Georgian Mathematical Journal 1(1994), No. 3, 287-302

ON SOME PROPERTIES OF MULTIPLE CONJUGATE TRIGONOMETRIC SERIES

D. LELADZE

Abstract. We have obtained the estimate in the terms of partial and mixed moduli of continuity of deviation of Cesáro (C, α) means (α = (α_{1}, …, α_{n}), α_{i} ∈ ℝ, α_{i} > −1, i = \overline{1, n}) of the sequence of rectangular partial sums of n-multiple (n > 1) conjugate trigonometric series from n-multiple truncated conjugate function. This estimate implies the result on the m_{λ}-convergence (λ ≥ 1) of (C, α) means (α_{i} > 0, i = \overline{1, n}), provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown, that the m_{λ}-convergence cannot be replaced by ordinary convergence.

1. Let f ∈ L([−π; π]^{n}), n ∈ ℕ, n > 1, be a function, 2π-periodic in each variable, σ_{n}[f] its n-multiple trigonometric Fourier series, and σ̄_{n}[f] its conjugate series with respect to n variables (see, e.g., [1]).

We set

m = (m_{1}, …, m_{n}) (m_{i} ∈ ℕ, i = \overline{1, n});
x = (x_{1}, …, x_{n}) (x_{i} ∈ ℝ, i = \overline{1, n});

α=(α_{1}, …, α_{n}) (α_{i} ∈ℝ, α_{i} >−1, i=\overline{1, n}); M=\{1, 2, …, n\}.

By M_{j} we denote the set of all subsets of M with j elements, by M^{(A)} a set M\A (A ⊂ M), and by M_{j}^{(A)} the set of all subsets of M^{(A)} with j elements.

For B = {i_{i}, …, i_{k}} ⊂ M we define m(B) = {1/m_{i_{1}}, …, 1/m_{i_{k}}} and the

1991 Mathematics Subject Classification. 42B08.

truncated conjugate function with respect to the corresponding variables

f̄_{m(B)}(x) =

= \frac{1}{(−2π)^{k}} ∫_{1/m_{i_{1}}}^{π}…∫_{1/m_{i_{k}}}^{π}(Δ_{k,s_{i_{k}}} (Δ_{k−1,s_{i_{k−1}}} ( … (Δ_{1,s_{i_{1}}} (f; x)) …)))×

× ∏_{j=1}^{k} ctg \frac{s_{i_{j}}}{2} ds_{i_{1}} … ds_{i_{k}}, f̄_{m}(x) = f̄_{m(M)}(x),

where

Δ_{i,h}(f, x) = f(x_{1}, …, x_{i−1}, x_{i} + h, x_{i+1}, …, x_{n}) −
− f(x_{1}, …, x_{i−1}, x_{i} − h, x_{i+1}, …, x_{n}), i = \overline{1, n}.

For f ∈ L^{q}([−π; π]^{n}), 1 ≤ q ≤ +∞ (L^{∞} = C), we consider its mixed modulus of continuity

ω_{B}(m(B); f)_{L^{q}} =

= sup_{|h_{1}|<1/m_{i_{1}} ,...,|h_{k}|<1/m_{i_{k}}} ∥Δ_{k}^{h_{k}}(Δ_{k−1}^{h_{k−1}}( … (Δ_{1}^{h_{1}}(f; x)) …))∥_{L^{q}},

where Δ_{i}^{h}(f; x) = f(x_{1}, …, x_{i−1}, x_{i} + h, x_{i+1}, …, x_{n}) − f(x_{1}, …, x_{n}). Let σ̄_{m}^{α}(x; f) be Cesáro means of σ̄_{n}[f]. In the sequel by A, B, A_{1}, B_{1}, C(α), C(β), C(α, β), etc. we will denote,

in general, different positive constants.

Finally, we set

λ(k, β) = \begin{cases}k^{−β}, & −1 < β < 0, \\ ln(k + 1), & β = 0, \\ 1, & β > 0, (k = 1, 2, …). \end{cases}

In the present paper we give the estimate of the deviation of n-multiple Cesáro means of the sequence of rectangular partial sums of σ̄_{n}[f] from f̄_{m} in the norm of L^{q}, q ∈ [1; +∞] (L^{∞} = C), in terms of partial and mixed moduli of continuity of f. This result generalizes the corresponding result of L.Zhizhiashvili (see [1]).

From this estimate ensues the result on the Cesáro summability of the sequence of rectangular partial sums of σ̄_{n}[f] and then the correctness of this result is shown.

2. The following is true:

Theorem 1. If f ∈ L^{q}([−π; π]^{n}), q ∈ [1; +∞] (L^{∞} = C), then

∥σ̄_{m}^{α}(x; f) − f̄_{m}(x)∥_{L^{q}} ≤ C(α) {∑_{k=1}^{n−1} ∑_{B∈M_{k}} ω_{B}(m(B); f)_{L^{q}}×

× ∏_{\begin{matrix}j=1 \\ \{i_{1},...,i_{k}\}∈M_{k} \end{matrix}}^{k} λ(m_{i_{j}}, α_{i_{j}}) + ∑_{i=1}^{n−1} ∑_{k=1}^{n−i} ∑_{A∈M_{i}} ∑_{B∈M_{k}^{(A)}} ω_{B}(m(B); f̄_{m(A)})_{L^{q}} ×

× ∏_{\begin{matrix}j=1 \\ \{t_{1},...,t_{k}\}∈M_{k}^{(A)} \end{matrix}}^{k} λ(m_{t_{j}}, α_{t_{j}}) + ∏_{i=1}^{n} λ(m_{i}, α_{i})ω_{M}(m(M); f)_{L^{q}}\} . (1)

Proof. For simplicity, we will prove the theorem in the case n = 2 which is typical. We will use the method of L.Zhizhiashvili ([1], p. 160-191), this method proving to be true for n ≥ 3.

Let ω(δ_{1}, δ_{2}; f)_{L^{p}} be the mixed modulus of continuity of the function f(x, y), x, y ∈ ℝ, with respect to two variables. By ω_{i}(δ; f)_{L^{q}} (i ∈ ℕ, δ ∈ ℝ^{+}), as usual, we denote the modulus of continuity of f in L^{q} with respect to the corresponding variable. Let m, n ∈ ℕ, α, β ∈ ℝ, α, β > −1, f̄_{m,n}(x, y) be the two-dimensional truncated conjugate function and σ̄_{mn}^{α,β}(x, y; f) be the Cesáro means of the double conjugate series σ̄_{2}[f].

We have ([1], p.187-188):

σ̄_{mn}^{α,β}(x, y; f) = 1/π^{2} ∫_{0}^{π/m} ∫_{0}^{π/n} ψ_{x,y}(s, t; f)K̄_{m}^{α}(s)K̄_{n}^{β}(t) dt ds +

+ 1/(2π^{2}) ∫_{0}^{π/m} ∫_{π/n}^{π} ψ_{x,y}(s, t; f) ctg \frac{t}{2}K̄_{m}^{α}(s) dt ds +

+ 1/π^{2} ∫_{0}^{π/m} ∫_{π/n}^{π} ψ_{x,y}(s, t; f)K̄_{m}^{α}(s)H_{n,1}^{β}(t) dt ds +

+ 1/π^{2} ∫_{0}^{π/m} ∫_{π/n}^{π} ψ_{x,y}(s, t; f)K̄_{m}^{α}(s)H_{n,2}^{β}(t) dt ds +

+ 1/(2π^{2}) ∫_{π/m}^{π} ∫_{0}^{π/n} ψ_{x,y}(s, t; f) ctg \frac{s}{2}K̄_{n}^{β}(t) dt ds +

+ 1/π^{2} ∫_{π/m}^{π} ∫_{0}^{π/n} ψ_{x,y}(s, t; f)H_{m,1}^{α}(s)K̄_{n}^{β}(t) dt ds +

+ 1/π^{2} ∫_{π/m}^{π} ∫_{0}^{π/n} ψ_{x,y}(s, t; f)H_{m,2}^{α}(s)K̄_{n}^{β}(t) dt ds +

+ 1/(4π^{2}) ∫_{π/m}^{π} ∫_{π/n}^{π} ψ_{x,y}(s, t; f) ctg \frac{s}{2} ctg \frac{t}{2} dt ds +

+ 1/(2π^{2}) ∫_{π/m}^{π} ∫_{π/n}^{π} ψ_{x,y}(s, t; f) ctg \frac{s}{2}H_{n,1}^{β}(t) dt ds +

+ 1/(2π^{2}) ∫_{π/m}^{π} ∫_{π/n}^{π} ψ_{x,y}(s, t; f) ctg \frac{s}{2}H_{n,2}^{β}(t) dt ds +

+ 1/(2π^{2}) ∫_{π/m}^{π} ∫_{π/n}^{π} ψ_{x,y}(s, t; f) ctg \frac{t}{2}H_{m,1}^{α}(s) dt ds +

+ 1/π^{2} ∫_{π/m}^{π} ∫_{π/n}^{π} ψ_{x,y}(s, t; f)H_{m,1}^{α}(s)H_{n,1}^{β}(t) dt ds +

+ 1/π^{2} ∫_{π/m}^{π} ∫_{π/n}^{π} ψ_{x,y}(s, t; f)H_{m,1}^{α}(s)H_{n,2}^{β}(t) dt ds +

+ 1/(2π^{2}) ∫_{π/m}^{π} ∫_{π/n}^{π} ψ_{x,y}(s, t; f) ctg \frac{t}{2}H_{m,2}^{α}(s) dt ds +

+ 1/π^{2} ∫_{π/m}^{π} ∫_{π/n}^{π} ψ_{x,y}(s, t; f)H_{m,2}^{α}(s)H_{n,1}^{β}(t) dt ds +

+ 1/π^{2} ∫_{π/m}^{π} ∫_{π/n}^{π} ψ_{x,y}(s, t; f)H_{m,2}^{α}(s)H_{n,2}^{β}(t) dt ds =

= ∑_{j=1}^{16} P_{mn}^{(j)}(x, y; f),  (2)

where

ψ_{x,y}(s, t; f) = f(x + s, y + t) − f(x − s, y + t) −
− f(x + s, y − t) + f(x − s, y − t)  (3)

and H_{m,1}^{α}(s), H_{m,2}^{α}(s) are the summands of the conjugate Fejér kernel K̄_{m}^{α}(s) (see [2], pp.157-160)

K̄_{m}^{α}(s) = \frac{1}{2} ctg \frac{s}{2} + H_{m,1}^{α}(s) + H_{m,2}^{α}(s),  (4)
H_{m,1}^{α}(s) = −\frac{cos ((m + \frac{1}{2} + \frac{α}{2})s − \frac{πα}{2})}{A_{m}^{α}(2 sin \frac{s}{2})^{1+α}}.  (5)

Besides,

|K̄_{m}^{α}(s)| ≤ C(α)m, |s| ≤ π;  (6)
|H_{m,2}^{α}(s)| ≤ \frac{C(α)}{m^{2}s^{3}} , π/m ≤ |s| ≤ π, m > 1.  (7)

The estimate (7) is more precise than the estimate (2.1.14) in [1] and it can be proved by arguments analogous to those in [2], pp.157-160.

From (2) we obtain

σ̄_{mn}^{α,β}(x, y; f) − f̄_{mn}(x, y) = σ̄_{mn}^{α,β}(x, y; f) − P_{mn}^{(8)}(x, y; f) =
= \begin{matrix}16 \\ ∑ & ′ \\ k=1 \end{matrix} P_{mn}^{(k)}(x, y; f),  (8)

where ′ indicates that the eighth member is omitted (the replacement of 1/m (1/n) in f̄_{mn}(x, y) by π/m (π/n) does not matter).

In the sequel we will use the inequality (see [3], p.179)

\begin{matrix}\{ ∫_{a}^{b} | ∫_{c}^{d} f(x, y) dy|^{q}dx\}^{1/q} ≤ ∫_{c}^{d} \{ ∫_{a}^{b} |f(x, y)|^{q} dx\}^{1/q}dy, \\ 1 ≤ q < +∞. \end{matrix} (9)

Taking into account (6) and (9), we obtain

∥P_{mn}^{(1)}(x, y; f)∥_{L^{q}} ≤ C(α, β){ω_{1}(1/m; f)_{L^{q}} + ω_{2}(1/n; f)_{L^{q}}}. (10)

It is easy to see, that

P_{mn}^{(2)}(x, y; f) = −\frac{1}{π} ∫_{0}^{π/m} [f̄_{n}^{(2)}(x + s, y) − f̄_{n}^{(2)}(x − s, y)]K̄_{m}^{α}(s) ds,

P_{mn}^{(5)}(x, y; f) = −\frac{1}{π} ∫_{0}^{π/n} [f̄_{m}^{(1)}(x, y + t) − f̄_{m}^{(1)}(x, y − t)]K̄_{n}^{β}(t) dt,

where

f̄_{m}^{(1)}(x, y)=−\frac{1}{2π} ∫_{π/m}^{π}[f(x + s, y)−f(x − s, y)] ctg \frac{s}{2} ds,  (11)
f̄_{n}^{(2)}(x, y)=−\frac{1}{2π} ∫_{π/n}^{π}[f(x, y + t)−f(x, y − t)] ctg \frac{t}{2} dt.  (12)

Hence, using again (6) and (9), we obtain:

∥P_{mn}^{(2)}(x, y; f)∥_{L^{q}} ≤ C(α, β)ω_{1}(1/m; f̄_{n}^{(2)})_{L^{q}};  (13)
∥P_{mn}^{(5)}(x, y; f)∥_{L^{q}} ≤ C(α, β)ω_{2}(1/n; f̄_{m}^{(1)})_{L^{q}}.  (14)

Now, it is easy to see that for estimation of P_{mn}^{(3)}(x, y; f) it suffices to estimate the integral

I(m, n) = n^{−β} ∫_{0}^{π/m} ∫_{π/n}^{π} ψ_{x,y}(s, t; f)K̄_{m}^{α}(s)ω_{β}(t) cos nt dt ds,

where

ω_{β}(t) = \frac{cos \frac{1+β}{2} t}{(sin \frac{t}{2})^{1+β}} .

292 D. LELADZE

We have 2I(m, n) = n^{−β}{∫_{0}^{π/m}∫_{π/n}^{π}[ψ_{x,y}(s, t; f)−ψ_{x,y}(s, t+π/n; f)] ×

×K̄_{m}^{α}(s)ω_{β}(t) cos nt dt ds+∫_{0}^{π/m}∫_{π/n}^{π}ψ_{x,y}(s, t+π/n; f)[ω_{β}(t)−

−ω_{β}(t+π/n)]K̄_{m}^{α}(s) cos nt dt ds+

+ ∫_{0}^{π/m} ∫_{π−π/n}^{π} ψ_{x,y}(s, t + π/n; f)K̄_{m}^{α}(s)ω_{β}(t + π/n) cos nt dt ds −

− ∫_{0}^{π/m} ∫_{0}^{π/n} ψ_{x,y}(s, t + π/n; f)K̄_{m}^{α}(s)ω_{β}(t + π/n) cos nt dt ds}. Now we note that (see [1], p.56, (2.1.18))

|ω_{β}(t) − ω_{β}(t + π/n)| ≤ C(β)/(nt^{2+β}), \frac{π}{n} ≤ t ≤ π.  (15)

(6) and (15) yield ∥I(m, n)∥_{L^{q}} ≤ C(α, β)λ(n, β)ω_{2}(1/n; f)_{L^{q}} and hence

∥P_{mn}^{(3)}(x, y; f)∥_{L^{q}} ≤ C(α, β)λ(n, β)ω_{2}(1/n; f)_{L^{q}}.  (16)

Analogously

∥P_{mn}^{(6)}(x, y; f)∥_{L^{q}} ≤ C(α, β)λ(m, α)ω_{1}(1/m; f)_{L^{q}}.  (17)

Furthermore, using (6),(7) and (9), we obtain

\begin{matrix}∥P_{mn}^{(k)}(x, y; f)∥_{L^{q}} ≤ C(α, β)\{ω_{1}(1/m; f)_{L^{q}} + ω_{2}(1/n; f)_{L^{q}}\} \\ (k = 4, 7, 16). \end{matrix} (18)

Analogously, taking into account that

P_{mn}^{(9)}(x, y; f) = −\frac{1}{π} ∫_{π/n}^{π} [f̄_{m}^{(1)}(x, y + t) − f̄_{m}^{(1)}(x, y − t)]H_{n,1}^{β}(t) dt,

P_{mn}^{(11)}(x, y; f) = −\frac{1}{π} ∫_{π/m}^{π} [f̄_{n}^{(2)}(x + s, y) − f̄_{n}^{(2)}(x − s, y)]H_{m,1}^{α}(s) ds,

we can prove

∥P_{mn}^{(9)}(x, y; f)∥_{L^{q}} ≤ C(α, β)λ(n, β)ω_{2}(1/n; f̄_{m}^{(1)})_{L^{q}};  (19)
∥P_{mn}^{(11)}(x, y; f)∥_{L^{q}} ≤ C(α, β)λ(m, α)ω_{1}(1/m; f̄_{n}^{(2)})_{L^{q}}.  (20)

Using the same arguments and applying (6), (7) and (9), we can prove

∥P_{mn}^{(10)}(x, y; f)∥_{L^{q}} ≤ C(α, β)ω_{2}(1/n; f̄_{m}^{(1)})_{L^{q}};  (21)

∥P_{mn}^{(14)}(x, y; f)∥_{L^{q}} ≤ C(α, β)ω_{1}(1/m; f̄_{n}^{(2)})_{L^{q}}.  (22)

Now we observe that the following lemma holds true (see [1], p.160, Lemma 10):

Lemma 1. Let f ∈ L^{q}([−π; π]^{2}), 1 ≤ q ≤ +∞, and

ϕ_{x,y}(s, t; f) = f(x + s, y + t) + f(x − s, y + t) + f(x + s, y − t) +
+ f(x − s, y − t) − 4f(x, y).  (23)

Then

∥m^{−α}n^{−β} ∫_{π/m}^{π} ∫_{π/n}^{π} ϕ_{x,y}(s, t; f)ω_{α}(s)ω_{β}(t) \begin{matrix}sin \\ cos \end{matrix} ms {sin \atop cos} nt dt ds∥_{L^{q}} =

= O{λ(m, α)λ(n, β)ω(1/m, 1/n; f)_{L^{q}} + λ(m, α)ω_{1}(1/m; f)_{L^{q}} +

+λ(n, β)ω_{2}(1/n; f)_{L^{q}}}.

There is another lemma in [1] (see p.171, Lemma 11), which can be corrected by means of (7) as follows:

Lemma 2. For f ∈ L^{q}([−π; π]^{2}), 1 ≤ q ≤ +∞, we have

∥m^{−α} ∫_{π/m}^{π} ∫_{π/n}^{π} ϕ_{x,y}(s, t; f)ω_{α}(s)H_{n,2}^{β}(t) \begin{matrix}sin \\ cos \end{matrix} ms dt ds∥_{L^{q}} =

= O{λ(m, α)ω_{1}(1/m; f)_{L^{q}} + ω_{2}(1/n; f)_{L^{q}}};

∥n^{−β} ∫_{π/m}^{π} ∫_{π/n}^{π} ϕ_{x,y}(s, t; f)ω_{β}(t)H_{m,2}^{α}(s) \begin{matrix}sin \\ cos \end{matrix} nt dt ds∥_{L^{q}} =

= O{λ(n, β)ω_{2}(1/n; f)_{L^{q}} + ω_{1}(1/m; f)_{L^{q}}}.

The lemmas and (3), (5), (7) and (23) yield

∥P_{mn}^{(12)}(x, y; f) + P_{mn}^{(13)}(x, y; f) + P_{mn}^{(15)}(x, y; f)∥_{L^{q}} ≤

≤ C(α, β){λ(m, α)λ(n, β)ω(1/m, 1/n; f)_{L^{q}} + λ(m, α)ω_{1}(1/m; f)_{L^{q}} +

+λ(n, β)ω_{2}(1/n; f)_{L^{q}}}.  (24)

Finally, (2)–(24) yield

∥σ̄_{mn}^{α,β}(x, y; f) − f̄_{mn}(x, y)∥_{L^{q}} ≤

≤ C(α, β){λ(m, α)λ(n, β)ω(1/m, 1/n; f)_{L^{q}} + λ(m, α)ω_{1}(1/m; f)_{L^{q}} +

+λ(n, β)ω_{2}(1/n; f)_{L^{q}} + λ(m, α)ω_{1}(1/m; f̄_{n}^{(2)})_{L^{q}} +
+λ(n, β)ω_{2}(1/n; f̄_{m}^{(1)})_{L^{q}}},  (25)

which is the formula (1) in the case n = 2.

Corollary. If f ∈ C([−π; π]^{n}) (n ≥ 2) and

ω_{i}(δ; f)_{C} = o(1/ ln^{n−1}(1/δ)) (i = \overline{1, n})  (26)

as δ → 0+, then lim_{m_{λ}→∞} ∥σ̄_{m}^{α}(x; f)−f̄_{m}(x)∥_{C} = 0, α = (α_{1}, …, α_{n}), α_{i} > 0

(i = \overline{1, n}), λ ≥ 1.

Now we will prove that the m_{λ}-summability in the corollary is essential. Namely, the following result holds true:

Theorem 2. There exists a function f ∈ C([−π; π]^{n}) which satisfies (26) and

\overline{lim_{m→∞}}|σ̄_{m}^{α}(0; f) − f̄_{m}(0)| = +∞,  (27)

α = (α_{1}, …, α_{n}), α_{i} > 0 (i = \overline{1, n}), 0 = (0, 0, …, 0).

Proof. We will prove the theorem for n = 2, this case being quite typical. First, let α, β ∈ (0; 1). We set

m_{k} = 2^{2^{2^{k}}} , n_{k} = m_{k}^{m_{k}^{m{m_{k} \atop k}}} , (k ∈ ℕ);  (28)
ln^{(2)} x = ln ln x, ln^{(k)} x = ln ln^{(k−1)} x, k ∈ ℕ, k ≥ 3;

g_{k}(x) = \begin{cases}\frac{1}{ln(1/(x−π/(6m_{k}))) ln^{(2)}(1/(x−π/(6m_{k})))}, & \frac{π}{6m_{k}} < x ≤ \frac{11π}{60m_{k}} , \\ \frac{1}{ln(1/(π/(5m_{k})−x)) ln^{(2)}(1/(π/(5m_{k})−x))}, & \frac{11π}{60m_{k}} ≤ x < \frac{π}{5m_{k}} , \\ 0, & x ∈ (0; π)\(\frac{π}{6m_{k}}; \frac{π}{5m_{k}} ). \end{cases}

Furthermore,

h(x) = ∑_{k=1}^{∞} g_{k}(x);

p(y) = \begin{cases}\frac{1}{ln(2π/y) ln^{(2)}(2π/y)}, & y ∈ (0; \frac{π}{2} ], \\ \frac{1}{ln(2π/(π−y)) ln^{(2)}(2π/(π−y))}, & y ∈ [\frac{π}{2} ; π); \end{cases}

f(x, y) = \begin{cases}h(x)p(y), & (x, y) ∈ (0; π)^{2}, \\ 0, & (x, y) ∈ [−π; π]^{2}\(0; π)^{2}. \end{cases}

Finally, outside the square [−π; π]^{2}, we extend the function f by periodicity with the period 2π in each variable. It is easy to see that f satisfies

ON SOME PROPERTIES OF TRIGONOMETRIC SERIES 295

From now on we set m = 2m_{k}, n = n_{k} + 1. We have σ̄_{mn}^{α,β}(0, 0; f) − f̄_{mn}(0, 0) = 1/π^{2} ∫_{0}^{π} ∫_{0}^{π} f(x, y)K̄_{m}^{α}(x)K̄_{n}^{β}(y) dy dx −

−1/(4π^{2}) ∫_{1/m}^{π} ∫_{1/n}^{π} f(x, y) ctg \frac{x}{2} ctg \frac{y}{2} dy dx =

= 1/π^{2} ∫_{0}^{1/m} ∫_{0}^{1/n} f(x, y)K̄_{m}^{α}(x)K̄_{n}^{β}(y) dy dx + +1/π^{2} ∫_{0}^{1/m} ∫_{1/n}^{π} f(x, y)K̄_{m}^{α}(x)K̄_{n}^{β}(y) dy dx + +1/π^{2} ∫_{1/m}^{π} ∫_{0}^{1/n} f(x, y)K̄_{m}^{α}(x)K̄_{n}^{β}(y) dy dx +

+1/π^{2} ∫_{1/m}^{π} ∫_{1/n}^{π} f(x, y)(K̄_{m}^{α}(x)K̄_{n}^{β}(y) − 1/4 ctg \frac{x}{2} ctg \frac{y}{2} ) dy dx =

= ∑_{j=1}^{4} R_{j}(m, n).  (29)

Obviously,

R_{1}(m, n) = o(1) (m, n → ∞).  (30)

Then,

R_{2}(m, n) = 1/π^{2} ∫_{0}^{1/m} ∫_{1/n}^{π} f(x, y)K̄_{m}^{α}(x) \frac{1}{2} ctg \frac{y}{2} dy dx −
− 1/π^{2} ∫_{0}^{1/m} ∫_{1/n}^{π} f(x, y)K̄_{m}^{α}(x)H_{n}^{β}(y) dy dx =
= R′_{2}(m, n) + R′′_{2}(m, n),  (31)

where

H_{n}^{α}(t) = \frac{1}{A_{n}^{α}} ∑_{ν=0}^{n} A_{ν}^{α−1} \frac{cos(ν + 1/2)t}{2 sin(t/2)}.  (32)

The following estimates hold true (see [2], (5.12)):

|K̄_{n}^{α}(t)| ≤ n, |t| ≤ π;  (33)
|H_{n}^{α}(t)| ≤ C(α)n^{−α}t^{−(α+1)}, 1/n ≤ |t| ≤ π.  (34)

296 D. LELADZE

Now we have

∫_{1/n}^{π} p(y) \frac{1}{2} ctg \frac{y}{2} dy = ∫_{1/n}^{π/2} p(y) \frac{1}{2} ctg \frac{y}{2} dy +

+ ∫_{π/2}^{π} p(y) \frac{1}{2} ctg \frac{y}{2} dy = U_{1}(n) + U_{2}(n).  (35)

Now,

U_{1}(n) = −C_{1} ∫_{1/n}^{π/2} \frac{d(2πy)}{2πy ln(2πy) ln | ln(2πy)|} ≤

≤ C_{2} ln^{(3)} n ≤ C_{3}m_{k} ln m_{k};  (36)
|U_{2}(n)| ≤ M.  (37)

(31) and (35)–(37) yield

|R′_{2}(m, n)| ≤ \frac{C}{ln m_{k+1} ln^{(2)} m_{k+1}} m_{k} ln m_{k}.  (38)

As to R′′_{2}(m, n), we have

|R′′_{2}(m, n)| ≤ \frac{C(β)}{ln m_{k+1} ln^{(2)} m_{k+1}} ∫_{0}^{1/m} ∫_{1/n}^{π} \frac{m}{n^{β}y^{β+1}} dy dx ≤

≤ \frac{C(β)}{ln m_{k+1} ln^{(2)} m_{k+1}} .  (39)

(31), (38) and (39) yield

|R_{2}(m, n)| ≤ \frac{C(β)}{ln m_{m+1} ln^{(2)} m_{k+1}} m_{k} ln m_{k}.  (40)

Analogously

R_{3}(m, n) = R′_{3}(m, n) + R′′_{3}(m, n).  (41)

Now,

|R′_{3}(m, n)| ≤ \frac{C}{ln n ln^{(2)} n} ln m,  (42)
|R′′_{3}(m, n)| ≤ \frac{C(α)}{ln n ln^{(2)} n}.  (43)

(41)–(43) yield

|R_{3}(m, n)| = o(1) (m, n → ∞).  (44)

Now let us consider R_{4}(m, n). We break it into 4 parts as follows

R_{4}(m, n) = ( ∫_{1/m}^{1/m^{τ}} ∫_{1/n}^{1/n^{τ}} + ∫_{1/m}^{1/m^{τ}} ∫_{1/n^{τ}}^{π} + ∫_{1/m^{τ}}^{π} ∫_{1/n}^{1/n^{τ}} +
+ ∫_{1/m^{τ}}^{π} ∫_{1/n^{τ}}^{π} )1/π^{2}f(x, y)(K̄_{m}^{α}(x)K̄_{n}^{β}(y) −

−\frac{1}{4} ctg \frac{x}{2} ctg \frac{y}{2} ) dy dx = ∑_{j=1}^{4} I_{j}(m, n),  (45)

where 1/2 ≤ τ < 1. We have

|I_{4}(m, n)| ≤

≤ C(α, β) ∫_{1/m^{τ}}^{π} ∫_{1/n^{τ}}^{π} (\frac{1}{m^{α}x^{α+1}y} + \frac{1}{n^{β}y^{β+1}x} + \frac{1}{m^{α}n^{β}x^{α+1}y^{β+1}} ) dy dx.

It is easy to see that

|I_{4}(m, n)| = o(1) (m, n → ∞).  (46)

Now we estimate I_{2}(m, n).

I_{2}(m, n) = ∑_{k=1}^{3} Q_{k}(m, n).  (47)

|Q_{2}(m, n)| = |1/π^{2} ∫_{1/m}^{1/m^{τ}} ∫_{1/n^{τ}}^{π} f(x, y)\frac{−H_{n}^{β}(y)}{2 tg(x/2)} dy dx| ≤

≤ C(β) max |f| ln m\frac{n^{βτ}}{n^{β}}

and

|Q_{2}(m, n)| = o(1) (m, n → ∞);  (48)
|Q_{3}(m, n)| = |1/π^{2} ∫_{1/m}^{1/m^{τ}} ∫_{1/n^{τ}}^{π} f(x, y)H_{m}^{α}(x)H_{n}^{β}(y) dy dx| = o(1) (49)
(m, n → ∞).

Next we will show that |Q_{1}(m, n)| → +∞ as m, n → ∞. We have \frac{1}{m} < \frac{π}{6m_{k}} < \frac{π}{5m_{k}} < \frac{1}{m^{τ}}. Therefore

Q_{1}(m, n) = 1/π^{2} ∫_{π/6m_{k}}^{π/5m_{k}} h(x)(−H_{m}^{α}(x)) dx ×

× ∫_{1/n^{τ}}^{π} p(y)\frac{1}{2 tg(y/2)} dy = Q_{1}^{(1)}(m)Q_{1}^{(2)}(n).  (50)

Since m = 2m_{k}, we have cos(i + 1/2)x > 0 for i = 0, 1, …, m and x ∈ [\frac{π}{6m_{k}}; \frac{π}{5m_{k}}]. Hence (−H_{m}^{α}(x)) < 0 (see (32)). Therefore we have

|Q_{1}^{(1)}(m)| = 1/π^{2} ∫_{π/6m_{k}}^{π/5m_{k}} H_{m}^{α}(x)h(x) dx ≥

≥ \frac{C}{A_{m}^{α}} ∫_{21π/120m_{k}}^{23π/120m_{k}} ∑_{i=0}^{m} A_{i}^{α−1} \frac{cos \frac{9π}{20}}{x} h(x) dx ≥

≥ \frac{C}{A_{m}^{α}} A_{m}^{α} \frac{1}{ln m_{k} ln^{(2)} m_{k}} ∫_{21π/120m_{k}}^{23π/120m_{k}} \frac{dx}{x} ≥ \frac{C}{ln m_{k} ln^{(2)} m_{k}} .  (51)

As to Q_{1}^{(2)}(n), we have (analogously to (36))

Q_{1}^{(2)}(n) ≥ Cm_{k} ln m_{k}.  (52)

(50)–(52) imply

|Q_{1}(m, n)| ≥ \frac{C}{ln m_{k} ln^{(2)} m_{k}} m_{k} ln m_{k}.  (53)

(28), (40) and (53) yield

|Q_{1}(m, n)| − R_{2}(m, n) → +∞  (54)

as m, n → ∞ (for m = 2m_{k}, n = n_{k} + 1).

From (47)–(49) and (54) we obtain

|I_{2}(m, n)| − R_{2}(m, n) → +∞ (m, n → ∞).  (55)

Now let us consider I_{3}(m, n). As in the case of I_{2}(m, n), we have

I_{3}(m, n) = ∑_{ν=1}^{3} J_{ν}(m, n).  (56)

Then

|J_{1}(m, n)| = 1/π^{2}| ∫_{1/m^{τ}}^{π} h(x)(−H_{m}^{α}(x)) dx ∫_{1/n}^{1/n^{τ}} \frac{p(y)}{2 tg(y/2)} dy| ≤

≤C(α)\frac{m^{ατ}}{m^{α}} (ln^{(3)} n^{τ} −ln^{(3)} n)= \frac{C(α)}{m^{α(1−τ)}} ln \frac{ln^{(2)} n^{τ}}{ln^{(2)} n} ;  (57)

|J_{2}(m, n)| = |1/π^{2} ∫_{1/m^{τ}}^{π} \frac{h(x)}{2 tg(x/2)} dx ∫_{1/n}^{1/n^{τ}} p(y)(−H_{n}^{β}(y)) dy| ≤

≤\frac{C(β) ln m}{ln n ln^{(2)} n} ∫_{1/n}^{1/n^{τ}} \frac{dy}{n^{β}y^{β+1}} ;  (58)

|J_{3}(m, n)| = |1/π^{2} ∫_{1/m^{τ}}^{π} ∫_{1/n}^{1/n^{τ}} h(x)p(y)H_{m}^{α}(x)H_{n}^{β}(y) dy dx| ≤
≤C(α, β)\frac{m^{ατ}}{m^{α}} \frac{n^{β}}{n^{β}} .  (59)

(56)–(59) yield

I_{3}(m, n) = o(1) (m, n → ∞).  (60)

Now we consider I_{1}(m, n). As above,

I_{1}(m, n) = ∑_{j=1}^{3} T_{j}(m, n).  (61)

We have

|T_{1}(m, n)| = 1/π^{2}| ∫_{1/m}^{1/m^{τ}} ∫_{1/n}^{1/n^{τ}} h(x)p(y)(−H_{m}^{α}(x))\frac{1}{2} ctg \frac{y}{2} dy dx| ≤
≤ \frac{C_{1}(α)}{ln m_{k} ln^{(2)} m_{k}} \frac{C_{2}(α)}{ln n_{k} ln^{(2)} n_{k}} \frac{m^{α}}{m^{α}} ln n;  (62)
|T_{2}(m, n)| = 1/π^{2}| ∫_{1/m}^{1/m^{τ}} ∫_{1/n}^{1/n^{τ}} h(x)p(y)(−H_{n}^{β}(y))\frac{1}{2} ctg \frac{x}{2} dy dx| ≤
≤ \frac{C_{1}(β)}{ln m_{k} ln^{(2)} m_{k}} \frac{C_{2}(β)}{ln n_{k} ln^{(2)} n_{k}} \frac{n^{β}}{n^{β}} ln m;  (63)
|T_{3}(m, n)| = 1/π^{2}| ∫_{1/m}^{1/m^{τ}} ∫_{1/n}^{1/n^{τ}} h(x)p(y)H_{m}^{α}(x)H_{n}^{β}(y) dy dx| ≤
≤ \frac{C_{1}(α, β)}{ln m_{k} ln^{(2)} m_{k}} \frac{C_{2}(α, β)}{ln n_{k} ln^{(2)} n_{k}} \frac{m^{α}n^{β}}{m^{α}n^{β}} .  (64)

(61)–(64) yield

I_{1}(m, n) = o(1) (m, n → ∞)  (65)

(we remind once more that m = 2m_{k}, n = n_{k} + 1).

Finally, (29), (30), (44)–(46), (55), (60) and (65) prove the Theorem 2 in the case n = 2 and α, β ∈ (0; 1).

For α = 1 we have

H_{n}^{1}(t) = \frac{sin(n + 1)t}{(n + 1)(2 sin \frac{t}{2})^{2}}  (66)

and an estimate analogous to (34) holds true.

Now we consider the case when α > 1. Using the method represented in [4], p.507, we obtain

H_{n}^{α}(t) = \frac{1}{A_{n}^{α}2 sin \frac{t}{2}} Re\{\frac{e^{i(n+1/2)t}}{(1 − e^{−it})^{α}} − e^{− \frac{1}{2} it} ∑_{m=1}^{d} A_{n}^{α−m}(1 − e^{−it})^{−m} −

− ∑_{m=n+1}^{∞} A_{m}^{α−d−1}e^{−i(m−n−1/2)t}(1 − e^{−it})^{−d}},  (67)

where d = [α].

Taking the real part we obtain that the first term of the finite sum is 0. Therefore if [α] = 1 we apply once more the Abel transformation to the infinite sum in (67) and obtain

H_{n}^{α}(t) = \frac{cos((n + \frac{1}{2} + \frac{α}{2})t − \frac{πα}{2})}{A_{n}^{α}(2 sin \frac{t}{2})^{1+α}} − \frac{(1 − α)α cos(t/2)}{8(n + 1)(n + α)(sin \frac{t}{2})^{3}} −

− \frac{1}{A_{n}^{α}} ∑_{m=1}^{∞} A_{m+n+1}^{α−3} \frac{sin(m − 1)t}{(2 sin \frac{t}{2})^{3}}.  (68)

Then, again, we have an estimate analogous to (34), which enables us to fulfil the proof. Namely,

|H_{n}^{α}(t)| ≤ \frac{C_{1}(α)}{n^{α}t^{α+1}} + \frac{C_{2}(α)}{n^{2}t^{3}}.  (69)

If [α] = 2 without further transformation we obtain

H_{n}^{α}(t) = \frac{cos((n + \frac{1}{2} + \frac{α}{2})t − \frac{πα}{2})}{A_{n}^{α}(2 sin \frac{t}{2})^{1+α}} + \frac{(α − 1)α cos(t/2)}{8(n + α − 1)(n + α)(sin \frac{t}{2})^{3}} +

+ \frac{1}{A_{n}^{α}} ∑_{m=1}^{∞} A_{m+n}^{α−3} \frac{cos(m − 3/2)t}{(2 sin \frac{t}{2})^{3}}  (70)

and, again, (69) holds true.

Analogous equations may be obtained if [α] ≥ 3.

Now, when α = 1 and β = 1, we use (66). If α = 1 and β < 1 (or vice versa), we use (34) and (66). If α = 1 and β > 1 (or vice versa), we use (66) and (67) (for the corresponding d). If α > 1 and β < 1 (or vice versa), we use again (67) (for the corresponding d) and (34).

In the n-dimensional case we define f as follows. We set

m_{k} = 2^{2^{2^{k}}} , (k ∈ ℕ);

g_{k}(x) = \begin{cases}\frac{1}{ln^{n−1}(1/(x−π/(6m_{k}))) ln^{(n)}(1/(x−π/(6m_{k})))}, & \frac{π}{6m_{k}} < x ≤ \frac{11π}{60m_{k}} , \\ \frac{1}{ln^{n−1}(1/(π/(5m_{k})−x)) ln^{(n)}(1/(π/(5m_{k})−x))}, & \frac{11π}{60m_{k}} ≤ x < \frac{π}{5m_{k}} , \\ 0, & x ∈ (0; π)\(\frac{π}{6m_{k}}; \frac{π}{5m_{k}} ), \end{cases}

Again

h(x) = ∑_{k=1}^{∞} g_{k}(x).

Then,

p(x_{2}, …, x_{n}) =
= \frac{1}{ln^{n−1} \frac{(2π)^{n−1}}{∏_{i=2}^{n}(π/2−|π/2−x_{i}|)} ln^{(n)} \frac{(2π)^{n−1}}{∏_{i=2}^{n}(π/2−|π/2−x_{i}|)}}  (71)

for (x_{2}, …, x_{n}) ∈ (0; π)^{n−1}.

And, finally

f(x_{1}, …, x_{n}) = \begin{cases}h(x_{1})p(x_{2}, …, x_{n}), & (x_{1}, …, x_{n}) ∈ (0; π)^{n}, \\ 0, & (x_{1}, …, x_{n}) ∈ [−π; π]^{n}\(0; π)^{n}. \end{cases}

Outside [−π; π]^{n} we extend the function f by periodicity with the period 2π in each variable.

We observe, that functions of the p(x_{2}, …, x_{n})-type were for the first time introduced and applied in the works of L.Zhizhiashvili (see [1], [5]).

Remark 1. For the function f(x_{1}, …, x_{n}) a stronger condition than (26) holds true, namely,

ω_{i}(δ; f)_{C} ≤ \frac{C(f, n)}{ln^{n−1}(1/δ) ln^{(n)}(1/δ)}, i = \overline{1, n}.

Remark 2. Results analogous to Theorem 1, the corollary and Theorem 2 hold true for the n-multiple Abel–Poisson summability method.

References

1. L.V. Zhizhiashvili, Conjugate functions and trigonometric series. (Russian) Tbilisi University Press, Tbilisi, 1969.

2. A. Zygmund, Trigonometric series, v.1. (Translated into the Russian) ”Mir”, Moscow, 1965; English original: Cambridge, 1959.

3. G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. (Translated into the Russian) ”IL”, Moscow, 1949; English original: Cambridge, 1934.

4. M.M. Lekishvili, On approximation of periodic functions by (C, α) means. (Russian) Mat.Sb. 121(163)(1983), No.4(8), 499-509.

5. L.V. Zhizhiashvili, On some problems from the theory of simple and multiple trigonometric and orthogonal series. (Russian) Uspekhi Mat.Nauk 28(1973), No.2, 65-119.

(Received 25.02.1993, revized version 13.05.1993)

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