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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 24. Выпуск 2.
УДК 517 DOI 10.22405/2226-8383-2023-24-2-38-62
О скорости сходимости средних Чезаро двойного ряда Фурье функций обобщенной ограниченной вариации
Р. К. Бера, Б. Л. Годадра
Бера Радж Кумар — факультет математики, Бародский университет Махараджи Саяджи-
рао (г. Гуджарат, Индия).
e-mail: rameshkbera8080@gmail. com
Годадра Бхикха Лила — факультет математики, Бародский университет Махараджи Са-яджирао (г. Гуджарат, Индия). e-mail: bhikhu_ ghodadra@yahoo.com
Аннотация
В этой статье оценивается скорость сходимости средних Чезаро двойного ряда Фурье для 2^-периодической функции по каждой переменной и обобщенной ограниченной вариации. Полученный результат является обобщением результата С. М. Мажара для одного ряда Фурье и нашего более раннего результата для функции двух переменных.
Ключевые слова: двойной ряд Фурье, обобщенная ограниченная вариация, поточечная сходимость, скорость сходимости, среднее значение Чезаро.
Библиография: 20 названий. Для цитирования:
Р. К. Бера, Б. Л. Годадра. О скорости сходимости средних Чезаро двойного ряда Фурье функций обобщенной ограниченной вариации // Чебышевский сборник, 2023, т. 24, вып. 2, с.38-62.
CHEBYSHEVSKII SBORNIK Vol. 24. No. 2.
UDC 517 DOI 10.22405/2226-8383-2023-24-2-38-62
On the rate of convergence of Cesaro means of double Fourier SGFIGS of functions of generalized bounded variation
R. K. Bera, B. L. Ghodadra
Bera Raj Kumar — department of mathematics, Maharaja Savajirao University of Baroda (Gujarat, India).
email: rameshkbera8080@gmail. com
Ghodadra Bhikha Lila — department of mathematics, Maharaja Savajirao University of Baroda (Gujarat, India).
email: bhikhu_ ghodadra@yahoo.com
Abstract
In this paper, the rate of convergence of Cesaro means of the double Fourier series of a 2^-periodic function in each variable and of generalized bounded variation, is estimated. The result obtained is a generalization of a result of S. M. Mazhar for a single Fourier series and of our earlier result for a function of two variables.
Keywords: double Fourier series, generalized bounded variation, pointwise convergence, rate of convergence, Cesaro mean.
Bibliography: 20 titles. For citation:
R. К. Bera, B. L. Ghodadra, 2023, "On the rate of convergence of Cesaro means of double Fourier series of functions of generalized bounded variation" , Chebyshevskii sbornik, vol. 24, no. 2, pp. 38-62.
1. Introduction
The Dirichlet-Jordan theorem (see [11] or [17, p. 57]) asserts that the Fourier series of a 2^-periodic function f of bounded variation on [-к, к] converges at each point and the convergence is uniform on closed intervals of continuity of f. Bojanic [4], and Bojanic and Mazhar [6] have quantified this result by estimating the rate of convergence of the Fourier series and of Cesaro means of the Fourier series at each point, respectively. Also, Bojanic and Waterman [5], and Mazhar [13] have generalize the results of Bojanic [4], and Bojanic and Mazhar [6], respectively, for functions of generalized bounded variation. Hardy [10] proved the extension of the Dirichlet-Jordan theorem from single to double Fourier series. Similar to Bojanic [4], and Bojanic and Waterman [5], Moricz [14] and, Bera and Ghodadra [7] have quantified the result of Hardy, by estimating the rate of convergence of double Fourier series of functions of bounded variation and of generalized bounded varition, respecctivelv. Here we shall give an estimate of the rate of convergence of Cesaro means of the double Fourier series of a function f, 2^-periodic in each variable and of generalized bounded variation. Our result of this paper is a generalization of a result of Mazhar [13] for a single Fourier series and of our earlier result [7] for a double Fourier series.
2. Single Fourier Series
Here we shall recall certain results for pointwise convergence and rate of convergence of a single Fourier series. We need the following definitions.
Definition 1. Let f : R ^ С be a 2n-periodic function, which is Lebesgue integrable over T := [-к, ж). The Fourier series of f, denoted by S(f,x), is defined by
<x
S(f,x) = £ спегпх,
n=—<x
where
1 г*
cn = — f (u)e-mudu, n e Z. 2K J-TT
The nth symmetric partial sum of the Fourier series of f, denoted by Sn(f,x), is defined as
n
Sn(f,x)= ^ Cj , n = 0,1, 2,....
j=-n
Definition 2. The (ordinary) oscillation of a function h : [a, b] ^ C over a subinterval J of [a, b] is defined as
osci(h, J) = sup{|h(i) - h(t')l : t, t' e J}.
In the sequel, we will distinguish the subintervals of the non-negative half of the one-dimensional torus T = [-k, k] : Ij,m = [0 j,m, dj+i,m], wher e dj,m = for j = 0,1, 2,...m;m e N u {0}.
Definition 3. Let f be a real-valued function defined on an interval [a, b] and A = { Ara}^=1 be a non-decreasing sequence of positive numbers such that j- diverges. Then the function f is said to be of A-bounded variation (f e ABV) on [a, b] if there exists a positive constant M such that
^ i f(ak) - f(b k )l <M
¿=1 Xk <
k=1
for every choice {Ik} of non-overlapping interv als with Ik = [dk, bk ] c [ a, b], k = 1,...,n. If f e ABV [a, b] , the A-variation of f is defined by
Tr(t\ ^ if(ak) -f(bk)| Va(f, [a, b\) = sup -t-,
k=1 k
where the supremum is extended over all sequences {Ik} as above.
Note that for A = {1} ABV=BV, the set of all functions of bounded variation on [ a, b]. Also, note that if f is of A-bounded variation, then f(x + 0) and f(x - 0) exist at every point x of [a, b] (see, e.g., [16, Theorem 4]). We define, for x e [a, b],
s(f,x) = \{f(x + 0) + f(x - 0)} (1)
and
0x(t) = f(x + t) + f(x - t) - 2f(x), te [a, b]. (2)
Jordan [11] proved that if f is a 2-r-periodic function of bounded variation on [-k,k], then its
( , x) x
x
Theorem 1. If f is a 2k-periodic function and is of bounded variation on [-k,k], then for all x n
IM f,x) - s( f,x)I < n ¿V(V
fc=i
0-1
A B V
A = {n1}, 0 < ^ < 1, and denoted that class by 7BV and the corresponding variation by (f, [a, b}). Their result (including their Lemmas 1 and 2) is as follows.
Theorem 2. Let f e 7BV (T) 0 <7 < 1, and let Vj (<fix,v) denote the generalized variation of on [0, v]. Then
IM f, x) - s( f, x)I < 2 jr T^osci (^ Ik,n) < (2(2+ -T-l jr k) ,
k=0 + (n + 1) k=1
where s(f,x) and <x(t) are as in (1) and (2); respectively.
In order to obtain a result for Cesaro means, we first recall the following definition and properties, which can be found in ([17, pp. 94-95], [13, Theorem 1], or [6]).
Definition 4. Let K?(t) denote the (С, a) kernel and a?( f,x) the (C, a) mean of S (f,x) for -1 < a ^ 0. Then
Kn(t) = (v,
n v=0
1 г
<( f,x) = - f(x + t)K(t)dt,
К J-ж
and
l r
f,x) - s(f,x) = - <x(t)K-(t)dt,
n Jo
where Dv (t) and A-- are defined as
Dv (t) = - E
Jjt =
=- v
sin ( V + §)t
2 sin §
and
n + a\ r(n + a + 1)
in + a\ n
AS = 1, A? = | - ' " 1 = ' .,' " ' , n e N.
Г(п + 1)Г(а + 1)'
Some properties of K?(t) are as follows:
2 ^
2 Г
- Kan(t)dt = 1, К J 0
|K£(i)| < n + -, 0 <t<K,
fek+1,n Ci
IK(t)ldt< -jC?, к = 1,2,... ,n, |a| < 1,
and
K?(u)du
<
C§
( n )
l+a '
(3)
(4)
(5)
(6)
(7)
where C\ and C2 are constants.
Mazhar [13] generalize the result of Bojanic and Waterman [5] by proving the following more general theorem (including their Lemmas 1 and 2).
Theorem 3. Let, f € 7BV(T), 0 ^7 < l and let V1 (<x,v) denote the generalized variation of <x on [0, v]. Then for a > 7 — — —l < a ^ 0;
K( f, x) — s(i, x)| < C- £ {k + 1)l+a osci «x^ Ik,n) < E k--^ D , (8)
where C™ = + C\ + ^C^bd s(f,x) and <x(t) are as in (1) and (2); respectively.
ж
t
3. Double Fourier Series
In this section, we shall recall certain results for pointwise convergence and rate of convergence of a Double Fourier series. We need the following definitions.
Definition 5. Let f : R2 ^ C be a function, 2ir-periodic in each variable and integrable over T2. The double Fourier series of f is defined by
oo oo
where
S(f,x, y)=^ E Cjkei(jx+ky), (9)
1 fK fK
cjk = -^1 J f(u,v)e-i(jU+kV)dudv, i, k g Z. (10)
We consider the double sequence of symmetric rectangular partial sums
m n
J(jx+ky)
Sm,n(f,x, y)= E E CjkelUx+ky),m,n = 0,1,2,.... (11)
j=-m k=-n
The Cesaro ( C, a, ft)-means of the double Fourier series (9) for —1 < a, ft ^ 0 are defined (see, e.g., [9, p. 106]) by
1 1 m n
am!n( f, x, V) = 1 E E W(f, x, y), m,n = 0, 1, 2, . . . .
Am An M=0 v=0
Definition 6. A function f defined on a rectangle R := [a, b] x [c, d] is said to be of bounded variation in the sense of Vitali, in symbol, f £ BVy ( R), if
m n
suP E E1 f(xj, yk) — f(xj-u y k) — f(xj, yk-\) + f(xj-l, yk-i)l < ^ (12)
where the su,prem,u,m, is extended over all partitions
V1 : a = x0 < x1 < ■ ■ ■ <xm = b and V2 : c = y0 < y1 < ■ ■ ■ < yn = d
of [a, b] and [c, d] respectively. The su,prem,u,m, in (12) denoted by V(f, [a, b], [c,d]) is called the total variation of f over R.
If a function f £ BVy ( R) is such that the marginal functions ]'(■, c) and f(a, ■) are of bounded
[ a, ] [ , d]
the sense of Hardy and Krause, in symbols, f £ BVh( R)-
Definition 7. The rectangular oscillation of a function f : [a, b] x [c, d] ^ C over a subrectangle J x K of [a, b] x [c, d] is defined as
osc2(f, J, K) = sup{| f(u, v) — f(u', v) — f (u, v') + f(u', v')l : u,u' £ J, v, v' £ K}.
We also recall that the modulus of continuity of a function f on T2 is defined by
f, ¿1, h) := sup | f(u, v) — f(u', v) — f(u, v') + f(u', v%
|u—¿i, ¿2
Ux(f, 5):= sup | f(u, v) — f(u', v)l,
|u—Si, ^GT
and
and also
Uy(f, S) := sup | f(u, v) - f(u, v%
^(f, Si) = sup {| f(x + hi) — f(x) | : x € Tn}, i = 1, 2,...,n. Si
Definition 8. Let f (x + 0,y + 0) := limst^0+ f (x + s,y +1) be the limiting value of f as ( x, ) ( x, )
f (x — 0,y + 0),f (x + 0,y — 0) and f (x — 0,y — 0) can be defined analogously.
on T2, 2n-periodic in each variable, then its Fourier series (9) converges to s(f,x,y) at each point ( x, )
Moricz [14] quantified Hardy's result by estimating the rate of convergence of double Fourier series of f at (x, y) by proving following theorems.
Theorem 4 ([14, Theorem 2]). If f is a bounded, measurable function on T2; 2n-periodic in each variable, such that the four limits f (x ± 0,y ± 0) exist at a certain point (x, y), and the four limit functions f (x ± 0, ■) and f (-,y ± 0) exist, then for any m,n ^ 0 we have
2 m n
ISm,n( f,X, y) - s(f,X,y)l < 1 + -
1
)2 m n £ £
1
-) — — (j + 1)(k + 1)
osc2 (Фху, Ij,m, h,n)
j=0 k=0 1 \ m 1 + ^ +-) g йггу
OSCi(^xy(•, 0), Ij,m)
1 n 1 + ( 1 + ^ E (к + 1)0Ш'1(ФхУ(0, •), h'n),
where
and
s( f,x, y) = ~{f(x + 0,y + 0) + f(x - 0,y + 0) + f(x + 0,y - 0) + f(x - 0,y - 0)] (13)
'f(x + u,,y + v) + f(x u,y + v) + f(x + u,,y - v)
+f(x u,y - v) - 4s( f,x, y), ifu,v> 0; f (x + 0,y + v) + f(x - 0,y + v) + f(x + 0,y - v)
Фху (u, v) = {+f(x - 0,y - v) - 4s( f,x, y), if u = 0 and v> 0; (14) ( x + u, + 0) + ( x - u, + 0) + ( x + u, - 0)
+f(x -u,y - 0) - 4s(f, x,y), if u > 0 and v = 0;
0, if u = v = 0.
Theorem 5 ([14, Theorem 3]). If f(x, y) is 2--periodic in each variable and of bounded variation over T2 in the sense of Hardy and Krause, then for all m,n ^ 0; we have
|Sm,n(f,x,y) - S(f,x, y)l ^
4 (1 + 1)
1 \ 2 m n ж
(m + 1)(n + 1)
m n
(фху, j=l k=l ^
0,-
+ jrvUxy (•, 0),
m + 1
j=l l \ n
+ -.±V(фху(0.) [0.-])
k=l
where s(f,x, y) and $xy(u, v) are as in (13) and (14), respectively.
In [20], Zhizhiashvili have proved the following theorem for function of several variables. Theorem 6. (a) If f e C(Tn) and
{K r]
Ut(f, 5i) = 0{ (log-) ¡> (Si ^ 0, i = 1, 2,...,n), e > 0,
(b) If
1X -n
(K)-]
f, 5i) = o{(log-) } (6Z ^ 0, i = 1, 2,...,n),
Moricz [15] have also proved the similar type of result for function of two variables, which is as follows
Theorem 7 ([15, Corollary 1.2]). If f is continuous on T2,
u( f, Si, ¿2) = oj(log^ (logl^j I (á i, 02 ^ 0),
(f, 5) = oj(log^ I (á ^ 0), and uy (/;5) = oj(log l) | (á ^ 0),
In [9], D'vachenko have constructed a continuos function of 2 m variables (m £ N) with modulus of continuity
l \ \ —m 1
(KD) 1
f, 5i) = 0[ (log(-)) ) (15)
and its Fourier series is divergent almost everywhere in the Pringsheim sense based on example of Bakhbukh and Nikishin [8]. Similar results for A-divergent Fourier series are also constructed by Bakhvalov (see [1],[2]).
Definition 9. Let f be a measurable function defined on the rectangle [a, b] x [c, d] and A = {An}?= 1 and A' = {An}^J=i be non-decreasing sequences of positive numbers such that An, A'n ^ to and t~> diverges. Then the function f is said to be of (A, A')-bounded variation
An ^n
(f e (A, A')BV) on [a, b] x [c,d], if
(1) f(-, c) e ABV[a, b] and f(a, ■) e A'BV[c, d], and
(2) if X1 and I2 are the sets of finite collections of non-overlapping intervals Ij = [aj, bj], j = 1,2,...,m, and Jk = [ ck ,dk ], k = 1,2,... ,n, in [a, b] and [ c,d] respectively, and f(Ij x Jk) = f(aj, Ck) - f(aj, dk) - f(bj, Ck) + f(bj, dk), then
^^ |f(Ij x Jk)| . .
sup^^ —^- < TO. (16)
j=ik=l AjAk
We denote the supremum in (16) by V(a,a')(f, [a, b], [c,d]).
Here we shall consider the class (A, A') BV, where A = [ny} and A' = [ns}, for 7,8 ^ 0, 7 + 5 ^ 1; denote this class bv (7, S)BV and the corresponding variations by Vy(/(■, c), [a, b}), Vs (f(a, ■), [ c,d]) and Vys (f, [a, b], [c ,d]) respectively. The present authors have proved (see [7, Theorem 7]) that if f(x, y) £ (7, S) BV(T2), then all the four limits f(x ± 0,y ± 0) exist at every point (x,y). They have also generalized Theorem 5 of Moricz and proved the following (see [7, Theorem 8]).
Theorem 8. Let f £ (7,5)BV(T2), 7,8 ^ 0 7 + 5 < 1, and let V-(cpxy(■, 0),u), Vs(<xy(0, ■), v), and Vys (<xy ,u, v) denote the generalized variation of $xy on [0,u], [0,v] and [0,u] x [0,v], respectively. Then
where s(f,x, y) and <fixy(u, v) are as in (13) and (14), respectively.
We note that if the four quadrant limits f(x ± 0,y ± 0) exist at each point (x, y), then in view of (10) and (11), we have the representation
Zhizhiashvili (see [18, Theorem A] or [19, p. 233]) redicovered this result with the supplement that if f is continuous on a rectangle E, then its Fourier series (9) converges to f (x, y) uniformly on any rectangle R\ inside R. He also proved that Hardy's result remains valid if convergence is replaced by (C,a,@)-summabilitv, where a, ft > —1 are fixed real numbers. Bakhvalov [3] generalized the Zhizhiashivli's theorem for larger class of several variable function (see [3, Theorem 1]). In particular, Bakhvalov proved the following theorem (see [3, Corollary 1]).
Theorem 9. Let a, ft £ (-2,, 0) and 7 = a + 1, 5 = ft + 1. Then, for any function f £ (7, 5)BV(T2) its Fourier series is (C, a, fi)-summable to s(f,x, y) and the summability is uniform on any compact set in the neighborhood of which the function is continuous.
4. Main Results
The main results of this paper are as follows.
Theorem 10. If f is a bounded, measurable function on T2, 2ir-periodic in each variable, such that the four limits f (x ± 0,y ± 0) exist at a certain point (x, y), and the four limit functions f(x ± 0, ■) and f(, y ± 0) exist, then for any m,n ^ 0 and —1 < a, ft ^ 0, we have
\Sm,n(f,x, y) - s(f,x, y)\ <
(1 + 1 )2 (2 -7)(2 - 6) (m + 1)1-7 (n + 1)1-s
m n 1 / ч
EE ,j,l)
j=i k=i
(17)
(f,x,y) - S(f,x,y) ^CaCp^Yl
1
ОвС2(фХу, Ij,m, Ik,n)
j=0 k=0
(j + 1)1+a(k + 1)1+p
where constants Ca and Cp are as in Theorem 3.
Our second result, which is a particular case of Theorem 10, reads as follows.
Theorem 11. Let f e (7, 5)BV(T2), 7,5 ^ 0, 7 + 5 < I, aril let V) (<xy(•, 0),u), Vs (<xy(0, • ), v), and Vjs (<xy ,u, v) denote the generalized variation of <xy on [°,-u]; [0, v] and [0, u] x [0, v], respectively. Then for a > 7 — 1, ft > 5 — 1, and —1 <a,fi ^ 0;
a^(fx y)-S(f xy) ^i+^jm+i^ccEyy 1 Vi (<
.(2 +a — 7)(2 + i3 — S)CaCp ^ 1 ^ k k
j=i k=i
(2 +a — 7)Ca^ 1 „ n, k
3
(2 + p — S)Cp ^ 1 ^ fn , k k=1
where constants Ca and Cp are as in Theorem 3.
In particular, taking 7 = 5 = 0 in Theorem 11, we get the following corollary.
Corollary 1. Let f e bvh (T2), and let V (<xy (•, 0),u), V (<xy (0, •), v), and V (<xy ,u, v) denote the variation of <xy on [0,u]; [0, v] and [0,u] x [0, v], respectively. Then for —1 < a, ft ^ 0; we have
+ (2 + a — 7)C+" VJ<xy (•, 0)1)
(m + 1)a-)+1 f=1 j))-a )Yxy(, ),jj (2 + f3 — S)Cp A 1 V , k n
+ {n + 1)P-s+1 E ]^Vs {<xy(0, •),*), (19)
Vmi(f,X y) — S(f,X, y)
,p+1 j-a}
' j=1 k=1 J
m
(m + 1)»+1p,:i-» V"'']) + ^ ¿^ V (<xy ), (20)
where constants C» and Cp are as in Theorem 3.
Also, we can derive the following corollary from Theorem 10. Corollary 2. Let a, ^ e (—1,0).
(i) V
u(f, Si, 62) = 0 (V"(51,62 ^ 0),
Ux(f]5i) = o(6-a) (60), and Uy(f; 62) = 0 (Vp) (¿2 ^ 0),
then the Fourier series of the function f is uniformly (C,a,ft)- summable in the sense of Pringsheim.
(ii) If f e C(T2) and
Ux(f, 51) = O {5-2a+') and Uy(f; 62) = O (6-2p+^ (i = 1,2), e > 0,
then the Fourier series of the function f is uniformly (C,a,ft)- summable in the sense of Pringsheim.
(Ш) V
Шх(f, Si) = О (5-2a) and Шу(f; 82) = о (ё-2р) (6г ^ 0, i = 1, 2),
then the Fourier series of the function f is uniformly (C,a,ft)- summable in the sense of Pringsheim.
(iv) there exists a continuous function on T2 satisfying
Шх(f, Si) = О (S-a) (¿1 ^ 0) and Шу(f, 62) = О (S(¿2 ^ 0), (21)
and its (C,a,P)-mean of Fourier series diverges almost everywhere in restricted sense.
Замечание 2. Our Theorem, 10 is more general than the Theorem 4 of Moricz (except for exact constants). Our Theorem 11 is a two-dimensional analogue of Theorem 3 and in a particular case, we provide a quantitative version of Theorem 9. Also, setting a = ft = 0 in Theorem 11, we get our earlier result Theorem 8 (except for exact constants). In Corollary 1, we provide a quantitative version of Zhizhiashvili's result (see [18, Theorem A] or [19, p. 233]) for (C, a, ^)-summabilitv, for a, ft > —1. Our Corollary 2 is more general than the Theorem 7 and also for the case of function of two variable in Theorem 6.
5. Proofs
We need the partial summation formulas for single and double sequences, which are as follows.
n
Lemma 1. Consider n £ N. For j = 0,1,... ,n, let aj and bj be real numbers. Let Bj = ^ bk
k=
for j = 0,1, 2,... ,n, and Bn+1 = 0. Then
n n
E a3bi = E (ai - ai-!) Bi + a°B1.
3=1 3=1
Lemma 2 ([12, Proposition 7.37]). Consider ( m,n) £ N2. For j = 0,1,...,m and
m n
к = 0,1,... ,n, let aj,k and bj,k be real numbers, and let Bm,n = ^ bj,k- Then
j=0k=0
m n m- 1 n- 1
E E a3,kbj,k = am,nBm,n + ^ ^ (aj,k — aj+1,k — aj,k+1 + aj+1,k+1) Bj,k j=0 k=0 j=0 k=0
m- 1 n- 1
+ (aj,n — aj+1,n) Bj,n + ^^ (am,k — am,k+1) Bm,k. (22)
j=0 k=0
m n
Also, if we assume that Bj,k = bj',k' and Bm+1,n+1 = Bj,n+1 = Bm+1,k = 0, for
j'=j k'=k
j = 0,1,... ,m, к = 0,1,... ,n, then
m n m n
E E a3,k^3,k = E E (ai,k — a3,k-1 — a3-1,k + aj-1,k-1) Bj,k j=1 k=1 j=1 k=1
m n
+ ^2i(a3,0 — aj-1,0)Bj,1 + ^2(a0,k — a0,k-1)B1,k + a0,0B1,1. (23) j=1 k=1
The proof of our Theorem 10 is similar to that of a result of Moricz [14, Theorem 2] and the proof of Theorem 11 is similar to that of our earlier result [7, Theorem 8].
Proof of Theorem 10. Let m,n £ N be fixed. We start with the representation (17) of the difference of om%(f,x, y) and s(f,x, y). By writing < instead of <xy, in view of (4), it is clear that
f,x V) —s(f,x,y)
= —2 / [<(u, v) — <(u, 0) — <(0, v)}K£(u)K%(v)dudv k Jo Jo
1 fn 1 fn + 2KJ <(u, 0)K° (u)du + — jf <(0, v)Kn (v)dv
— Amn + Bm
+ Cn, say. (24)
Defining g(u, v) = <(u, v) — <(u, 0) — <(0, v), we decompose the double integral defining Amn as
K2Amn =j / g(u, v)Km(u)Kn(v)dudv
+ E / I [9(u,V) — 9(6j,m, v)}Km(u)Kn (v)dudv
j=\J 1j,rn J Iq
+ E/ / 9(0j,m, V)K\I(u)K%(v)dudv
j=l I j,rn Io,n
+ E/ / [g(u,v) — 9(u, 0k,n)}Km (u)Kn(v)dudv
k=l lQ,m J Ik,n
+ it [ / 9(u, ektn)K° (u)Knn (v)dudv
k=\ lQ,m J Ik,n
+ EE / / [g (u,v) — 9(u, 0k,n) — g(0j,m, v)+g(dj>m, 0k,n)} K^ (u)Kn (v)dudv
j=l k=l I3,^Jlk
+ EE / / [9 Wjm, v) — 9(0j,m, 0k,n)}Km (u)K* (v)dudv
j=l k=l I3,^Jlk
+ EE/ / [9(u, °kn) — 9(hm, 0k,n)}Km(u)K%(v)dudv
j=l k=l I3,m Ik
+ EE/ / 9( 0j,m, dk,n)Km (u)K,n (v)dudv
j=l k=l Ik,n
= Al + A2 +-----hAg, say. (25)
To estimate Al and A2, from (5) and by definition of g(u, v), as <(0,0) = 0, we have
lAll / lg(u, vmm(u)llKn(v)ldudv
J I Q ™ j I Q n
iq , m j iQ , n
\<(u, v) —<(u, 0) —<(0, v) + <(0,0)llKm(u)llKZ(v)ldudv
•-m (u)\\Kn
' Io,m Iq,
^ osc2(<, Io,m, !o,n) J J (m + + 0 dudv
^ n2OSC2«, Io,m, I0,n) (26)
and using (6), we have
Al = Ё/ / {9(u,v) - v)}K^(u)K?(v)dudv
1Ф(и, v) - ф(и, 0) -ф(в^т, ь)+ф( ej>m, 0)llKm(u)l K? (v) < E OSC2(ф, Ijm l0,n)(^ ^ J^^ ^ (n + 2)
т
< L
j=1 ^
j=i
1
^ ^^^ E "ЧТ^OSC2(Ф, Ij,m, Io,n) j=1 J
1
j=0
(j + 1)1+а
OOSC2 (Ф, Ij,m, Io,n).
Similalry we can get
Al < ^21+/3Ci E (fc + 1)1+/3 OSC2 (ф, Io,m, Ik,n).
Next, we estimate A3. Put
Rj,m = Km(u)du, j = 0,1,...,m + 1
J @j,m
and
R3 =
Rk,n =
f-K
/ КП(v)dv, k = 0,1,...,n + 1.
Then by (4) and (7), we have
2 \ 1+a
У
Rml -2 ) , i = 1, 2,...,m; Km = ^, R
Ж
2
m+1,m
and similarly
lRl,J < C2 (
1+1З
-k) , k = 1, 2,...,n; R0,n = Ж, Rn+1,n = 0.
Now, bv definition of A3, we have
u d
(27)
(28)
(29)
(30)
(31)
(32)
A3 = Е/ / 9(0j,m, v)Km(u)K?(v)dudv
j=1 I j,m Io,n
f { Ё 9(eim, v) i Km(u)du \ Kn(v)dv J lo ,n I j=1 I
J \ E 9(eim, v) Rm - Щ+im) \ Kn(v)dv.
n I j=1
0
Using the partial summation formula of Lemma 1 with aj = g(dj>m, v) and bj = Ram — RCj+l m) for j = 0,1,... ,m, we have
A3 =
/ \ E (9(9jm v) — 9(Oj-lm v)) (RZm — Ki+l,m) I Q,n l j=l
+ g(9o,m, v) (RZ m Rm+l,m
) (v)dv
/ \ E ( <( hm, V) — <( 0j,m, 0) — <(0j-l,m, ") + <( 0j-l,m, 0)) R*m \ K (v)dv, jiqt, „-_1
(33)
because g(0jym, v) — g(0j-l,m, v) = <(9j,m, v) — <(0jym, 0) — <(0j-l,m, v) + <(9j-l,m, 0^ g(0o,m, v) = = g(0, v) = 0 by definition of g(u, v), and Rm+l m = 0 by (31). From (5), (31), and (33), we conclude that
\A3\ ^ I \m,m, V) —<(0j,m, 0) — <( 0j-l,m, v)+<(dJ-l>m, 0)\\R*,m\\ K (v)\dv
JlQ.n „-_1
<
<
=l
c22i+z m 1
— l+Z _< n'
=l
C22l+a m-l
E -J+ Osc2(<, Ij-lm Io,n) jf (n + 0 dv
(
j=0
(j + 1)l+a
OSC2(<, Ij,m, I0,n).
(34)
Analogously, now using (32) instead of (31), we can see that
n l
IA , C22l+n ^
\ A5 \ < cVE
t^O (k + 1)l+n
OSC2(<, Io,m, Ik,n).
(35)
Next we estimate A^. By definition of g(u, v), we have
g(u, v) — g(u, dk,n) — g(0j,m, v) + g(Qj,m, °k,n)
= <(u, v) — <(u, dk,n) — <(Qj,m, V) + <(6j,m, °k,n),
and hence by definition of A6 and (6), we have
A\ = __
j=l k=l I3,m Ik,i m n „ „
" " \ <(u, V) — <(u, 0k,n) — <( ehm, v) + <(6jtm, 0k,n)\ \^ (u) \ K (V)
(36)
EE/ / [y(u,v) — 9(u, 0k,n) — g(0j,m, v) + g(Gj,m, &k,n)}K^(u)Kn(v)dudv
j=l k=l Ij.mJIk.n
m n
(,
j=l k=l 19,m 2k,n
m n
^ E E OSC2( <, Tj,m, h,n) j=l k=l
m n
d u d
k,n) I I Km
I j.m ^ I k.n
\ Km (u) \ Kn (v)
u d
<
Cl2
j=l k=l
1
jl+afcl+fi
OSC2(<, I j,m, h,n)
<
22+a+nC!^ -
=o k=o
(j + 1)l+a(k + 1)l+n
OSC2(<, Ij,m, h,n).
(37)
1
1
1
To estimate A7, using notation (29) and the partial summation formula of Lemma 1 with a3 = 9( Oj,m,v) — 9( Oj,m, Ok,n) and b3 = R<j,m — R'j+1,m j = 0,1,... ,m, we obtain
m n . .
A7 = EE/ / ^(0j,m, V) — 9(0itm, Ok,n)}Km(u)K,p(v)dudv j=1 k=1
I О л ■ I I о ■
E / \ T,(9(e3,m, V) - g(djtm, Ok,n))(Rim - Щ+1>т) \ ^(v)dv
k=1J Ik,n ^j=1 J
n „ , m
E / \ E(^(^^jm v) - 9(Qjm 9k,n)] - \9(v) - 9(Oj-1m Ok,n)])(Rlm - R^+1,m) k=1Jh,n I j=1
+ (g( 0o,m, V) - g( 0o,m, Ok,n))( Rim - Ri+1,m)}K% (v)dv
n I m I
E / ) E(Ф(d3>m, V) - Ф(03,m, Ok,n) - Ф^з-Х^, v) + Ф(0j-1,m, Ok,n))Rl,m > K?(v)dv,
k=1JIk,n I j=1
because g(do,m, v) = g(do,m, dk,n) = 0 by definition of g(u, v), R»+im = 0 by (31), and by (36) with u = dj-1,m. Now, in view of (6) and (31), it follows that
n , m n.
lA^l < E / \ E 1Ф( ^m, V) - Ф( Oj,m, Ok,n) -Ф( Oj-l,m, v) + Ф( 0j-1,m, в^Ш^ (v)ldv k=1 Ik,n у j=1 )
c1c221+i 1
<
E E П+ik1+? OSC2(Ф, ^ j-1,m, h,n)
j=1 k=1J
< ClC222+»+pm-.^ 1 UT r )
< -1++»- E E (j + 1)1+a(k + 1)1+p OSC2(l, ^m, Ik,n). (38)
Similarly, using (32) instead of (31), we can estimate
IA , C\C221+l3mA 1 /±T T ,
|A8| < Kl+p E E j1+ak1+p OSC2(<, hm, h-1,n) j=1 k=1 J
C\C222+»+p mn-1 1 UT T )
< -^- E E (j + 1)1+a(k + 1)1+p OSC2(<, 3<m, h,n). (39)
Keeping notation (29) and (30) in mind, we may write
m n
A9 = E E5'(Qk,n)(R»m — R)j+1,m)(Rk,n — Rк+l,n), j=1 k=1
whence a double summation by parts (see (23) of Lemma 2) with aj,k = g(0j,m, 0k,n) and b3,k = ( R(j,m — R'j+1,m)(Rk,n — Rk+1tn) ioT 3 = 0,1,..., m and k = 0,1,...,n, gives
m n
A9 = E E I 9(Ojm 0k,n) - 9(Ojm &k-1,n) - 9(Qj-im ® k,n) + 9(Qj-im ®k-1,n) \ j=1k=1 ^ '
x ( Rj,m - Rm+1,m)(Rk,n - Rn+1,n)
m n
E Е{Ф(^jm Qk,n) - Ф(9j,m, ®k-1,n) - Ф(Gj-1,m, ®k,n) + Ф(@j-1,m, &k-1,n)}Km^n j=1k=1
because aj,0 = a0kk = 0 by definition of g(u, v), for all j = 0,1,... ,m and k = 0,1,..., n, by (31), (32), and by (36) with u = 0j-\,m and v = dk-l,n. Thus, from (31) and (32), it follows that
jAyj «
EE
1
C222+Z+n
K2+a+n A^I A^I jl+afcl+n j=l k=l J
OSC2(<, Ij-l,m, h-l,n)
<
C222+ a+ n
m- l n- l (
-2+a+n j-^ ^ (j + 1)l+a (k + 1)l+n
OSC2 (<, I j,m, Ik,n).
(40)
Combining (25)-(28), (34)-(37), and (38)-(40) yields
\Amn\ CaCn ^ ^ ^ ^
=o k=o
(j + 1)l+a(k + 1)l+n
OSC2(<, Ij,m, h,n),
(41)
where Cv =(1 + ^Cl + ^C^or V = a,p.
■k2
In order to estimate Bm Cn in (24), it is enough to apply the first inequality of (8) of Theorem 3 with the equality (3), which gives
and
\Bm\
|C n| =
1 r ^ 1
— <(u,0)Km(u)du ^CaYl^j + 1)l+aOscl(0),ljm)
1 rK n 1
— <(v, 0)Kn(v)dv -— 1)l+n Oscl(<(0, ■), Ik,n). 0 k=0 ( + )
(42)
(43)
Now, using (41)-(43) in (24), we get (18) to be proved. ■
m n [0, 1, 2, . . . }
k
Mk = EEr
1
±3, k = ^^ f. . , , xS OSC2 (<xy, h,m, Il,n) ,
(i + 1)7 (l + 1)
=o =o
Mj = Et7
and
=o k
(i + 1)i (n + 1)s
Osc2 (<xy, Iiym, I'n;ri)
// v—>
M k = E
=o
1
(m + 1)1 (1 + 1)s^
Osc2 (<xy, Im,m, 1l,n) ,
where j = 0,1,..., m; k = 0,1, ...,n. Then we have
Mj,k ^ VyS (<xy, 6j+l,m, Gk+l,n) . Also, define functions M(u, v), M' (u^d M'' (v) on the rectangle
(44)
intervals
m+l'
^^d n+lrespectively, by
m+l
x [^
M( u, ) = M
|~(m+1)«j-t ^ (n + 1)^ -
M' (u) =M^ (m+i)tt
(45)
(46)
1
1
■
■
and
Note that
и ,
M (v) = Mi
(47)
(j + 1)ж <u< U + 2)ж
m + 1
m + 1
( m + 1) u j + 1 < (-)- <j+ 2
ж
( m + 1) u
ж
j + 1.
Similarly
(к + 1)ж (к + 2)Ж п+1 ' п+1
(п + 1)v
ж
= к + 1 .
Therefore, for each j = 0,1,..., m — 1; k = 0,1,..., n—1, and for each (u, v) in by (45), we have
m+1 , m+1
(к+1)ж ( к+2)ж n+1 , n+1
M(u, v)=Mjtk.
Now, using the double partial summation formula (see (22) of Lemma 2) with
(48)
n __1_
aj'k = U+1)1+a-'< (k+1)1+^-s к = 0,1,... ,п, we get
and bj к =
1
U+1)1 (k+1)
-OSC2 (ф:
xy, Ij,m, h,n) îor j = 0,1,...,m;
x
x
m n 1
У] I 1)1+д(к I 1)1+8 OSC2 (фхУ, Ij'm, Ik'n) m n 1 1
= 5 £ (з + 1)1+а-Чк + 1)1+ß-s и + 1)Чк + 1)s OSC2 (фху, Ij'm, h'n)
j=0 k=0 m 1 n 1
m- 1 n- 1
§ § M (i + (к + l)'+ß-s - (:j + 2)1+°-(к + l)l+"-s
M, ■ 1 1
11
+
(j + 1)1+a-1 (к + 2)1+ß-S (j + 2)1+a-J ( к + 2)1+ß-S, 1 m- M ( 1___1
+ (п + 1)1+ß-S M,n V (3 + 1)1+a-1 (3 + 2)1+a-1
+ (m + 1)1+a-^% Mm'k ( (к + 1)1+ß-S - ( к + 2)1+ß-\
+
M-m,n
(m + 1)1+0— (п + 1)1+ß-S = A + B + C + D, sav. (49)
We will use properties of the Riemann-Stieltjes integral to estimate A, B, and C. First, we estimate A. Since a > 7 — 1 and ft > 5 — 1, the functions —u-1-a+J and —v-1-p+s are continuous and
u, > 0
m l n l
A = EE MM
n— n i—n \
1
1
l+a—y
=o k=o
m l n l
)((
1
(j + 1)l+a-y (j + 2)l+a~yJ \(k + 1)l+n-S (k + 2)l+n-S
"1-t n-l / f j+2 \ / rk+2 . EEMj,k(/ d(—u-l-a+y))( d(—V-l-n+s))
=o k=o +l k+l
"-l n-J j+2 rk+2
E E Mjkk / (1 + a — 7)(1 +p — 5)u-2-a+yv-2-n+Sdudv „•_n ,, n ' jj+l Jk+l
(j + 2)l+a-yJ \(k + 1)J k+2
k+l
)
=o k=o
m- l n- l
= (1 + a — 7)(1+p — 5) EEMJtkl
+2
u
-2-a+ydu
=o k=o
Put s = "fr Then TTs = "T^u —3 + 1 &* —
+l
)(C,j-2-r'+Sd'j)■
(50)
(j + l)K
s — , u — j + 2 & s — ^m+T- Therefore
+2
u
-2-a+ydu =
+l
f^r /(m + 1)s\-2-a+y /m + 1 \ ju+dl \ -K j v k j
m + l
d
k
, \ -l-a+y , (J+2)l
m + 1 \ ' m+l _2_
'U + 1)l
a+ y
m + l
k
l+ a- y
, (j + 2)l
(m + 1)l+a-y Jg+i)L
m + l
u-2- a+ y d u.
Similarly,
k+2
k
l+ n-
( fe + 2)L
I V ~2-n+SdV = --I
Jk+l (n + 1)l+n-s J(j+±l
1 n + l
2
n+ d .
(51)
(52)
Using (51) and (52) in (50), we get
Mi i a 2+Z+8—y—S m-l n-l (3+2)l (k+2)L
A=(1 + a — 7)(} + fj — 5)K EE^ Z."1 M,ku-2-a+yv -2- n+s dudv.
(m + 1)l+a-y (n + 1)l+n-s
'(J+1)l (k+l)L j = 0 k=0 " m + l n + l
Since M(u, v) = Mj,k for all (u, v) £
(j+l)K (j+2)K m+l ' m+l
( k+l)K (k+2)K
n+l ' n+l
j, we get
I w, , A 2+a + R-y-S m-l n-l _ (3 + 2)l r (k + 2)L
A = (1 + a — 7)(}+<3 — 5)K f: EE [ m+1 ^ M(u, v)u-2-a+yv-2-fi+sdudv
' ' ' ' (j + l)L (k + l)L V ' 7
j=0 k=0 m + l n + l
(m + 1)l+a-y (n + 1)l+n-s (1 + a — 7)(1+p — §)K2+a+n-y-S rK
(m + 1)l+a-y (n + 1)l+n-s
/K fK
M( u, ) u-2- a+ y -2- n+ d u d .
m + l n+l
Put u = K and v = j. Then jS; = —ks-2, ^ = —Kt-2, u — m+l & s — m + 1 u — k & s — 1, v — u+t & t^n + ^d v —y k 1. Therefore
1
m+l
x
_ (1 + а --)(1+Р - 5)-2+i+l-y-& = ^ x
f1 f1 , Г(Ж Ж \ (Ж \-2-i+l (Ж \-2-Р+ь 2 2 2 , , x М -,- - - Ж2s-2t dsdt =
Jm+1 Jn+1 tJ \SJ \tJ
= + + H-У Г+1 Г+1 м (Ж,Ж-) ,ши =
(m + 1)1+i-y (n + 1)1+l- J1 J1 Ks'tJ (1 + a--)(1 + f3 - 6) Г+1 fk+\, f- -\i-irf-6.
+1 k+1
ЕЕ/ J М(Ж, Ж) si-1tl-&dsdt =
(m + 1)1+i-y(n + 1)1+l- j=1k=1Jj Jk ^s't
< (1+a---)(1+fj- q, £ E м () г-ч^ г Г dsdt=
(m + 1)1+i-y(n + 1)1+l-S p^f^ \j'k
k
(1 + a --)(1 + f3 - 6) ^ £ / жЖ\ f-1k?-&. (53)
(m + 1)1+i-y(n + 1)1+l-S \j'k
Now, we estimate B. Proceeding as above, we have
1 m-1 ( 1 1
B = (n + 1)1+l-s ]=0 Mj,n V (j + 1)1+i-1 - (j + 2)1+i-\ 1 m-1 / rj+2
m-1 / p+2 \ E Mj,n{l+i d (-u-1-i+y)j
(n + 1)1+l- ''n\Jj+1
1 m-1 ( i j+2 \
E {M3tn-1 + Mj) ^у (1 + a - --)u-2-i+yduj
(n + 1)1+l-s ^oy n- Jj \J j+1
I Л 1+i-y m-1 (i+2>
, (1+1a- Ф -+,, У М n-1 + М') m+1 u-2-i+ydu. (54)
(m + 1)1+i-y(n + 1)1+l- A^ v jn 1 'j J(3+i)* v ;
Note that if u e
^, j and v = П+г, then as [^] = [■ ] = n, by (48), we (u, пц) = Mj,n-ь Also, for u e [, )> [^^] - 1=h so that M'(u) = Mj.
have M
So from (54), we get
B =, »+.--»Т..,1¿J (M(u,n+T) + м'и)u 2-i+ydu
(m + 1)1+i~y(n + 1)1+l- A^ Jи+1)ж \ \>n + 1
j = o m +1 '
(1 + a --)-1+i-y Г ii пж , . , 2
--Ц-Г---ттт^г^ [M[u,-- )+M'(u])u-2-
(m + 1)1+i-y (n + 1)1+l- y^V V n + 1
^ (^u^ + M'(u)^ u-2-i+ydu
(1 + ar+1 (M (=.+=1) +M'
(m + 1)1+i-y(n + 1)1+l- J1 \ \s'n + 1) \s.
= mm n + £ I' + (M (f, n+r) + M' ( Ж ))
, (m + -n + \у1+?-> Z (M (^) + M' (Ж)) 3i-y, M
and similarly, we can prove
C -J++ V 6\ s E fM ) +M'' (K)) kns. (56)
(m + 1)l+a-y (n + 1)l+n-S/L^\ \m + Vk) \k) J '
In view of (44)-(47), we have
M{ H) =M[^ ]-l.[ ^ ]-l ^V-s{<xy, ^}", d[ ^ ],n) ^Vys(<xy (57)
and
m(+ M'( t) = mu+1-[ + m'r +,!
K n K K
n K
+ Myy =
[ m±i 1_1
L ] J n-l
1
g So ( i + m+1)SOsc2 (<xy,hn)
m-1 1
+ £ (1 + 1)1 (n + 1)s Osc2 (<xy, Iim In,n)
r1-1
L j \ n 1
= § £ (i+mi+ 1)S°X2 {4"y'IiM = Mi<fl H-
<Vys($xy ,-.,Ty (58)
In a similar way, we can prove the inequalities
m {m+T-A)+m■ © < v* (<xy,K,T) m
and
Mm,n ^ VyS (<xy, T, T . (60)
Using (57) into (53), (58) into (55), (59) into (56), and then the results and (60) into (49), we get
m n 1
E E (j + 1)J+a(ft + 1)1+nOsc2 (<xy, Ij'm, Ik'n)
< (1 + a — 7)(1+P — S) A A .a-ykn-SVJ6 T A ^ (m + 1)l+a-y (n + 1)T+n-S K VyS \<xy, j,k)
=1 k=1 m
+ (m + 1)l++y(n + 1)l+n-s lVyS {<xy, j,T)
+ (m + 1)1+Z-1(n +1)T+n-S E k^~&VyS {<xy, T -t) + (m + 1)T+a-l(n + 1)T+n-S V-S (<xy,T,T). (61)
Note that
т / ч т n / \
(фхуЛ,А <(фху,
3=1 V J J 3=1 k=1 V J J
n т n / \
£k3-% (фху< EE-r^k^VyS (фху 1,1)
k=1 j=1 k=1 V J /
and
т n
■a-y k.fi-&
т n
VlS (фху< EE^'k^VyS (фху,-,k) .
j=1 k=1 \ 3 J
yxy> j ' k
■j=i k=i
Therefore, from (61) we get
m n 1
E E (j + 1)i+a(ft + 1)1+nOsc2 (<xy, 1 ^ Ik'U
« (m +1)1^ i(n ^Ll £ Yf^VyS , T-k). m
j- 1 k- 1
Second, in view of the second inequality of (8) of Theorem 3, we get the inequalities
m 1 2 + — m 1 / N
E (j + 1)l+a Osci (<xy(■, 0), Ij,m) < ^O^ly E J-ZVy [<xy 0, 0),j) (63)
and
E {k +^1+13 OSC1 (фху(0, •), Ik,n) < ^^js E k-iK (фху(0, •), k) . (64)
k=0 V" 1 > k=l
Using (62)-(64) in the Inequality (18) of Theorem 10 we get (19). This completes the proof of Theorem 11. ■
l 2 m
n satisfving "T+2 ^ 2^ < "T+T n+2 ^ 2k < U+T respectively, implies that
ОЪС2(фху, Ij,m, h,n) ^ 4ш(f, 81, 52),
osc1(фху(•, 0), Ij,m) ^ 2шх( f, ¿1), and osc1(фху(0, •), h,n) ^ 2шу( f, 82). a £ (-1, 0)
т
1 т 1 т i3+1 1 * r+1 1 ,
> 7-r;— = > 7-T-T— dt ^ 1+y -T—dt = 1+ -T—dt
¿0 (j + 1)1+a (3 + 1)1+aJ3 ¿1Л t1+a J1 t1+a
1_ _1
a(m + 1)a a = \ (m + 1)
Similarly for ft £ (—1, 0), we have
= 1 - + 1 = О (-.-) = О(5').
£ W+W+3 = О^3)-
k=0
Now, using the inequality of Theorem 10, we have
m n 1
°m%( f, X, y) — S(f, X, y) | ^4CaCl3 E E (j + 1)1+'1(k + 1)1+p U( f, S1,
j=0 k=0 m
+ 2C"E 77
-w.
(f, 51)+2CpJ2
_ (j + 1)1+i"xKJ' ^ ' (k + 1)1+p =O(515p)w(f, 51,52) + O(51 )ux( f, 51) + O(5P)wy(f, 52).
om%( f, X, y) — s( f, X, y)l= O( 515P )o( 5-15-p) + O(5 ?)o(S-1) + O(5 p )o(5-p)
= o(1)^ as , 52 ^ 0.
For the proof of (ii) and (iii), first we have
w(f, 51, 52)= sup | f(u, v) — f(u', v) — f(u, v')+f (u', v')l
lu-u' ¿1, |u-¿2
< sup | f(u, v) — f(u', v)l + sup | f(u, v') — f(u', V')
lu-u'¿i,^GT lu-u'¿1, w'GT
= 2wx(f, ¿1),
similarly, we have w(f, 51, S2) ^ 2wy(f, S2), and
w(f, 61,52) = Vw(f, 61,52) Vw(f, 61, 52) < 2^Wx(f, 51)^Wy(f, 52).
^anl^(f,X,y) — s(f,X,y) <8C»Cp EE (j + 1)1+'1(kk + 1)1+p ^CA^/iCA^)
Wy(f, 52) (65) (66)
j=0 k=0 m
+ 2 C1 (
( + 1) 1+ 1
;(f, 51)+2Cp E
( k + 1)1+ p
wy (f, 52) (67)
j=0 w 7 k=0 =O( 5" S! )O( 5-1+e/25-p+e/2) + O( 5f)O( 5-2l+e) + O(5P) 5-2p+" =O(5e/25e2/2) + O(5-1+e) + O(5-p+€) = o(1), as 5x, 52 ^ 0.
ompn( f, X, y) — s( f, X, y) =O(515P)o(5-15-p) + O(5?)o(S-21) + O(5P)o(5-2p)
=0(1) + o(5-1) + 0 ( ¿2-p) = 0(1), as 51,52 ^ 0.
T2
(21), its Fourier series is (C,a,@) summable in restricted sense. Then it is (C, 0, 0)-summable in
m = 2
lim
-
(log (1))
TW=! = 0 (a > —1).
1
1
Therefore for any fixed e> 0 there is 50 > 0 such that for any 0 < 5T, S2 ^ S0, we have
^ * £('Og (1 )r v"*' ^ (hg (1 ))_1 •
Therefore, in view of (21), we have
^x( f, Si) = o (¿r*) =
and
Uy (f, 62) = 0 (5-) =
That means, for any function f £ C(T2) satisfying the conditions (68) and (69), its Fourier series converges in the Pringsheim sense. Which contradicts the theorem of D'vachenko [9, Theorem 1.2.4]. This completes the proof. ■
Funding
The first author would like to thank Council of Scientific and Industrial Research (CSIR) for financial support through JRF (File No. 09/114(0233)/2019-EMR-I).
Data availability
Not applicable.
Declarations Ethical conduct
This article is original, it has not been previously published, and it has not been simultaneously submitted for evaluation to another journal.
Informed consent
The research does not involve human participants and/or animals.
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Acknowledgements
The authors are thankful to the referee for his/her valuable suggestions for the present form of the paper.
О ( ( log I 1 ) ) ) (¿0 ^ ¿1 ^ 0) (68)
О [log hr (60 0). (69)
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Получено: 28.01.2023 Принято в печать: 14.06.2023
