GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 4, 1999, 307-322

ON ESTIMATING INTEGRAL MODULI OF CONTINUITY OF FUNCTIONS OF SEVERAL VARIABLES IN TERMS

OF FOURIER COEFFICIENTS

L. GOGOLADZE

Abstract. Inequalities are derived which enable one to estimate integral moduli of continuity of functions of several variables in terms of Fourier coefficients.

In the paper inequalities are derived which give estimates of integral moduli of continuity in terms of Fourier coefficients. These inequalities generalize S. A. Telyakovskiĭ’s results [1] to the multi-dimansional case. For simplicity and brevity we shall limit our discussion to the two-dimensional case.

First we give some one-dimensional theorems for the series

\frac{a_{0}}{2} + ∑_{i=1}^{∞} a_{i} cos ix,  (1)

∑_{i=1}^{∞} a_{i} sin ix.  (2)

Let

\begin{matrix}Δa_{i} = a_{i} − a_{i+1}, \\ δa_{i} = \begin{matrix}[ & \frac{i}{2} & ] \\ ∑ \\ ν=1 \end{matrix} \frac{Δa_{i−ν} − Δa_{i+ν}}{ν} , i ≥ 2; δa_{i} = 0, i < 2, \end{matrix}  (3)

α_{i,m}(p) = \begin{cases}(im^{−1})^{p}, & 1 ≤ i ≤ m, \\ 1, & i > m, \\ 0, & i < 1. \end{cases}  (4)

In [1] and [2] the following theorems are proved.

307 Theorem A_{1}. Let

1991 Mathematics Subject Classification. 42B05.

Key words and phrases. Integral modulus of continuity.

1072-947X/99/0700-0307$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

lim_{i→∞} a_{i} = 0,

∑_{i=0}^{∞} (|Δa_{i}| + |δa_{i}|) < ∞.

Then series (1) converges for all x ≠ 0 and for its sum f(x) we have

∫_{−π}^{π} |f(x)|dx = O( ∑_{i=0}^{∞} (|Δa_{i}| + |δa_{i}|)),

ω_{p}(f, \frac{1}{m})_{L} = O( ∑_{i=0}^{∞} α_{i,m}(p)(|Δa_{i}| + |δa_{i}|)),  (5)

where ω_{p}(f, \frac{1}{m})_{L} is an integral modulus of continuity of order p.

Theorem A_{2}. Let the conditions of Theorem A_{1} be fulfilled and

∑_{i=1}^{∞} \frac{|a_{i}|}{i} < ∞.

Then series (2) converges for all x ∈ (−π, π] and for its sum g(x) we have the estimates

∫_{\frac{π}{2μ+1}}^{π} |g(x)|dx = ∑_{i=1}^{μ} \frac{|a_{i}|}{i} + O( ∑_{i=0}^{∞} (|Δa_{i}| + |δa_{i}|)),

ω_{p}(g, \frac{1}{m}) = \frac{2^{p}}{π} ∑_{i=m}^{∞} \frac{|a_{i}|}{i} + O( ∑_{i=0}^{∞} α_{i,m}(p)(|Δa_{i}| + |δa_{i}|)).  (6)

Note that in formulas (5) and (6) the coefficients in the O-terms depend only on p.

Now let a double sequence of numbers a_{i,j}, i, j = 0, 1, …, be given and set

Δ_{1}a_{i,j} = a_{i,j} − a_{i+1,j}, Δ_{2}a_{i,j} = a_{i,j} − a_{i,j+1},

Δ_{1}Δ_{2}a_{i,j} = Δ_{1}(Δ_{2}a_{i,j}) = Δ_{2}(Δ_{1}a_{i,j}),

Δ_{1}^{2}a_{i,j} = Δ_{1}(Δ_{1}a_{i,j}), Δ_{2}^{2}a_{i,j} = Δ_{2}(Δ_{2}a_{i,j}),

δ_{1}a_{i,j} = \begin{matrix}[ & \frac{i}{2} & ] \\ ∑ \\ ν=1 \end{matrix} \frac{Δ_{1}a_{i−ν,j} − Δ_{1}a_{i+ν,j}}{ν} , i ≥ 2; δ_{1}a_{i,j} = 0, i < 2,

δ_{2}a_{i,j} = \begin{matrix}[ & \frac{j}{2} & ] \\ ∑ \\ μ=1 \end{matrix} \frac{Δ_{2}a_{i,j−μ} − Δ_{2}a_{i,j+μ}}{μ} , j ≥ 2; δ_{2}a_{i,j} = 0, j < 2,

δ_{1}Δ_{1}a_{i,j} = δ_{1}(Δ_{1}a_{i,j}) = Δ_{1}(δ_{1}a_{i,j}) = Δ_{1}δ_{1}a_{i,j}, δ_{1}Δ_{2}a_{i,j} = δ_{1}(Δ_{2}a_{i,j}) = Δ_{2}(δ_{1}a_{i,j}) = Δ_{2}δ_{1}a_{i,j}, δ_{2}Δ_{1}a_{i,j} = δ_{2}(Δ_{1}a_{i,j}) = Δ_{1}(δ_{2}a_{i,j}) = Δ_{1}δ_{2}a_{i,j}, δ_{2}δ_{1}a_{i,j} = δ_{2}(δ_{1}a_{i,j}) = δ_{1}(δ_{2}a_{i,j}) = δ_{1}δ_{2}a_{i,j}.

Given a function f ∈ L(T^{2}), T = (−π, π], consider the expressions

Δ_{t}^{(p,0)}f(x, y) = ∑_{μ=0}^{p}(−1)^{μ} (\begin{matrix}p \\ μ \end{matrix}) f(x + (p − 2μ)t, y),

Δ_{s}^{(0,q)}f(x, y) = ∑_{ν=0}^{q}(−1)^{ν} \binom{q}{ν}f(x, y + (p − 2ν)s),

Δ_{t,s}^{(p,q)}f(x, y) = Δ_{t}^{(p,0)}(Δ_{s}^{(0,q)}f(x, y)) = Δ_{s}^{(0,q)}(Δ_{t}^{(p,0)}f(x, y)),

ω^{(p,0)}(h, f) = sup_{|t|≤h} ∫_{T^{2}} |Δ_{t}^{(p,0)}f(x, y)|dx dy,

ω^{(0,q)}(η, f) = sup_{|s|≤η} ∫_{T^{2}} |Δ_{s}^{(0,q)}f(x, y)|dx dy,

ω^{(p,q)}(h, η, f) = sup_{\begin{matrix}|t|≤h \\ |s|≤η \end{matrix}} ∫_{T^{2}} |Δ_{t,s}^{(p,q)}f(x, y)|dx dy.

Consider the series

∑_{i=0}^{∞} ∑_{j=0}^{∞} λ_{i,j}a_{i,j} cos ix cos jy,  (7)

where λ_{0,0} = \frac{1}{4}, λ_{0,j} = λ_{i,0} = \frac{1}{2}.

Theorem 1. Let

lim_{i+j→∞} a_{i,j} = 0,  (8)
A_{i,j} = |Δ_{1}Δ_{2}a_{i,j}| + |δ_{1}Δ_{2}a_{i,j}| + |Δ_{1}δ_{2}a_{i,j}| + |δ_{1}δ_{2}a_{i,j}|,  (9)
∑_{i=0}^{∞} ∑_{j=0}^{∞} A_{i,j} < ∞.  (9′)

Then series (7) converges for all x ≠ 0, y ≠ 0 and for its sum f(x, y) we have

∫_{T^{2}} |f(x, y)|dx dy = O( ∑_{i=0}^{∞} ∑_{j=0}^{∞} A_{i,j}),  (10)

∫_{T} ( sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{s}^{(0,q)}f(x, y)|dy)dx=O(∑_{i=0}^{∞}∑_{j=0}^{∞} α_{j,n}(q)A_{i,j})=O(B_{n}), (11)

∫_{T} ( sup_{|t|≤ \frac{1}{m}} ∫_{T} |Δ_{t}^{(p,0)}f(x, y)|dx)dy =O(∑_{i=0}^{∞}∑_{j=0}^{∞} α_{i,m}(p)A_{i,j})=O(C_{m}), (12)

sup_{|t|≤ \frac{1}{m}} ∫_{T} ( sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{t,s}^{(p,q)}f(x, y)|dy)dx =

= O( ∑_{i=0}^{∞} ∑_{j=0}^{∞} α_{i,m}(p)α_{j,n}(q)A_{i,j}) = O(D_{m,n}),  (13)

sup_{|s|≤ \frac{1}{n}} ∫_{T} ( sup_{|t|≤ \frac{1}{m}} ∫_{T} |Δ_{t,s}^{(p,q)}f(x, y)|dx)dy = O(D_{m,n}).  (14)

Note that the coefficient in the O-term depends only on q in (11), only on p in (12), and only on p and q in (13) and (14).

Proof. (9) and (9′) imply

∑_{i=0}^{∞} ∑_{j=0}^{∞} |Δ_{1}Δ_{2}a_{i,j}| < ∞.  (15)

This and (8) give for each j ≥ 0

∑_{i=0}^{∞} |Δ_{1}a_{i,j}| = ∑_{i=0}^{∞} | ∑_{ν=j}^{∞} Δ_{2}Δ_{1}a_{i,ν} | ≤ ∑_{i=0}^{∞} ∑_{ν=j}^{∞} |Δ_{1}Δ_{2}a_{i,ν}| < ∞ (16)

and

lim_{j→∞} ∑_{i=0}^{∞} |Δ_{1}a_{i,j}| = 0.  (16′)

In a similar manner, for any i ≥ 0 we obtain

∑_{j=0}^{∞} |Δ_{2}a_{i,j}| < ∞

and

lim_{i→∞} ∑_{j=0}^{∞} |Δ_{2}a_{i,j}| = 0.  (17)

Further, applying the Hardy transformation, we have

∑_{i=0}^{m} ∑_{j=0}^{n} λ_{i,j}a_{i,j} cos ix cos jy = ∑_{i=0}^{m−1} ∑_{j=0}^{n−1} D_{i}(x)D_{j}(y)Δ_{1}Δ_{2}a_{i,j} +

+ ∑_{i=0}^{m−1} D_{i}(x) cos nyΔ_{1}a_{i,n} + ∑_{j=0}^{n−1} D_{j}(y) cos mxΔ_{2}a_{m,j} + D_{m}(x)D_{n}(y)a_{m,n}.

Now, using the property of Dirichlet kernels and (8), (15), (16′), (17), one can easily verify that series (7) converges for all x ≠ 0, y ≠ 0. Moreover,

(8) and (16) imply that the series

\frac{a_{0,j}}{2} + ∑_{i=1}^{∞} a_{i,j} cos ix = f_{j}(x)  (18)

converges for each j ≥ 0 and all x ≠ 0, y ≠ 0.

As it is known (see, e.g., [3], p. 14) if a double series ∑_{i,j=0}^{∞} u_{ij} is convergent

to S and the series ∑_{i=0}^{∞} u_{ij} is convergent for each j, then the double series is

repeatedly convergent to S, i.e.,

∑_{i=0}^{∞} ( ∑_{i=0}^{∞} u_{ij}) = S.

Therefore, series (7) will converge repeatedly and for its sum f(x, y) the equality

f(x, y) = \frac{f_{0}(x)}{2} + ∑_{j=1}^{∞} f_{j}(x) cos jy  (19)

will be fulfilled. Hence, in turn, it follows that for each x ≠ 0

lim_{j→∞} f_{j}(x) = 0  (20)

and for each fixed t ∈ [ − \frac{1}{n}, \frac{1}{n}] we have

Δ_{t}^{(p,0)}f(x, y) = \frac{Δ_{t}^{(p,0)}f_{0}(x)}{2} + ∑_{j=1}^{∞} Δ_{t}^{(p,0)}f_{j}(x) cos jy  (21)

for all (except the finite number of values of) x. From (18) we obtain

Δf_{j}(x) = \frac{Δ_{2}a_{0,j}}{2} + ∑_{i=1}^{∞} Δ_{2}a_{i,j} cos ix,  (22)

δf_{j}(x) = \frac{δ_{2}a_{0,j}}{2} + ∑_{i=1}^{∞} δ_{2}a_{i,j} cos ix,  (23)

while (9) and (9’) imply

∑_{i=0}^{∞} (|Δ_{1}Δ_{2}a_{i,j}| + |δ_{1}Δ_{2}a_{i,j}| + |Δ_{1}δ_{2}a_{i,j}| + |δ_{1}δ_{2}a_{i,j}|) < ∞ (24)

for each j ≥ 0.

Moreover, for each j ≥ 0 (8) yields

lim_{i→∞} Δ_{2}a_{i,j} = 0, lim_{i→∞} δ_{2}a_{i,j} = 0.  (25)

In view of the latter two relations Theorem A_{1} being applied to (22) and (23) results

∫_{T} |Δf_{j}(x)|dx = O( ∑_{i=0}^{∞} (|Δ_{1}Δ_{2}a_{i,j}| + |δ_{1}Δ_{2}a_{i,j}|)),  (26)

∫_{T} |δf_{j}(x)|dx = O( ∑_{i=0}^{∞} (|Δ_{1}δ_{2}a_{i,j}| + |δ_{1}δ_{2}a_{i,j}|)),  (27)

∫_{T} |Δ_{t}^{(p,0)}(Δf_{j}(x))|dx = O( ∑_{i=0}^{∞} α_{i,m}(p)(|Δ_{1}Δ_{2}a_{i,j}| +

+ |δ_{1}Δ_{2}a_{i,j}|)), |t| ≤ \frac{1}{m},  (28)

∫_{T} |Δ_{t}^{(p,0)}(δf_{j}(x))|dx = O( ∑_{i=0}^{∞} α_{i,m}(p)(|Δ_{1}δ_{2}a_{i,j}| +

+ |δ_{1}δ_{2}a_{i,j}|)), |t| ≤ \frac{1}{m}.  (29)

By summing (26) and (27) we obtain

∫_{T} ∑_{j=0}^{∞} (|Δf_{j}(x) + |δf_{j}(x)|)dx = O( ∑_{i=0}^{∞} ∑_{j=0}^{∞} (|Δ_{1}Δ_{2}a_{i,j}| +

+|δ_{1}Δ_{2}a_{i,j}| + |Δ_{1}δ_{2}a_{i,j}| + |δ_{1}δ_{2}a_{i,j}|)).

Since the right-hand side of this inequality is finite by virtue of (9) and (9′), we have

∑_{j=0}^{∞} (|Δf_{j}(x)| + |δf_{j}(x)|) < ∞

for almost all x ∈ T. Thus for each fixed |t| ≤ \frac{1}{m} we obtain

∑_{j=0}^{∞} (|Δ(Δ_{t}^{(p,0)}f_{j}(x))| + |δ(Δ_{t}^{(p,0)}f_{j}(x))|) < ∞

for almost all x.

Using the latter two relations and (20), by virtue of Theorem A_{1} we have for series (19), (21)

∫_{T} |f(x, y)|dy = O( ∑_{j=0}^{∞} (|Δf_{j}(x)| + |δf_{j}(x)|)),

sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{s}^{(0,q)}f(x, y)|dy = O( ∑_{j=0}^{∞} α_{j,n}(q)(|Δf_{j}(x)| + |δf_{j}|)),

sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{s}^{(0,q)}(Δ_{t}^{(p,0)}f(x, y))|dy = sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{t,s}^{(p,q)}f(x, y))|dx dy =

= O( ∑_{j=0}^{∞} α_{j,n}(q)(|Δ(Δ_{t}^{(p,0)}f_{j}(x))| + |δ(Δ_{t}^{(p,0)}f_{j}(x))|)).

Integrating these inequalities with respect to x and using (26)–(29), we obtain the validity of (10), (11), and (13).

If instead of series (18) we consider the series

\frac{a_{i,0}}{2} + ∑_{j=1}^{∞} a_{i,j} cos jy,

then by applying arguments similar to those above we shall obtain the validity of (12), (14).

Theorem 2. Let the conditions of Theorem 1 be fulfilled and

∑_{i=1}^{∞} ∑_{j=1}^{∞} (\frac{|a_{i,j}|}{ij} + \frac{|Δ_{1}a_{i,j}| + |δ_{1}a_{ij}|}{j} + \frac{|Δ_{2}a_{i,j}| + |δ_{2}a_{i,j}|}{i}) < ∞. (30)

314 L. GOGOLADZE

Then the series

∑_{i=1}^{∞} ∑_{j=1}^{∞} a_{i,j} sin ix sin jy  (31)

converges for all x ∈ T^{2} and for its sum g(x, y) the following estimates are valid:

∫_{\frac{π}{2μ+1}}^{π} ∫_{\frac{π}{2ν+1}}^{π} |g(x, y)|dx dy = ∑_{i=1}^{μ} ∑_{j=1}^{ν} \frac{|a_{i,j}|}{ij} + O( ∑_{i=1}^{∞} ∑_{j=1}^{ν} \frac{|Δ_{1}a_{i,j}| + |δ_{1}a_{i,j}|}{j}) +

+ O( ∑_{i=1}^{μ} ∑_{j=1}^{∞} \frac{|Δ_{2}a_{i,j}| + |δ_{2}a_{i,j}|}{i}) + O( ∑_{i=1}^{∞} ∑_{j=1}^{∞} A_{i,j}),  (32)

∫_{\frac{π}{2μ+1}}^{π} ( sup_{|s|≤ \frac{1}{n}} ∫_{T} Δ_{s}^{(0,q)}g(x, y)|dy)dx = \frac{2^{q}}{π} ∑_{i=1}^{μ} ∑_{j=n}^{∞} \frac{|a_{i,j}|}{ij} +

+ O( ∑_{i=1}^{∞} ∑_{j=n}^{∞} \frac{|Δ_{1}a_{i,j}| + |δ_{1}a_{i,j}|}{j}) +

+ O( ∑_{i=1}^{μ} ∑_{j=1}^{∞} α_{j,n}(q)\frac{|Δ_{2}a_{i,j}| + |δ_{2}a_{i,j}|}{i}) + O(B_{n}) = O(E_{n,μ}),  (33)

∫_{\frac{1}{2ν+1}}^{π} ( sup_{|t|≤ \frac{1}{m}} ∫_{T} |Δ_{t}^{(p,0)}g(x, y)|dx)dy = \frac{2^{p}}{π} ∑_{i=m}^{∞} ∑_{j=1}^{ν} \frac{|a_{i,j}|}{ij} +

+ O( ∑_{i=m}^{∞} ∑_{j=1}^{∞} \frac{|Δ_{2}a_{i,j}| + |δ_{2}a_{i,j}|}{i}) +

+ O( ∑_{i=1}^{∞} ∑_{j=1}^{ν} α_{i,m}(p)\frac{|Δ_{1}a_{i,j}| + |δ_{1}a_{i,j}|}{j}) + O(C_{m}) = O(F_{m,ν}),  (34)

sup_{|t|≤ \frac{1}{m}} ∫_{T} ( sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{t,s}^{(p,q)}g(x, y)|dy)dx = \frac{2^{p+q}}{π^{2}} ∑_{i=m}^{∞} ∑_{j=n}^{∞} \frac{|a_{i,j}|}{ij} +

+ O( ∑_{i=1}^{∞} ∑_{j=n}^{∞} α_{i,m}(p)\frac{|Δ_{1}a_{i,j}| + |δ_{1}a_{i,j}|}{j}) +

+ O( ∑_{i=m}^{∞} ∑_{j=1}^{∞} α_{j,n}(q)\frac{|Δ_{2}a_{i,j}|+|δ_{2}a_{i,j}|}{i})+O(D_{m,n})=O(G_{m,n}),  (35)

sup_{|t|≤ \frac{1}{n}} ∫_{T} ( sup_{|t|≤ \frac{1}{m}} ∫_{T} |Δ_{t,s}^{(p,q)}g(x, y)|dx)dy = O(C_{m,n}).  (36)

Note that the coefficient in the O-term depends only on q in (33), only on p in (34) and only on p and q in (35) and (36).

Proof. The convergence of the series

∑_{i=1}^{∞} a_{i,j} sin ix

and series (31) is proved in the same manner as that of series (18) and (7). Denote their sums by g_{j}(x) and g(x, y), respectively. The proof of the equalities

g_{j}(x) = ∑_{i=1}^{∞} a_{i,j} sin ix,  (18′)

g(x, y) = ∑_{j=1}^{∞} g_{j}(x) sin jy  (19′)

repeats that of equalities (18) and (19).

The validity of the equalities

lim_{j→∞} g_{j}(x) = 0,  (20′)
Δ_{t}^{(p,0)}g(x, y) = ∑_{j=1}^{∞} Δ_{t}^{(p,0)}g_{j}(x) sin jy, |t| ≤ \frac{1}{m},  (21′)

Δg_{j}(x) = ∑_{i=1}^{∞} Δ_{2}a_{i,j} sin ix,  (22′)

δg_{j}(x) = ∑_{i=1}^{∞} δ_{2}a_{i,j} sin ix,  (23′)

is obtained by using the same arguments as for equalities (20)–(23). Inequality (30) implies that for each j ≥ 1

∑_{i=1}^{∞} \frac{|Δ_{2}a_{i,j}|}{i} < ∞, ∑_{i=1}^{∞} \frac{|δ_{2}a_{i,j}|}{i} < ∞,  (37)

∑_{i=1}^{∞} \frac{|a_{i,j}|}{i} < ∞, ∑_{i=1}^{∞} |Δ_{1}a_{i,j}| < ∞, ∑_{i=1}^{∞} |δ_{1}a_{i,j}| < ∞.  (38)

Using (8), (24), (25), (37), and (38), by virtue of Theorem A_{2} we obtain for series (18′), (22′), (23′)

∫_{\frac{π}{2μ+1}}^{π} |g_{j}(x)|dx = ∑_{i=1}^{μ} \frac{|a_{i,j}|}{i} + O( ∑_{i=1}^{∞} (|Δ_{1}a_{i,j}| + |δ_{1}a_{i,j}|)),  (39)

∫_{\frac{π}{2μ+1}}^{π} |Δg_{j}(x)|dx=∑_{i=1}^{μ} \frac{|Δ_{2}a_{i,j}|}{i} +O(∑_{i=1}^{∞} (|Δ_{1}Δ_{2}a_{i,j}|+|δ_{1}Δ_{2}a_{i,j}|)),  (40)

∫_{\frac{π}{2μ+1}}^{π} |δg_{j}(x)|dx=∑_{i=1}^{μ} \frac{|δ_{2}a_{i,j}|}{i} +O( ∑_{i=1}^{∞} (|Δ_{1}δ_{2}a_{i,j}|+|δ_{1}δ_{2}a_{i,j}|)),  (41)

∫_{T} |Δ_{t}^{(p,0)}g_{j}(x)|dx = \frac{2^{p}}{π} ∑_{i=m}^{∞} \frac{|a_{i,j}|}{i} + O( ∑_{i=1}^{∞} α_{i,m}(p)(|Δ_{1}a_{i,j}| +

+ |δ_{1}a_{i,j}|)), |t| ≤ \frac{1}{m},  (42)

∫_{T} |Δ_{t}^{(p,0)}(Δg_{j}(x))|dx = \frac{2^{p}}{π} ∑_{i=m}^{∞} \frac{|Δ_{2}a_{i,j}|}{i} + O( ∑_{i=1}^{∞} α_{i,m}(p)(|Δ_{1}Δ_{2}a_{i,j}| +

+ |δ_{1}Δ_{2}a_{i,j}|)), |t| ≤ \frac{1}{m},  (43)

∫_{T} |Δ_{t}^{(p,0)}(δg_{j}(x))|dx = \frac{2^{p}}{π} ∑_{i=m}^{∞} \frac{|δ_{2}a_{i,j}|}{i} + O( ∑_{i=1}^{∞} α_{i,m}(p)(|Δ_{1}δ_{2}a_{i,j}| +

+ |δ_{1}δ_{2}a_{i,j}|)), |t| ≤ \frac{1}{m}.  (44)

After multiplying (39) by 1/j, passing to the limit in (39)–(41) as μ → ∞, and then summing them with respect to the index j, we obtain

∫_{T} ( ∑_{j=1}^{∞} (\frac{|g_{j}|}{j} + |Δg_{j}(x)| + |δg_{j}(x)|))dx = ∑_{i=1}^{∞} ∑_{j=1}^{∞} \frac{|a_{i,j}|}{ij} +

+ ∑_{i=1}^{∞} ∑_{j=1}^{∞} \frac{|Δ_{2}a_{i,j}| + |δ_{2}a_{i,j}|}{i} + O( ∑_{i=1}^{∞} ∑_{j=1}^{∞} \frac{|Δ_{1}a_{i,j}| + |δ_{1}a_{i,j}|}{j}) +

+ O( ∑_{i=1}^{∞} ∑_{j=1}^{∞} (|Δ_{1}Δ_{2}a_{i,j}| + |δ_{1}δ_{2}a_{i,j}| + |Δ_{1}δ_{2}a_{i,j}| + |δ_{1}δ_{2}a_{i,j}|)).

Since by virtue of (9), (9′), and (30) the right-hand side of this inequality is finite, we have

∑_{j=1}^{∞} (\frac{|g_{j}|}{j} + |Δg_{j}(x)| + |δg_{j}(x)|) < ∞

for almost all x ∈ T. Hence for each fixed |t| ≤ \frac{1}{m} we have

∑_{j=1}^{∞} (\frac{|Δ_{t}^{(p,0)}g_{j}|}{j} + |Δ(Δ_{t}^{(p,0)}g_{j}(x))| + |δ(δ_{t}^{(p,0)}g_{j}(x))|) < ∞

for almost all x.

By virtue of Theorem A_{2} the latter two relations and (20′) imply for series (19′) and (21′) that

∫_{\frac{π}{2μ+1}}^{π} |g(x, y)|dy = ∑_{j=1}^{ν} \frac{|g_{j}|}{j} + O( ∑_{j=1}^{∞} (|Δg_{j}(x)| + |δg_{j}|)),

sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{s}^{(0,q)}g(x, y)|dy = \frac{2^{q}}{π} ∑_{j=n}^{∞} \frac{|g_{j}|}{j} + O( ∑_{j=1}^{∞} α_{j,n}(q)(|Δg_{j}| + |δg_{j}|)),

sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{t,s}^{(p,q)}g(x, y)|dy = sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{s}^{(0,q)}(Δ_{t}^{(p,0)}g(x, y))|dy =

= \frac{2^{q}}{π} ∑_{j=n}^{∞} \frac{|Δ_{t}^{(p,0)}g_{j}(x)|}{j} + O( ∑_{j=1}^{∞} α_{j,n}(q)(|Δ(Δ_{t}^{(p,0)}g_{j}(x))| +

+ |δ(Δ_{t}^{(p,0)}g_{j}(x)|)).

After integrating these inequalities with respect to x and using (39)–(44), we obtain the validity of (32), (33), (35).

If instead of series (18′) we consider the series

∑_{i=1}^{∞} ∑_{j=1}^{∞} a_{i,j} sin jy,

then by a similar reasoning we shall obtain the validity of (34) and (36).

Corollary 1. Let the conditions of Theorem 1 be fulfilled. Then

ω^{(p,0)}(\frac{1}{m}, f) = O(C_{m}), ω^{(0,q)}(\frac{1}{n}, f) = O(B_{n}),

ω^{(p,q)}(\frac{1}{m}, \frac{1}{n}, f) = O(D_{m,n}).

Corollary 2. Let the conditions of Theorem 2 be fulfilled. Then

ω^{(p,0)}(\frac{1}{m}, g) = O(F_{m}), ω^{(0,q)}(\frac{1}{n}, f) = O(E_{n}),

ω^{(p,q)}(\frac{1}{m}, \frac{1}{n}, f) = O(G_{m,n}).

where

E_{n} = lim_{μ→∞} E_{n,μ}, F_{m} = lim_{ν→∞} F_{m,ν}.

It is easy to verify that for each function φ ∈ L(T^{2})

ω^{(p,0)}(\frac{1}{m}, φ) = O( ∫_{T} ( sup_{|t|≤ \frac{1}{m}} ∫_{T} |Δ_{t}^{(p,0)}φ(x, y)|dx)dy),  (45)

ω^{(0,q)}(\frac{1}{n}, φ) = O( ∫_{T} ( sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{s}^{(0,q)}φ(x, y)|dy)dx),  (46)

ω^{(p,q)}(\frac{1}{m}, \frac{1}{n}, φ) = O( sup_{|t|≤ \frac{1}{m}} ∫_{T} ( sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{t,s}^{(p,q)}φ(x, y)|dy)dx). (47)

These equalities and (11)–(13) imply the validity of Corollary 1. Corollary 2 is proved by a similar reasoning, but first we must pass to the limit in (33) as μ → ∞ and in (34) as ν → ∞.

Corollary 3. Let equality (8) be fulfilled and

∑_{i=0}^{∞} ∑_{j=0}^{∞}(i + 1)(j + 1)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}| < ∞.

Then

∫_{T^{2}} |f(x, y)|dx dy = O( ∑_{i=0}^{∞} ∑_{j=0}^{∞}(i + 1)(j + 1)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|),  (48)

∫_{T} ( sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{s}^{(0,q)}f(x, y)|dy)dx =

= O( ∑_{i=0}^{∞} ∑_{j=0}^{∞}(i + 1)(j + 1)α_{j,n}(q)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|) = O(H_{n}),  (49)

∫_{T} ( sup_{|s|≤ \frac{1}{m}} ∫_{T} |Δ_{t}^{(p,0)}f(x, y)|dx)dy =

= O( ∑_{i=0}^{∞} ∑_{j=0}^{∞}(i + 1)(j + 1)α_{i,m}(p)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|) = O(K_{n}),  (50)

sup_{|t|≤ \frac{1}{m}} ∫_{T} ( sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{t,s}^{(p,q)}f(x, y)|dy)dx =

= O( ∑_{i=0}^{∞} ∑_{j=0}^{∞}(i + 1)(j + 1)α_{i,m}(p)α_{j,n}(q)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|) = O(L_{m,n}), (51)

sup_{|s|≤ \frac{1}{n}} ∫_{T} ( sup_{|t|≤ \frac{1}{m}} ∫_{T} |Δ_{s,t}^{(p,q)}f(x, y)|dx)dy = O(L_{m,n}).  (52)

As one can easily verify, to prove Corollary 3 it suffices to estimate the right-hand sides of relations (10)–(14) by the right-hand sides of (48)–(52), respectively.

In [1] it is shown that

∑_{i=0}^{∞} α_{i,m}(p)|Δa_{i}| = O( ∑_{i=0}^{∞}(i + 1)α_{i,m}(p)|Δ^{2}a_{i}|),  (53)

∑_{i=0}^{∞} α_{i,m}(p)|δa_{i}| = O( ∑_{i=0}^{∞}(i + 1)α_{i,m}(p)|Δ^{2}a_{i}|)  (54)

are valid. In a similar manner one can prove

∑_{i=0}^{∞} |Δa_{i}| = O( ∑_{i=0}^{∞}(i + 1)|Δ^{2}a_{i}|),  (55)

∑_{i=0}^{∞} |δa_{i}| = O( ∑_{i=0}^{∞}(i + 1)|Δ^{2}a_{i}|).  (56)

Applying (53) and (55) successively, we obtain

∑_{i=0}^{∞} ∑_{j=0}^{∞} α_{j,n}(q)|Δ_{1}Δ_{2}a_{i,j}| = O( ∑_{i=0}^{∞} ∑_{j=0}^{∞}(j + 1)α_{j,n}(q)|Δ_{1}Δ_{2}^{2}a_{i,j}|) =

= O( ∑_{i=0}^{∞} ∑_{j=0}^{∞}(i + 1)(j + 1)α_{j,n}(q)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|).

By a similar technique we derive the estimates

∑_{i=0}^{∞} ∑_{j=0}^{∞} α_{j,n}(q)|δ_{1}Δ_{2}a_{i,j}| = O( ∑_{i=0}^{∞} ∑_{j=0}^{∞}(i + 1)(j + 1)α_{j,n}(q)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|),

∑_{i=0}^{∞} ∑_{j=0}^{∞} α_{j,n}(q)|Δ_{1}δ_{2}a_{i,j}| = O( ∑_{i=0}^{∞} ∑_{j=0}^{∞}(i + 1)(j + 1)α_{j,n}(q)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|),

∑_{i=0}^{∞} ∑_{j=0}^{∞} α_{j,n}(q)|δ_{1}δ_{2}a_{i,j}| = O( ∑_{i=0}^{∞} ∑_{j=0}^{∞}(i + 1)(j + 1)α_{j,n}(q)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|).

The latter four relations enable us to estimate (see (9)) the right-hand side of (11) by the right-hand side of (49). In the same manner one can prove the validity of relations (48), (50)–(52).

Let

β_{i,m}(p) = \begin{cases}(im^{−1})^{p}, & 1 ≤ i ≤ m, \\ 1 + ln \frac{i}{m}, & i > m, \\ 0, & i < 1. \end{cases}

Corollary 4. Let (8) be fulfilled and

∑_{i=1}^{∞} ∑_{j=1}^{∞} ij ln(i + 1) ln(j + 1)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}| < ∞.

Then

∫_{T^{2}} |g(x, y)|dx dy = O( ∑_{i=1}^{∞} ∑_{j=1}^{∞} ij ln(i + 1) ln(j + 1)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|),

∫_{T} ( sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{s}^{(0,q)}g(x, y)|dy)dx =

= O( ∑_{i=1}^{∞} ∑_{j=1}^{∞} ij ln(i + 1)β_{j,n}(q)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|) = O(M_{n}),

∫_{T} ( sup_{|t|≤ \frac{1}{m}} ∫_{T} |Δ_{t}^{(p,0)}g(x, y)|dx)dy =

= O( ∑_{i=1}^{∞} ∑_{j=1}^{∞} ij ln(i + 1)β_{i,m}(p)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|) = O(N_{m}),

sup_{|t|≤ \frac{1}{m}} ∫_{T} ( sup_{|s|≤ \frac{1}{n}} ∫_{T} |Δ_{t,s}^{(p,q)}g(x, y)|dy)dx =

= O( ∑_{i=1}^{∞} ∑_{j=1}^{∞} ijβ_{i,m}(p)β_{j,n}(q)|Δ_{1}^{2}Δ_{2}^{2}a_{i,j}|) = O(R_{m,n}),

sup_{|s|≤ \frac{1}{n}} ∫_{T} ( sup_{|t|≤ \frac{1}{m}} ∫_{T} |Δ_{t,s}^{(p,q)}g(x, y)|dx)dy = O(R_{m,n}).

We have

∑_{i=m}^{∞} \frac{|a_{i}|}{i} = ∑_{i=m}^{∞} \frac{1}{i}| ∑_{μ=i}^{∞} ∑_{ν=μ}^{∞} Δ^{2}a_{ν} | ≤ ∑_{i=m}^{∞} \frac{1}{i} ∑_{ν=i}^{∞} ∑_{μ=i}^{ν} |Δ^{2}a_{ν}| ≤

≤ ∑_{i=m}^{∞} ∑_{ν=i}^{∞} \frac{ν}{i} |Δ^{2}a_{ν}| = ∑_{ν=m}^{∞} ∑_{i=m}^{ν} \frac{ν}{i} |Δ^{2}a_{ν}| =

= O( ∑_{ν=m}^{∞} ν ln \frac{ν + 1}{m}|Δ^{2}a_{ν}|).  (57)

The proof of Corollary 4 is obtained from Theorem 2 and (53)–(57) by the reasoning similar to the proof of Corollary 3 by means of Theorem 1 and (53)–(56), but first one must pass to the limit in (33) as μ → ∞ and in (34) as ν → ∞.

Corollary 5. Let the conditions of Corollary 3 be fulfilled. Then

ω^{(p,0)}(\frac{1}{m}, f) = O(K_{m}), ω^{(0,q)}(\frac{1}{n}, f) = O(H_{n}),

ω^{(p,q)}(\frac{1}{m}, \frac{1}{n}) = O(L_{m,n}).

Corollary 6. Let the conditions of Corollary 4 be fulfilled. Then

ω^{(p,0)}(\frac{1}{m}, g) = O(M_{m}), ω^{(0,q)}(\frac{1}{n}, g) = O(N_{n}),

ω^{(p,q)}(\frac{1}{m}, \frac{1}{n}, g) = O(R_{m,n}).

Corollaries 5 and 6 follow from Corollaries 3 and 4, respectively, if we use (45), (46), (47). Concerning these corollaries, see [4], pp. 6–86.

One can also consider the series

∑_{i=0}^{∞} ∑_{j=1}^{∞} λ_{i,j} cos ix sin jy, ∑_{i=1}^{∞} ∑_{j=0}^{∞} λ_{i,j} sin ix cos jy.

The interested reader will readily know how to formulate and simultaneously prove the corresponding theorems for these series by the repeated application of Theorems A_{1} and A_{2}.

Acknowledgment

This research was supported by the Grant of the Georgian Department of Science and Technology (Contract No. 13–1).

References

1. S. A. Telyakovskiĭ, The integrability of trigonometric series. Estimation of an integral modulus of continuity. (Russian) Mat. Sb. 91(133) (1973), No. 4(8), 537–553.

2. S. A. Telyakovskiĭ, Conditions for the integrability of trigonometric series and their application to the study of linear methods of summation of Fourier series. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 28(1964), No. 6, 1209–1236.

3. V. G. Chelidze, Some methods of summation of double series and double integrals. (Russian) Tbilisi University Press, Tbilisi, 1977.

4. T. Sh. Tevzadze, Special multiple trigonometric series. (Russian) Tbilisi University Press, Tbilisi, 1998.

(Received 22.09.1997; revised 31.08.1998)

Author’s address: Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia