <t,T< 83(£). By the condition u G BB f A2 and Lemma 3 (ii), we have iMN = o(MmNnu(2n/M, 2n/N)), M,N ^ ro, whence

iMN ^ C\£u(t, t), 0 <t,T< 84(£). (33)

From estimates (31)-(33) we find that \AmT-nh(r's)(x, y)\ ^ C5£u(t,T) for all t,T < 8(£) := minKK4 8,(£), i.e. h G Wr'shm'n(u).

(ii) The assertion is proved similarly to the proof of (ii) from Theorem 4. We may substitute in all steps of this proof C1 instead of n and get at the end of Steps I-IV an estimate of type KnMmNnu(2n/M, 2n/N), where K depends on m, n. Setting Krq = £, we finish the proof. □

Remark. It is interesting to obtain a variant of Theorem 4 without the condition u G BmBn. The first attempt in this direction may be found in [27, Theorem 3], but this result gives only sufficient conditions for an odd in each argument functions to belong to a Nikol'skii class.

Acknowledgement. Author thanks all referees for their valuable comments improving the paper.

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Received July 03, 2019. In revised form, September 20, 2019. Accepted September 24, 2019. Published online October 7, 2019.

Saratov State University 83 Astrakhanskaya St., Saratov 410012, Russia E-mail: VolosivetsSS@mail.ru