GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 4, 1996, 379-400

ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS OF AN EVEN FUNCTION OF SEVERAL

VARIABLES WITH QUASICONVEX FOURIER COEFFICIENTS

T. TEVZADZE

Abstract. The estimate of the modulus of smoothness of an even function of several variables with quasiconvex Fourier coefficients obtained in this paper extends one result of S. A. Telyakovski.

1. Let (a_{i,j})_{i,j≥0} be a double numerical sequence. Denote by Δ^{2}a_{i,j} the expression Δ_{12}(Δ_{12}a_{i,j}), where

Δ_{12}a_{i,j} = Δ_{1}(Δ_{2}a_{i,j}) = Δ_{1}(a_{i,j} − a_{i,j+1}) = a_{i,j} − a_{i+1,j} − a_{i,j+1} + a_{i+1,j+1}.

Definition 1. The double sequence (a_{i,j})_{i,j≥0} will be called quasiconvex if the series

∑_{i=0}^{∞} ∑_{j=0}^{∞} [(i + 1)(j + 1)|Δ^{2}a_{i,j}| + (i + 1)|Δ_{1}(Δ_{12}a_{ij})| + (j + 1)|Δ_{2}(Δ_{12}a_{ij})|]

converges.

As can be easily shown, if the sequence (a_{i,j})_{i,j≥0} is quasiconvex and

lim_{i+j→∞} a_{i,j} = 0,

then the series

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

where λ_{0,0} = \frac{1}{4}, λ_{0,j} = λ_{i,0} = \frac{1}{2}, i, j = 1, 2, …, λ_{i,j} = 1 for i, j > 0, converges on (0, 2π)^{2} to some function f ∈ L(T^{2}) and is its Fourier series with T^{2} = [0, 2π]^{2}.

379 The expression

1991 Mathematics Subject Classification. 42B05.

Key words and phrases. Integral modulus of smoothness, quasiconvex Fourier coefficients.

1072-947X/96/0700-0379$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

ω_{m,n}(δ, ρ; f)_{1} =

= sup_{\begin{matrix}|h|≤δ \\ |η|≤ρ \end{matrix}} ∫_{T^{2}} | ∑_{μ=0}^{m} ∑_{ν=0}^{n}(−1)^{μ+ν} \binom{m}{μ}\binom{n}{ν}f[x + (m − 2μ)h, y + (n − 2ν)η]|dx dy

is usually called an integral modulus of smoothness of order (m, n) of a function f ∈ L(T^{2}). Here and in what follows it is assumed that δ ∈ [0, π], ρ ∈ [0, π].

Aljančič and Tomič [1], [2] considered an estimate of the integral modulus of continuity in terms of Fourier coefficients of the function f for some classes of sequences in the one-dimensional case. In 1963 M. and S. Izumi [3] proved that if the Fourier coefficients a_{n} → 0, n → ∞, form a quasiconvex sequence, then the estimate

ω(δ; f)_{1} ≤ cδ ∑_{i≤ \frac{1}{δ}} i^{2}|Δ_{1}^{2}a_{i}| + ∑_{i> \frac{1}{δ}} i|Δ_{1}^{2}a_{i}|^{1}

holds for the integral modulus of continuity ω(δ; f)_{1}.

This result was later generalized by Telyakovski [4] who proved a theorem giving an estimate of an integral modulus of smoothness of order m, m ∈ ℕ.

In this paper an estimate is obtained for the integral modulus of smoothness for a function of several variables which is even with respect to each variable.

2. For a further discussion we need

Lemma 1. Let 2mh ≤ t ≤ π. Then

|T_{p}^{m}| = | ∑_{i=0}^{m}(−1)^{i}\binom{m}{i}K_{p}[t + (m − 2i)h]| ≤ c(m)h^{m}p^{m−1}t^{−2},

|T_{p}^{m}| ≤ c(m)h^{m}p^{m+1},

where K_{p}(t) is the Fejer kernel, p ∈ ℕ.

Proof. It is easy to show that

T_{p}^{m} = ∑_{i=0}^{m}(−1)^{i}\binom{m}{i}K_{p}[t + (m − 2i)h] =

= ∑_{i=0}^{m−1}(−1)^{i}\binom{m − 1}{i}\{K_{p}[t + (m−2i)h]−K_{p}[t+(m−2(i+1))h]\}=

1Here and in what follows, c, c(m), c(m, n), … denote, generally speaking, various positive constants depending only on the parameters indicated in the brackets.

= −2h ∑_{i=0}^{m−1}(−1)^{i}\binom{m − 1}{i}K′_{p}[t + (m − 2(i + 1 − Q_{1}))h] = ⋯ =

= (−1)^{m}h^{m}K_{p}^{(m)}(t + c(m)h),

where c(m) = m − 2(m − Q_{1} − ⋯ − Q_{m}), 0 < Q_{i} < 1, i = 1, …, m. Since −m < c(m) < m, we have

K_{p}^{(m)}(t + c(m)h) = \frac{1}{p + 1} ∑_{i=0}^{p} D_{i}^{(m)}(t + c(m)h) =

= ±\frac{1}{p + 1} ∑_{i=0}^{p} ∑_{j=1}^{i} j^{m} \begin{cases}cos j(t + c(m)h), & 2|m, \\ sin j(t + c(m)h), & 2 ∤ m, \end{cases}

where D_{i}(t) is the Dirichlet kernel.

Let 2|m (the case 2 ∤ m is considered similarly). Then applying twice the Abel transformation and taking into account the estimates

K_{p}(t) ≤ \frac{c}{pt^{2}} , 0 < |t| ≤ π,

K_{p}(t) ≤ p, |t| ≤ π,

we find

K_{p}^{(m)}(t + c(m)h) = \frac{1}{p + 1} ∑_{i=0}^{p} ∑_{j=1}^{i} |j^{m} − 2(j + 1)^{m} +

+ (j + 2)^{m}|jK_{j}(t + c(m)h) +

+ \frac{1}{p + 1} ∑_{i=0}^{p} |(i − 1)^{m} − i^{m}|iK_{i}(t + c(m)h) +

+ \frac{1}{p + 1} ∑_{i=0}^{p} |i^{m} − (i + 1)^{m}|iK_{i}(t + c(m)h) +

+ \frac{1}{p + 1} · p^{m+1}K_{p}(t + c(m)h) ≤

≤ \begin{cases}c(m)p^{m−1}t^{−2}, & 2mh ≤ t ≤ π, \\ c(m)p^{m+1}, & −π ≤ t ≤ π. \end{cases}

Therefore

|T_{p}^{m}| ≤ c(m)h^{m}p^{m−1}t^{−2}, 2mh ≤ t ≤ π,
|T_{p}^{m}| ≤ c(m)h^{m}p^{m+1}, −π ≤ t ≤ π.

The two-dimensional analogue of Lemma 1 given below is proved similarly.

382 T. TEVZADZE

Lemma 2. Let 2mh ≤ x ≤ π, 2nη ≤ y ≤ π. Then

|T_{p,q}^{m,n}| = | ∑_{μ=0}^{m} ∑_{ν=0}^{n}(−1)^{μ+ν} \binom{m}{μ}\binom{n}{ν}K_{p}[x+(m−2μ)h]K_{q}[y+(n−2ν)η]|≤

≤ c(m, n) \frac{h^{m}η^{n}p^{m−1}q^{n−1}}{(xy)^{2}} ,

|T_{p,q}^{m,n}| ≤ c(m, n)h^{m}η^{n}p^{m+1}q^{n+1}, (x, y) ∈ T^{2}.

Lemma 3 ([5]). Let 2mh ≤ x ≤ π. Then

| ∑_{i=0}^{m}(−1)^{i}\binom{m}{i}D_{p}[x + (m − 2i)h]| ≤ c(m)h^{m}p^{m}x^{−1},

where D_{p} is the Dirichlet kernel, p ∈ ℕ.

Theorem 1. Let a double sequence (a_{i,j})_{i,j≥0} be quasiconvex and

lim_{i+j→∞} a_{i,j} = 0.

Then for the sum f of the series (∗) we have

ω_{m,n}(h, η; f)_{1} ≤ c(m, n)\{h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m+1}j^{n+1}|Δ^{2}a_{i,j}| +

+ h^{m} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} i^{m+1}j|Δ^{2}a_{i,j}| +

+ η^{n} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} ∑_{j=1}^{[ \frac{1}{2nη}]} ij^{n+1}|Δ^{2}a_{i,j}| +

+ ∑_{i=[ \frac{1}{2mh}]+1}^{∞} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} ij|Δ^{2}a_{i,j}|\} =

= ∑_{s=1}^{4} P_{s}(m, n, h, η).

Proof. We write

A(m, n, f) = 4 ∫_{0}^{π} ∫_{0}^{π} |Δ^{m,n}f| dx dy = 4{ ∫_{0}^{2mh} ∫_{0}^{2nη}+ ∫_{2mh}^{π} ∫_{0}^{2nη}+ ∫_{0}^{2mh}∫_{2nη}^{π} +

ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS 383

+ ∫_{2mh}^{π} ∫_{2nη}^{π} }|Δ^{m,n}f| dx dy ≡ ∑_{s=1}^{4} A_{s}(m, n, f),

where

Δ^{m,n}f = ∑_{μ=0}^{m} ∑_{ν=0}^{n}(−1)^{μ+ν} \binom{m}{μ}\binom{n}{ν}f[x + (m − 2μ)h, y + (n − 2ν)η] =

= ∑_{μ=0}^{m} ∑_{ν=0}^{n}(−1)^{μ+ν} \binom{m}{μ}\binom{n}{ν}\{ ∑_{i=1}^{[ \frac{1}{x}]} ∑_{j=1}^{[ \frac{1}{y}]} + ∑_{i=[ \frac{1}{x}]+1}^{∞} ∑_{j=1}^{[ \frac{1}{y}]} + ∑_{i=1}^{[ \frac{1}{x}]} ∑_{j=[ \frac{1}{y}]+1}^{∞} +

+ ∑_{i=[ \frac{1}{x}]+1}^{∞} ∑_{j=[ \frac{1}{y}]+1}^{∞} \}a_{i,j} cos i[x + (m − 2μ)h] cos[y + (n − 2ν)η] ≡

≡ ∑_{s=1}^{4} T_{s}(m, n).

It is easy to show that

T_{1} ≡ T_{1}(m, n) = 2^{m+n} ∑_{i=1}^{[ \frac{1}{x}]} ∑_{j=1}^{[ \frac{1}{y}]} a_{i,j} sin^{m} ih sin^{n} jη ×

× \begin{cases}(−1)^{\frac{m+n}{2}} cos ix cos jy, & 2|m, 2|n, \\ (−1)^{\frac{m+n−1}{2}} sin ix cos jy, & 2 ∤ m, 2|n, \\ (−1)^{\frac{m+n−1}{2}} cos ix sin jy, & 2|m, 2 ∤ n, \\ (−1)^{\frac{m+n−2}{2}} sin ix sin jy, & 2 ∤ |m, 2 ∤ n. \end{cases}

Further,

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{1}|dx dy ≤ c(m, n)h^{m}η^{n} ∫_{2mh}^{π} ∫_{2nη}^{π} ∑_{i=1}^{[ \frac{1}{x}]} ∑_{j=1}^{[ \frac{1}{y}]} i^{m}j^{n}|a_{i,j}|dx dy ≤

≤ c(m, n)h^{m}η^{n} ∫_{1}^{2mh} ∫_{1}^{2nη}(xy)^{−2} ∑_{i=1}^{[x]} ∑_{j=1}^{[y]} i^{m}j^{n}|a_{i,j}|dx dy ≤

≤ c(m, n)h^{m}η^{n} ∑_{r=1}^{[ \frac{1}{2mh}]} ∑_{s=1}^{[ \frac{1}{2nη}]}(rs)^{−2} ∑_{i=1}^{r} ∑_{j=1}^{s} i^{m}j^{n}|a_{i,j}| ≤

≤ c(m, n)h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m}j^{n}|a_{i,j}| ∑_{r=i}^{[ \frac{1}{2mh}]} ∑_{s=j}^{[ \frac{1}{2nη}]}(rs)^{−2} ≤

≤ c(m, n)h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m−1}j^{n−1}|a_{i,j}|.  (2.1)

Applying twice the Hardy transformation [6], p = [\frac{1}{2mh}], q = [\frac{1}{2nη}], which is a two-dimensional analog of the Abel transformation, we find

∑_{i=1}^{p} ∑_{j=1}^{q} i^{m−1}j^{n−1}|a_{i,j}| = ∑_{i=1}^{p} i^{m−1}{ ∑_{j=1}^{q−1} (|a_{i,j}| − |a_{i,j+1}|) ∑_{s=1}^{j} s^{n−1} +

+ ∑_{s=1}^{q} s^{n−1}|a_{i,q}|} ≤ ∑_{j=1}^{q−1} j^{n}{ ∑_{i=1}^{p} i^{m−1}|Δ_{2}a_{i,j}| + ∑_{i=1}^{p} i^{m−1}q^{n}|a_{i,q}|} =

= ∑_{j=1}^{q−1} j^{n}{ ∑_{i=1}^{p−1} (|Δ_{2}a_{i,j}| − |Δ_{2}a_{i+1,j}|) ∑_{r=1}^{i} r^{m−1} +

+ ∑_{i=1}^{p} i^{m−1}|Δ_{2}a_{p,j}|} + q^{n} ∑_{i=1}^{p−1} (|a_{i,q}| − |a_{i+1,q}|) ∑_{r=1}^{i} r^{m−1} +

+q^{n}|a_{p,q}| ∑_{i=1}^{p} i^{m−1} ≤ ∑_{i=1}^{p−1} ∑_{j=1}^{q−1} |Δ_{12}a_{i,j}|i^{m}j^{n} +

+p^{m} ∑_{j=1}^{q−1} j^{n}|Δ_{2}a_{p,j}| + q^{n} ∑_{i=1}^{p−1} i^{m}|Δ_{1}a_{i,q}| + p^{m}q^{n}|a_{p,q}| ≤

≤ ∑_{i=1}^{p−1} ∑_{j=1}^{q−1} i^{m+1}j^{n+1}|Δ^{2}a_{i,j}| + q^{n+1} ∑_{i=1}^{p−2} i^{m+1}|Δ_{12}a_{i,q} − Δ_{12}a_{i+1,q}| +

+p^{m+1} ∑_{j=1}^{q−2} j^{n+1}|Δ_{12}a_{p,j} − Δ_{12}a_{p,j+1}| + p^{m+1}q^{n+1}|Δ_{12}a_{p,q}| +

+q^{n} ∑_{i=1}^{p−2} i^{m+1}|Δ_{1}(a_{i,q} − a_{i+1,q})| + p^{m} ∑_{j=1}^{q−2} j^{n+1}|Δ_{2}(a_{p,j} − a_{p,j+1})| +

+q^{n}p^{m+1}|Δ_{1}a_{p,q}| + p^{m}q^{n+1}|Δ_{2}a_{p,q}| + p^{m}q^{n}|a_{p,q}| ≡
≡ ∑_{α=1}^{9} I_{α}(m, n, p, q).  (2.2)

For I_{2} we obtain

I_{2} = q^{n} ∑_{i=1}^{p−2} i^{m+1}q|Δ_{1}(Δ_{12}a_{i,q})| = q^{n} ∑_{i=1}^{p−2} i^{m+1}q| ∑_{j=q}^{∞} Δ^{2}a_{i,j}| ≤

ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS 385

≤ q^{n} ∑_{i=1}^{p−2} i^{m+1} ∑_{j=q}^{∞} j|Δ^{2}a_{i,j}|.  (2.3)

Similarly,

I_{3} ≤ p^{m} ∑_{i=p}^{∞} ∑_{j=1}^{q−2} ij^{n+1}|Δ^{2}a_{i,j}|.  (2.4)

Let us now estimate I_{5}. We have

I_{5} ≤ q^{n} ∑_{i=1}^{p−2} i^{m+1}|Δ_{1}^{2}a_{i,q}|,

where

Δ_{1}^{2}a_{i,q} = a_{i,q} − a_{i+1,q} + a_{i+2,q}.

It is easy to show that

Δ_{1}^{2}a_{i,q} = ∑_{s=q}^{∞} Δ_{2}(Δ_{1}^{2}a_{i,s}) = ∑_{j=q}^{∞} Δ_{2}( ∑_{s=j}^{∞} Δ_{2}(Δ_{2}a_{i,s})) = ∑_{j=q}^{∞} ∑_{s=j}^{∞} Δ^{2}a_{i,s}.

Therefore

|Δ_{1}^{2}a_{i,q}| ≤ ∑_{j=q}^{∞} ∑_{s=j}^{∞} |Δ^{2}a_{i,s}| ≤ ∑_{s=q}^{∞} s|Δ^{2}a_{i,s}|.  (2.5)

Thus

I_{5} ≤ q^{n} ∑_{i=1}^{p−2} ∑_{s=q}^{∞} i^{m+1}s|Δ^{2}a_{i,s}|.  (2.6)

Similarly,

I_{6} ≤ p^{m} ∑_{i=p}^{∞} ∑_{j=1}^{q−2} ij^{n+1}|Δ^{2}a_{i,j}|.  (2.7)

Next,

I_{7} = p^{m}q^{n} · p|Δ_{1}a_{p,q}|.

Since

p|Δ_{1}a_{p,q}| = p| ∑_{i=p}^{∞} Δ_{1}^{2}a_{i,q}| ≤ ∑_{i=p}^{∞} i|Δ_{1}^{2}a_{i,q}|,

by repeating the arguments used for I_{5} we obtain

I_{7} ≤ p^{m}q^{n} ∑_{i=p}^{∞} ∑_{j=q}^{∞} ij|Δ^{2}a_{i,j}|.  (2.8)

386 T. TEVZADZE

The same estimate holds for I_{8}. For I_{4} we have

I_{4} ≤ p^{m}q^{n} ∑_{i=p}^{∞} ∑_{j=q}^{∞} ij|Δ^{2}a_{i,j}|.  (2.9)

In the same way,

I_{9} ≤ p^{m}q^{n}| ∑_{i=p}^{∞} ∑_{j=q}^{∞} ∑_{r=i}^{∞} ∑_{s=j}^{∞} Δ^{2}a_{r,s}| ≤ p^{m}q^{n} ∑_{r=p}^{∞} ∑_{s=q}^{∞} rs|Δ^{2}a_{r,s}|. (2.10)

Using (2.2)–(2.10), we find

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{1}| dx dy ≤ ∑_{α=1}^{4} P_{α}(m, n, h, η).  (2.11)

If p = [\frac{1}{x}]+1, q = [\frac{1}{y}]+1, we again apply twice the Hardy transformation for T_{4} and obtain

T_{4} = ∑_{μ=0}^{m} ∑_{ν=0}^{n}(−1)^{μ+ν} \binom{m}{μ}\binom{n}{ν}\{a_{p,q}D_{p}[x+(m−2μ)h]D_{q}[y+(n−2ν)η]−

− D_{p}[x + (m − 2μ)h]Δ_{2}a_{p,q}(q + 1)K_{q}[y + (n − 2ν)η] − − D_{q}[y + (n − 2ν)η]Δ_{1}a_{p,q}(p + 1)K_{p}[x + (m − 2μ)h] +

+ D_{p}[x + (m − 2μ)h] ∑_{j=q}^{∞} Δ_{2}^{2}a_{p,j}(j + 1)K_{j}[y + (n − 2ν)η] +

+ D_{q}[y + (n − 2ν)η] ∑_{i=p}^{∞} Δ_{1}^{2}a_{i,q}(i + 1)K_{i}[x + (m − 2μ)h] +

+ Δ_{12}^{2}a_{p,q}(p + 1)(q + 1)K_{p}[x + (m − 2μ)h]K_{q}[y + (n − 2ν)η] −

− (p + 1)K_{p}[x+(m − 2μ)h] ∑_{j=q}^{∞} Δ_{2}(Δ_{12}a_{p,j})(j + 1)K_{j}[y+(n − 2ν)η] −

− (q + 1)K_{q}[y+(n − 2ν)η] ∑_{i=p}^{∞} Δ_{1}(Δ_{12}a_{i,q})(i + 1)K_{i}[x+(m − 2μ)h] +

+ ∑_{i=p}^{∞} ∑_{j=q}^{∞} Δ^{2}a_{i,j}(i + 1)(j + 1)K_{i}[x+(m − 2μ)h]K_{j}[y+(n − 2ν)η]}=

= ∑_{α=1}^{9} T_{4}^{(α)}(m, n, p, q).  (2.12)

By Lemma 2 (see (2.1)) we have

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(1)}| dx dy ≤ c(m, n)h^{m}η^{n} ∫_{1}^{\frac{1}{2mh}} ∫_{1}^{\frac{1}{2nη}}|a_{[x],[y]}|x^{m−1}y^{n−1} dx dy ≤

≤ c(m, n)h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m−1}j^{n−1}|a_{i,j}| dx dy ≤

≤ ∑_{s=1}^{4} P_{s}(m, n, h, η).  (2.13)

Applying Lemmas 3 and 1 for T_{4}^{(2)} we find

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(2)}| dx dy ≤ c(m, n)h^{m}η^{n} ∫_{1}^{\frac{1}{2mh}} ∫_{1}^{\frac{1}{2nη}}x^{m−1}y^{n}|Δ_{2}a_{[x]+1,[y]+1}| dx dy ≤

≤ c(m, n)h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m−1}j^{n}|Δ_{2}a_{i,j}|.

Using the Abel transformation leads to

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(2)}| dx dy ≤ c(m, n)\{h^{m}η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1} ∑_{i=1}^{[ \frac{1}{2mh}]} i^{m−1}|Δ_{2}^{2}a_{i,j}| +

+ h^{m}η^{−1} ∑_{i=1}^{[ \frac{1}{2mh}]} i^{m−1}|Δ_{2}a_{i,[ \frac{1}{2nη}]}|\}.

Again applying twice the Abel transformation we obtain

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(2)}| dx dy ≤ c(m, n)\{h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m+1}j^{n+1}|Δ^{2}a_{i,j}| +

+η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1}h^{−1}| ∑_{i=[ \frac{1}{2mh}]}^{∞} Δ_{1}(Δ_{1}(Δ_{2}^{2}a_{i,j}))| +

+η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1}| ∑_{r=[ \frac{1}{2mh}]}^{∞} ∑_{i=r}^{∞} Δ_{1}(Δ_{1}(Δ_{2}^{2}a_{i,j}))| +

388 T. TEVZADZE

+h^{m} ∑_{i=1}^{[ \frac{1}{2mh}]} i^{m+1}η^{−1}| ∑_{j=[ \frac{1}{2nη}]}^{∞} Δ^{2}a_{i,j}| + η^{−1}| ∑_{j=q}^{∞} Δ_{2}^{2}a_{[ \frac{1}{2mh}],j}|\} ≤

≤ c(m, n)\{h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m+1}j^{n+1}|Δ^{2}a_{i,j}| +

+η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} i|Δ^{2}a_{i,j}| + η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} i|Δ^{2}a_{i,j}| +

+h^{m} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} i^{m+1}j|Δ^{2}a_{i,j}| + ∑_{i=[ \frac{1}{2mh}]+1}^{∞} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} ij|Δ^{2}a_{i,j}|\} ≤

≤ ∑_{s=1}^{4} P_{s}(m, n, h, η).  (2.14)

One can estimate T_{4}^{(3)} quite similarly. Let us now consider T_{4}^{(4)}. We have

T_{4}^{(4)} = ∑_{μ=0}^{m}(−1)^{μ}\binom{m}{μ}D_{[ \frac{1}{x}]}[x + (m − 2μ)h] ∑_{ν=0}^{n}(−1)^{ν} \binom{n}{ν} ×

× ( ∑_{j=[ \frac{1}{y}]+1}^{[ \frac{1}{2nη}]} + ∑_{j=[ \frac{1}{2nη}]+1}^{∞} )(j + 1)Δ_{2}^{2}a_{[ \frac{1}{x}],j}K_{j}[y + (n − 2ν)η] =

= T_{4}^{(4)}(1) + T_{4}^{(4)}(2).  (2.15)

Since

|T_{4}^{(4)}(1)| ≤ c(m, n)h^{m}η^{n}x^{−m−1}y^{−2} ∑_{j=[ \frac{1}{y}]+1}^{[ \frac{1}{2nη}]} j^{n}|Δ_{2}^{2}a_{[ \frac{1}{x}],j}|,

we obtain

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(4)}(1)| dx dy ≤ c(m, n)h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} i^{m−1} ∑_{s=1}^{[ \frac{1}{2nη}]} ∑_{j=s+1}^{[ \frac{1}{2nη}]} j^{n}|Δ_{2}^{2}a_{i,j}| ≤

≤ c(m, n)h^{m}η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1} ∑_{i=1}^{[ \frac{1}{2mh}]} i^{m−1}|Δ_{2}^{2}a_{i,j}|.

ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS 389

Repeating the arguments used in estimating T_{4}^{(2)}, we find

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(4)}(1)| dx dy ≤ P_{1}(m, n, h, η) + P_{3}(m, n, h, η). (2.16)

Similarly,

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(4)}(2)| dx dy ≤ c(m, n)h^{m}η^{−1} ∑_{i=1}^{[ \frac{1}{2mh}]} i^{m−1} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} |Δ_{2}^{2}a_{i,j}| ≤
≤ ∑_{s=1}^{4} P_{s}(m, n, h, η).  (2.17)

By (2.16) and (2.17) we conclude that

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(4)}| dx dy ≤ ∑_{s=1}^{4} P_{s}(m, n, h, η).  (2.18)

Quite similarly we obtain

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(5)}| dx dy ≤ ∑_{s=1}^{4} P_{s}(m, n, h, η).  (2.19)

By Lemma 1 and analysis of the arguments used in estimating I_{2} we have

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(6)}| dx dy ≤ c(m, n)h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m}j^{n}|Δ_{12}a_{i,j}| ≤

≤ c(m, n)h^{m}η^{n}\{ ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m+1}j^{n+1}|Δ^{2}a_{i,j}| +

+h^{−m−1} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1}|Δ_{2}(Δ_{12}a_{[ \frac{1}{2mh}],j})| +

+η^{−n−1} ∑_{i=1}^{[ \frac{1}{2mh}]} i^{m+1}|Δ_{1}(Δ_{12}a_{i,[ \frac{1}{2nη}]}| + h^{−m−1}η^{−n−1}|Δ_{12}a_{[ \frac{1}{2mh}],[ \frac{1}{2nη}]}| ≤

≤ c(m, n)\{h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m+1}j^{n+1}|Δ^{2}a_{i,j}| +

+h^{m}η^{−1} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} i^{m+1}|Δ_{1}(Δ_{12}a_{i,[ \frac{1}{2nη}]})| +

+h^{−1}η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1}|Δ_{2}(Δ_{12}a_{[ \frac{1}{2mh}],j})| +

+(hη)^{−1}|Δ_{12}a_{[ \frac{1}{2mh}], \frac{1}{2nη}]}| ≤≤ ∑_{s=1}^{4} P_{s}(m, n, h, η).  (2.20)

For T_{4}^{(8)} we obtain

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(8)}| dx dy ≤c(m, n)h^{m}η^{n} ∫_{1}^{\frac{1}{2mh}} ∫_{1}^{\frac{1}{2nη}}y^{n} ∑_{i=[x]+1}^{[ \frac{1}{2mh}]}i^{m}|Δ_{1}(Δ_{12}a_{i,[y]})|dxdy+

+c(m, n)η^{n}∫_{1}^{\frac{1}{2nη}}y^{n} ∫_{mh}^{π+mh}x^{−2} ∑_{i=[ \frac{1}{2mh}]+1}^{∞}|Δ_{1}(Δ_{12}a_{i,[y]})|dxdy ≤

≤ c(m, n)\{h^{m}η^{n} ∑_{r=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n} ∑_{i=r+1}^{[ \frac{1}{2mh}]} i^{m}|Δ_{1}(Δ_{12}a_{i,j})| +

+ η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n} ∑_{r=1}^{[ \frac{1}{mh}]} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} |Δ_{1}(Δ_{12}a_{i,j})|\} ≤

≤ c(m, n)\{h^{m}η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n} ∑_{r=1}^{[ \frac{1}{2mh}]} i^{m+1}|Δ_{1}(Δ_{12}a_{i,j})| +

+ η^{n} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} ∑_{j=1}^{[ \frac{1}{2nη}]} ij^{n}|Δ_{1}(Δ_{12}a_{i,j})|\} =

≤ c(m, n)\{h^{m}η^{n} ∑_{r=1}^{[ \frac{1}{2mh}]} i^{m+1} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n}|Δ_{1}(Δ_{12}a_{i,j})| +

+ η^{n} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} i ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n}|Δ_{1}(Δ_{12}a_{i,j})|\}.

ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS 391

Using the Abel transformation leads to

∑_{j=1}^{[ \frac{1}{2nη}]} j^{n}|Δ_{1}(Δ_{12}a_{i,j})| ≤ ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1}|Δ^{2}a_{i,j}| + η^{−n−1}|Δ_{1}(Δ_{12}a_{i,[ \frac{1}{2nη}]})|.

Hence (see (2.8)) it is easy to show that

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(8)}| dx dy ≤ ∑_{α=1}^{4} P_{α}(m, n, h, η).  (2.21)

Similarly,

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(7)}| dx dy ≤ ∑_{α=1}^{4} P_{α}(m, n, h, η).  (2.22)

Further,

T_{4}^{(9)} = ∑_{μ=0}^{m} ∑_{ν=0}^{n}(−1)^{μ+ν} \binom{m}{μ}\binom{n}{ν}\{ ∑_{i=[ \frac{1}{x}]+1}^{[ \frac{1}{2mh}]} ∑_{j=[ \frac{1}{y}]+1}^{[ \frac{1}{2nη}]} + ∑_{i=[ \frac{1}{2mh}]+1}^{∞} ∑_{j=[ \frac{1}{y}]}^{[ \frac{1}{2nη}]} +

+ ∑_{i=[ \frac{1}{x}]+1}^{[ \frac{1}{2mh}]} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} \}(i + 1)(j + 1) ×

× K_{i}[x + (m − 2μ)h]K_{j}[y + (n − 2ν)η] = ∑_{s=1}^{4} T_{4}^{(9)}(s).  (2.23)

By Lemma 2 we have

|T_{4}^{(9)}(1)| ≤ c(m, n)h^{m}η^{n} ∑_{i=[ \frac{1}{x}]+1}^{[ \frac{1}{2mh}]} ∑_{j=[ \frac{1}{y}]+1}^{[ \frac{1}{2nη}]} i^{m}j^{n}(xy)^{−2}|Δ^{2}a_{i,j}|.

Therefore

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(9)}(1)|dxdy ≤ c(m, n)h^{m}η^{n} ∑_{r=1}^{[ \frac{1}{2mh}]} ∑_{s=1}^{[ \frac{1}{2nη}]} ∑_{i=r}^{[ \frac{1}{2mh}]} ∑_{j=s}^{[ \frac{1}{2nη}]} i^{m}j^{n}|Δ^{2}a_{i,j}|≤

≤c(m, n)h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]}i^{m+1}j^{n+1}|Δ^{2}a_{i,j}|.  (2.24)

392 T. TEVZADZE

The remaining terms of (2.22) are estimated by similar arguments. Thus we conclude that

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}^{(9)}| dx dy ≤ ∑_{α=1}^{4} P_{α}(m, n, h, η).  (2.25)

Taking into account (2.13), (2.14), (2.18)–(2.22), and (2.25) we obtain

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{4}| dx dy ≤ ∑_{α=1}^{4} P_{α}(m, n, h, η).  (2.26)

For T_{2} we have

|T_{2}| ≤ c(n)η^{n} ∑_{j=1}^{[ \frac{1}{y}]} j^{n}| ∑_{μ=0}^{m}(−1)^{μ}\binom{m}{μ} ∑_{i=[ \frac{1}{x}]+1}^{∞} a_{i,j} cos i[x + (m − 2μ)h]|.

Applying twice the Abel transformation, we find

∑_{μ=0}^{m}(−1)^{μ}\binom{m}{μ} ∑_{i=[ \frac{1}{x}]+1}^{∞} a_{i,j} cos i[x + (m − 2μ)h] =

= ∑_{μ=0}^{m}(−1)^{μ}\binom{m}{μ}\{ ∑_{i=[ \frac{1}{x}]+1}^{∞}(i + 1)Δ_{1}^{2}a_{i,j}K_{i}[x + (m − 2μ)h] −

−Δ_{1}a_{[ \frac{1}{x}],j}[\frac{1}{x}]K_{[ \frac{1}{x}]}[x + (m − 2μ)h] − a_{[ \frac{1}{x}],j}D_{[ \frac{1}{x}]}[x + (m − 2μ)h]\}.

By virtue of Lemmas 3 and 1 we obtain

|T_{2}| ≤ c(m, n)h^{m}η^{n}\{ ∑_{j=1}^{[ \frac{1}{y}]} j^{n}( ∑_{i=[ \frac{1}{x}]+1}^{[ \frac{1}{2mh}]} i^{m}|Δ_{1}^{2}a_{i,j}|x^{−2} +

+ x^{−m−2}|Δ_{1}a_{[ \frac{1}{x}],j}| + |a_{[ \frac{1}{x}],j}|x^{−m−1})\} +

+ c(n)η^{n} ∑_{j=1}^{[ \frac{1}{y}]} j^{n}( ∑_{i=[ \frac{1}{2mh}]}^{∞} i|Δ_{1}^{2}a_{i,j}| ×

× | ∑_{μ=0}^{m}(−1)^{μ}\binom{m}{μ}K_{i}[x + (m − 2μ)h]|.

ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS 393

Hence

σ_{m,n} ≡ ∫_{2mh}^{π} ∫_{2nη}^{π} |T_{2}| dx dy ≤

≤ c(m, n)h^{m}η^{n} ∫_{2mh}^{π} x^{−2} ∫_{2nη}^{π} ∑_{j=1}^{[ \frac{1}{y}]} j^{n} ∑_{i=[ \frac{1}{x}]+1}^{[ \frac{1}{2mh}]} i^{m}|Δ_{1}^{2}a_{i,j}| dx dy +

+ c(m, n)h^{m}η^{n} ∫_{2mh}^{π} x^{−m−2} ∫_{2nη}^{π} ∑_{j=1}^{[ \frac{1}{y}]} j^{n}|Δ_{1}a_{[ \frac{1}{x}],j}| dx dy +

+ c(m, n)h^{m}η^{n} ∫_{2mh}^{π} x^{−m−1} ∫_{2nη}^{π} ∑_{j=1}^{[ \frac{1}{y}]} j^{n}|a_{[ \frac{1}{x}],j}| dx dy +

+ c(n)η^{n} ∫_{2mh}^{π} ∫_{2nη}^{π} ∑_{j=1}^{[ \frac{1}{y}]} j^{n} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} i|Δ_{1}^{2}a_{i,j}| ×

× | ∑_{μ=0}^{m}(−1)^{μ}\binom{m}{μ}K_{i}[x + (m − 2μ)h]| dx dy ≤

≤ c(m, n)h^{m}η^{n}\{ ∫_{1}^{\frac{1}{2mh}} ∫_{1}^{\frac{1}{2nη}}y^{−2} ∑_{j=1}^{[y]} j^{n} ∑_{i=[x]+1}^{[ \frac{1}{2mh}]} i^{m}|Δ_{1}^{2}a_{i,j}| dx dy +

+ ∫_{1}^{\frac{1}{2mh}}x^{m} ∫_{1}^{\frac{1}{2nη}}y^{−2} ∑_{j=1}^{[y]} j^{n}|Δ_{1}a_{[x],j}| dx dy +

+ ∫_{1}^{\frac{1}{2mh}}x^{m−1} ∫_{1}^{\frac{1}{2nη}}y^{−2} ∑_{j=1}^{[y]} j^{n}|a_{[x],j}| dx dy\} +

+ c(m, n)η^{n} ∫_{1}^{\frac{1}{2mh}} ∫_{1}^{\frac{1}{2nη}}y^{−2} ∑_{j=1}^{[y]} j^{n} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} |Δ_{1}^{2}a_{i,j}| dx dy.

Next, as is easy to show,

σ_{m,n} ≤ c(m, n)\{h^{m}η^{n} ∑_{r=1}^{[ \frac{1}{2mh}]} ∑_{s=1}^{[ \frac{1}{2nη}]} s^{−2} ∑_{j=1}^{s} j^{n} ∑_{i=r+1}^{[ \frac{1}{2mh}]} i^{m}|Δ_{1}^{2}a_{i,j}| +

+ h^{m}η^{n} ∑_{r=1}^{[ \frac{1}{2mh}]} r^{m} ∑_{s=1}^{[ \frac{1}{2nη}]} s^{−2} ∑_{j=1}^{s} j^{n}|Δ_{1}a_{r,j}| +

+ h^{m}η^{n} ∑_{r=1}^{[ \frac{1}{2mh}]} r^{m−1} ∑_{s=1}^{[ \frac{1}{2nη}]} s^{−2} ∑_{j=1}^{s} j^{n}|a_{r,j}| +

+ η^{n}h^{−1} ∑_{s=1}^{[ \frac{1}{2nη}]} s^{−2} ∑_{j=1}^{s} j^{n} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} |Δ_{1}^{2}a_{i,j}|\}.

Therefore, after simple calculations, we find

σ_{m,n} ≤ c(m, n)h^{m}η^{n}\{ ∑_{i=1}^{[ \frac{1}{2mh}]} i^{m+1} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n−1}|Δ_{1}^{2}a_{i,j}| +

+ ∑_{r=1}^{[ \frac{1}{2mh}]} r^{m} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n−1}|Δ_{1}a_{r,j}| + ∑_{r=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} r^{m−1}j^{n−1}|a_{r,j}|\} +

+c_{m,n}η^{n} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} i ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n−1}|Δ_{1}^{2}a_{i,j}|.

The analysis of estimates (2.14) and (2.18) gives

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{2}| dx dy ≤ ∑_{α=1}^{4} P_{α}(m, n, h, η).  (2.27)

Similarly,

∫_{2mh}^{π} ∫_{2nη}^{π} |T_{3}| dx dy ≤ ∑_{α=1}^{4} P_{α}(m, n, h, η).  (2.28)

Therefore by virtue of (2.11), (2.26), (2.27) and (2.28) we obtain

A_{4}(m, n; f) ≤ ∑_{α=1}^{4} P_{α}(m, n, h, η).  (2.29)

For A_{1}(m, n; f) we have

A_{1}(m, n; f) = ∫_{0}^{2mh} ∫_{0}^{2nη}|Δ^{m,n}f| dx dy =

= ∫_{0}^{2mh} ∫_{0}^{2nη}|( ∑_{i=1}^{r} ∑_{j=1}^{s} + ∑_{i=r+1}^{∞} ∑_{j=1}^{s} + ∑_{i=1}^{r} ∑_{j=s+1}^{∞} +

+ ∑_{i=r+1}^{∞} ∑_{j=s+1}^{∞} )(i + 1)(j + 1)Δ^{2}a_{i,j} ×

× ∑_{μ=0}^{m} ∑_{ν=0}^{n}(−1)^{μ+ν} \binom{m}{μ}\binom{n}{ν}K_{i}[x + (m − 2μ)h] ×

×K_{j}[y+(n−2ν)η]|dx dy = ∑_{α=1}^{4}A_{1}^{(α)}(m, n, h, η),  (2.30)

where r = [\frac{1}{2mh}], s = [\frac{1}{2nη}]. By Lemma 2 we obtain

A_{1}^{(1)} ≤ c(m, n)h^{m}η^{n} ∫_{0}^{2mh} ∫_{0}^{2nη}∑_{i=1}^{r} ∑_{j=1}^{s} i^{m+2}j^{n+2}|Δ^{2}a_{i,j}|dx dy ≤

≤ c(m, n)h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m+1}j^{n+1}|Δ^{2}a_{i,j}|.  (2.31)

Further,

A_{1}^{(4)} = ∫_{0}^{2mh} ∫_{0}^{2nη}| ∑_{i=r+1}^{∞} ∑_{j=s+1}^{∞}(i + 1)(j + 1)Δ^{2}a_{i,j}|( ∑_{m>2μ} ∑_{n>2ν} +

+ ∑_{m≤2μ} ∑_{n>2ν} + ∑_{m>2μ} ∑_{n≤2ν} + ∑_{m≤2μ} ∑_{n≤2ν} )(−1)^{μ+ν} \binom{m}{μ}\binom{n}{ν} ×

× K_{i}[x + (m − 2μ)h]K_{j}[y + (n − 2ν)η]| dx dy = ∑_{α=1}^{4} A_{1}^{(4)}(α).

Obviously, it is sufficient to estimate A_{1}^{(4)}(4). For the term of this sum A_{1;μ,ν}^{(4)}(4), where μ ([\frac{m}{2}] ≤ μ ≤ m) and ν ([\frac{n}{2}] ≤ ν ≤ n) are fixed, a = 2μ−m, b = 2ν − n, we obtain, after passing to the variables t = x − ah, τ = y − bη,

A_{i;μ,ν}^{(4)}(4) ≤ c(m, n) ∑_{i=[ \frac{1}{2mh}]+1}^{∞} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} |Δ^{2}a_{i,j}| ∑_{r=1}^{3} ∑_{s=1}^{3} F_{r,s},

where

F_{r,s} = ∫_{u_{r}} ∫_{v_{s}} (i + 1)(j + 1)K_{i}(t)K_{j}(τ) dt dτ,

u_{1} = [ − ah, −\frac{1}{i}], u_{2} = [ − \frac{1}{i}, \frac{1}{i}], u_{3} = [\frac{1}{i}, (2m − a)h],

v_{1} = [ − bη, −\frac{1}{j}], v_{2} = [ − \frac{1}{j}, \frac{1}{j}], v_{3} = [\frac{1}{j}, (2n − b)η],

[−ah, −(2m − a)h] × [−bη, (2n − b)η] = ⋃_{r,s=1}^{3} u_{r} × v_{s}.

Taking into account the properties of Fejer kernels in the respective intervals, we have

F_{1,1} ≤ ij ∫_{u_{1}} ∫_{v_{1}} \frac{dt dτ}{ij(tτ)^{2}} , F_{2,1} ≤ i^{2}j^{2} \frac{2}{i} ∫_{v_{1}} \frac{dτ}{jτ^{2}} ,

F_{3,1} ≤ ij ∫_{u_{3}} ∫_{v_{1}} \frac{dt dτ}{ij(tτ)^{2}} , F_{1,2} ≤ ij \frac{2}{j} ∫_{u_{1}} \frac{dt}{it^{2}} ,

F_{2,2} ≤ i^{2}j^{2} \frac{4}{ij} , F_{3,2} ≤ ij^{2} \frac{2}{j} ∫_{u_{3}} \frac{dt}{it^{2}} ,

F_{1,3} ≤ ij ∫_{u_{1}} ∫_{v_{3}} \frac{dt dτ}{ij(tτ)^{2}} , F_{2,3} ≤ i^{2}j \frac{2}{i} ∫_{v_{3}} \frac{dτ}{jτ^{2}} ,

F_{3,3} ≤ ij ∫_{u_{3}} ∫_{v_{3}} \frac{dt dτ}{ij(tτ)^{2}} .

Therefore

A_{1}^{(4)}(4) ≤ c(m, n) ∑_{i=[ \frac{1}{2mh}]+1}^{∞} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} ij|Δ^{2}a_{i,j}|.  (2.32)

A_{1}^{(2)} and A_{1}^{(3)} are estimated by the same scheme as used in deriving (2.31) and (2.32). Therefore for A_{1} we have

A_{1} ≤ ∑_{s=1}^{4} P_{s}(m, n, h, η).  (2.33)

It remains to consider A_{2}(m, n; f) (A_{3} is estimated similarly). We have

A_{2} = ∫_{2mh}^{π} ∫_{0}^{2nη}|Δ^{m,n}f| dx dy = ∫_{2mh}^{π} ∫_{0}^{2nη}|\{ ∑_{i=0}^{[ \frac{1}{x}]} ∑_{j=0}^{[ \frac{1}{2nη}]} + ∑_{i=[ \frac{1}{r}]+1}^{∞} ∑_{j=0}^{[ \frac{1}{2nη}]} +

+ ∑_{i=0}^{[ \frac{1}{x}]} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} + ∑_{i=[ \frac{1}{x}]+1}^{∞} ∑_{j=[ \frac{1}{2nη}]+1}^{∞} \}(i + 1)(j + 1)Δ^{2}a_{i,j} ×

ESTIMATION OF THE INTEGRAL MODULUS OF SMOOTHNESS 397

× ∑_{μ=0}^{m} ∑_{ν=0}^{n}(−1)^{μ+ν} \binom{m}{μ}\binom{n}{ν}K_{i}[x + (m − 2μ)h] ×

× K_{j}[y + (n − 2ν)η]| dx dy = ∑_{α=1}^{4} A_{2}^{(α)}(m, n, h, η).  (2.34)

Again, by Lemma 2 we obtain

A_{2}^{(1)} ≤ c(m, n)h^{m}η^{n} ∫_{2mh}^{π} ∫_{0}^{2nη}∑_{i=1}^{[ \frac{1}{x}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m+2}j^{n+2}|Δ^{2}a_{i,j}|dx dy ≤

≤ c(m, n)h^{m}η^{n} ∫_{1}^{\frac{1}{2mh}}x^{−2} ∑_{i=1}^{[x]} i^{m+2} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1}|Δ^{2}a_{i,j}|dx ≤

≤ c(m, n)h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m+1}j^{n+1}|Δ^{2}a_{i,j}|.  (2.35)

For A_{2}^{(2)} we have

A_{2}^{(2)} = ∫_{2mh}^{π} ∫_{0}^{2nη}|( ∑_{i=[ \frac{1}{x}]+1}^{[ \frac{1}{2mh}]} + ∑_{i=[ \frac{1}{2mh}]+1}^{∞} ) ∑_{j=0}^{[ \frac{1}{2nη}]}(i + 1)(j + 1)|Δ^{2}a_{i,j}| ×

× ∑_{μ=0}^{m} ∑_{ν=0}^{n}(−1)^{μ+ν} \binom{m}{μ}\binom{n}{ν}K_{i}[x + (m − 2μ)h] ×

× K_{j}[y + (n − 2ν)η]| dx dy = A_{2}^{(2)}(1) + A_{2}^{(2)}(2).  (2.36)

Further,

A_{2}^{(2)}(1) ≤ c(m, n)h^{m}η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1} ∫_{1}^{\frac{1}{2mh}}x^{−2} ∑_{i=[x]+1}^{[ \frac{1}{2mh}]} i^{m+2}|Δ^{2}a_{i,j}|dx dy ≤

≤ c(m, n)h^{m}η^{n} ∑_{j=1}^{[ \frac{1}{2nη}]} j^{n+1} ∑_{r=1}^{[ \frac{1}{2mh}]} r^{−2} ∑_{i=r+1}^{[ \frac{1}{2mh}]} i^{m+2}|Δ^{2}a_{i,j}|dx ≤

≤ c(m, n)h^{m}η^{n} ∑_{i=1}^{[ \frac{1}{2mh}]} ∑_{j=1}^{[ \frac{1}{2nη}]} i^{m+1}j^{n+1}|Δ^{2}a_{i,j}|.  (2.37)

Similarly (see (2.32)) we find

A_{2}^{(2)}(2) ≤ c(m, n)η^{n} ∑_{i=[ \frac{1}{2mh}]+1}^{∞} ∑_{j=1}^{[ \frac{1}{2nη}]} ij^{n+1}|Δ^{2}a_{i,j}|.  (2.38)

By (2.35)–(2.37) we obtain

A_{2}^{(2)} ≤ P_{1}(m, n, h, η) + P_{3}(m, n, h, η).  (2.39)

It is likewise easy to show that

A_{2}^{(3)} ≤ P_{2}(m, n, h, η) + P_{4}(m, n, h, η), A_{2}^{(4)} ≤ P_{4}(m, n, h, η). (2.40)

Keeping in mind (2.33), (2.34), (2.38)–(2.40), we find

A_{2} ≤ ∑_{s=1}^{4} P_{s}(m, n, h, η).  (2.41)

With regard to (2.29), (2.33), and (2.41) we conclude that

A ≤ ∑_{s=1}^{4} P_{s}(m, n, h, η).

3. Let now ℝ^{k} be a k-dimensional Euclidean space of points x=(x_{1},…,x_{k}) with ordinary linear operations and the Euclidean norm ∥x∥. The product of two vectors x, y ∈ ℝ^{k} is understood componentwise and so are the inequalities x < y, x ≤ y. It is assumed that T^{k} = [0, 2π]^{k}.

If n = (n_{1}, …, n_{k}) is a multi-index with non-negative integral components, 0 = (0, …, 0), 1 = (1, …, 1), and (a_{n})_{n≥0} is a k-multiple numerical sequence, then

Δ_{i}a_{n} = a_{n} − a_{n+(δ_{i1},...,δ_{ik})},

where δ_{ij} is the Kronecker symbol, i, j ∈ M = {1, …, k}.

We introduce the notation

Δ_{M}a_{n} ≡ Δ_{1···k}a_{n} = Δ_{1}(Δ_{2}(⋯ (Δ_{k}a_{n}) ⋯)),

for each B, ∅ ≠ B ⊂ M, Δ_{B}a_{n} is defined similarly, and assume that λ(n) is the number of zero coordinates of the vector n (see [7]). We will also use the notation B′ = M \ B, where B ⊂ M.

Definition 2. A sequence (a_{n})_{n≥0} will be called quasiconvex if the series

∑_{n≥0} ∑_{B⊂M, B≠∅} ∏_{i∈B}(n_{i} + 1)|Δ_{B}(Δ_{M}a_{n})|

converges.

If the sequence (a_{n})_{n≥0} is quasiconvex and

lim_{∥n∥→∞} a_{n} = 0,

then the series

∑_{n≥0} 2^{−λ(n)}a_{n} ∏_{i=1}^{k} cos n_{i}x_{i}  (∗∗)

converges on (0, 2π)^{k} to some function f : ℝ^{k} → ℝ summable on T^{k} and is its Fourier series.

Let m, i, δ, h ∈ ℝ^{k}, where m and i have non-negative integral components, while δ has positive components. Consider

ω_{m}(δ; f)_{1} = sup_{−δ≤h≤δ} ∫_{T^{k}} | ∑_{0≤i≤m}(−1)^{∑_{j=1}^{k} i_{j}} ∏_{j=1}^{k} \binom{m_{j}}{i_{j}}f[x + (m − 2i)h]| dx

as an integral modulus of smoothness of order m of the function f ∈ L(T^{k}). The validity of the following analogue of Lemmas 1 and 2 is obvious.

Lemma 4. Let p ∈ ℝ^{k} have natural components, h > 0 and 2mh ≤ x ≤ π1. Then

|T_{p}^{m}| = | ∑_{0≤i≤m}(−1)^{∑_{j=1}^{k} i_{j}} ∏_{j=1}^{k} [\binom{m_{j}}{i_{j}}K_{p_{j}}[x_{j} + (m_{j} − 2i_{j})h_{j}]]| ≤

≤ c(m) ∏_{j=1}^{k} x_{j}^{−2}h_{j}^{m_{j}}P_{j}^{m_{j}−1}, |T_{p}^{m}| ≤ c(m) ∏_{j=1}^{k} h_{j}^{m_{j}}P_{j}^{m_{j}+1}.

We have

Theorem 2. Let a sequence (a_{n})_{n≥0} be quasiconvex and lim_{∥n∥→∞} a_{n} = 0.

Then for the sum f of the series (∗∗) we have

ω_{1}^{(m)}(δ; f) ≤ c(m) ∑_{B:B⊂M} \{ ∑_{\begin{matrix}1≤n_{ν}≤N_{ν} \\ ν∈B \end{matrix}} ∑_{\begin{matrix}N_{μ}+1≤n_{μ}≤∞ \\ μ∈B′ \end{matrix}} ×

× ∏_{ν∈B} N_{ν}^{−m_{ν}} i_{ν}^{m_{ν}+1} ∏_{μ∈B′} i_{μ}|Δ_{B}(Δ_{M}a_{n})|}

where N_{ν} = [\frac{1}{δ_{ν}}], ν ∈ M.

(As usual, the empty product is assumed to be zero.) To prove the theorem note that the Hardy transformation is defined in ℝ^{k}

for k > 2 too and its structure becomes more complicated as the dimension increases. Nevertheless, the estimates from the proof of Theorem 1 can hold for the case k > 2 as well.

References

1. S. Aljanĉiĉ and M. Tomiĉ, Sur le module de continuite integral des séries de Fourier a coefficients convexes. C. R. Acad. Sci. Paris 259(1964), No. 9, 1609–1611.

2. S. Aljanĉiĉ and M. Tomiĉ, Über den Stetigkeitsmodul von Fourier-Reihen mit monotonen Koeffizienten. Math. Z. 88(1965), No. 3, 274–284.

3. M. and S. Izumi, Modulus of continuity of functions defined by trigonometric series. J. Math. Anal. and Appl. 24(1968), 564–581.

4. S. A. Telyakovski, The integrability of trigonometric series. Estimation of the integral modulus of continuity. (Russian) Mat. Sbornik 91(133)(1973), No. 4(8), 537–553.

5. T. Sh. Tevzadze, Some classes of functions and trigonometric Fourier series. (Russian) Some questions of function theory (Russian), v. II, 31–92, Tbilisi University Press, 1981.

6. E. H. Hardy, On double Fourier series and especially those which represent the double zeta-function with real and incommensurable parameters. Quart. J. Math. 37(1906), 53–79.

7. L. V. Zhizhiashvili, Some questions of the theory of trigonometric Fourier series and their conjugates. (Russian) Tbilisi University Press, 1993.

(Received 29.12.1993; revised version 15.06.1995)

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