Zhizhiashvili proved sufficient condition for the Cesáro summability by negative order of N-multiple trigonometric Fourier series in the spaceLp,1≤p

http://jipam.vu.edu.au/

Volume 7, Issue 3, Article 86, 2006

CESÁRO MEANS OF N-MULTIPLE TRIGONOMETRIC FOURIER SERIES

USHANGI GOGINAVA

DEPARTMENT OFMECHANICS ANDMATHEMATICS

TBILISISTATEUNIVERSITY

CHAVCHAVADZE STR. 1 TBILISI0128, GEORGIA

[email protected]

Received 06 March, 2006; accepted 08 March, 2006 Communicated by L. Leindler

ABSTRACT. Zhizhiashvili proved sufficient condition for the Cesáro summability by negative order of N-multiple trigonometric Fourier series in the spaceL^p,1≤p≤ ∞. In this paper we show that this condition cannot be improved .

Key words and phrases: Trigonometric system, Cesáro means, Summability.

2000 Mathematics Subject Classification. 42B08.

Let R^N be N-dimensional Euclidean space. The elements of R^N are denoted by x = (x₁, . . . , x_N), y = (y₁, . . . , y_N), ... . For any x, y ∈ R^N the vector (x₁+y₁, . . . , x_N +y_N) of the spaceR^N is denoted byx+y. Letkxk=

PN

i=1x²_i1/2

. Denote byC

[0,2]^N

the space of continuous on[0,2π]^N,2π-periodic relative to each vari- able functions with the following norm

kfk_C = sup

x∈[0,2π]^N

|f(x)|

and L^p

[0,2π]^N

, (1≤p≤ ∞) are the collection of all measurable, 2π-periodic relative to each variable functionsf defined on[0,2π]^N, with the norms

kfk_p = Z

[0,2π]^N

|f(x)|^pdx ¹_p

<∞.

For the casep=∞, byL^p

[0,2π]^N

we meanC

[0,2π]^N .

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

068-06

Let M := {1,2, . . . , N}, B := {s₁, . . . , s_r}, s_k < s_k+1, k = 1, . . . , r − 1, B ⊂ M, B⁰ :=M\B.Let

∆^{si}(f, x, h_si) :=f(x₁, . . . , x_si−1, x_si+h_si, x_si+1, . . . , x_N)

−f(x₁, . . . , x_si−1, x_si, x_si+1, . . . , x_N). The expression we get by successive application of operators ∆^{s1}(f, x, h_s1), . . . ,

∆^{sr}(f, x, hsr)will be denoted by∆^B(f, x, hs1, . . . , hsr),i. e.

∆^B(f, x, h_s1, . . . , h_sr) := ∆^{sr} ∆^B\{sr}, x, h_sr . Letf ∈L^p

[0,2π]^N

.The expression ω_B(δ_s1, . . . , δ_sr;f) := sup

|^h_si|^≤δ_si^,i=1,...,r

∆^B(f,·, h_s1, . . . , h_sr) p

is called a mixed or a particular modulus of continuity in theL^p norm, when card(B) ∈[2, N]

or card(B) = 1.

The total modulus of continuity of the functionf ∈ L^p

[0,2π]^N

in theL^pnorm is defined by

ω(δ, f)_p = sup

khk≤δ

kf(·+h)−f(·)k_p (1≤p≤ ∞).

Suppose that f is a Lebesgue integrable function on [0,2π]^N, 2π periodic relative to each variable.Then itsN-dimensional Fourier series with respect to the trigonometric system is de- fined by

∞

X

i1=0

· · ·

∞

X

iN=0

2^−λ(i) X

B⊂M

a^(B)_i1,...,iN Y

j∈B⁰

cosijxj

Y

k∈B

sinikxk, where

a^(B)_i1,...,i

N = 1 π^N

Z

[0,2π]^N

f(x)Y

j∈B⁰

cosi_jx_j Y

k∈B

sini_kx_kdx

is the Fourier coefficient of f and λ(i) is the number of those coordinates of the vector i :=

(i₁, . . . , i_N)which are equal to zero.

LetS_p1,...,pN(f, x)denote the(p₁, . . . , p_N)-th rectangular partial sums of theN-dimensional Fourier series with respect to the trigonometric system, i. e.

S_p1,...,pN(f, x) :=

p1

X

i1=0

· · ·

pN

X

iN=0

A_i1,...,iN(f, x), where

A_i1,...,iN(f, x) := 2^−λ(i) X

B⊂M

a^(B)_i

1,...,iN

Y

j∈B⁰

cosi_jx_j Y

k∈B

sini_kx_k.

The Cesáro(C;α₁, . . . , α_N)-means ofN-multiple trigonometric Fourier series defined by σ_m^α1,...,αN

1,...,mN(f, x) =

N

Y

i=1

A^α_mⁱ

i

!⁻¹ _m1 X

p1=0

· · ·

mN

X

pN=0 N

Y

j=1

A^α_m^j_j−⁻¹_p

jS_p1,...,pN(f, x), where

A^α_n = (α+ 1) (α+ 2)· · ·(α+n)

n! , α6=−1,−2, . . . , n = 0,1, . . . .

It is well-known that [4]

(1) c1(α)n^α ≤A^α_n ≤c2(α)n^α.

For the uniform summability of Cesáro means of negative order of one-dimensional trigono- metric Fourier series the following result of Zygmund [3] is well-known: if

ω(δ, f)_C =o(δ^α)

and α ∈ (0,1), then the trigonometric Fourier series of the function f is uniformly (C,−α) summable tof.

In [2] Zhizhiashvili proved sufficient conditions for the convergence of Cesáro means of nega- tive order ofN-multiple trigonometric Fourier series in the spaceL^p

[0,2π]^N

,(1≤p≤ ∞).

The following is proved.

Theorem A (Zhizhiashvili). Letf ∈L^p

[0,2π]^N

for somep∈[1,+∞]andα₁+· · ·+α_N <

1, whereα_i ∈(0,1),i= 1,2, . . . , N. If

ω(δ, f)_p =o δ^{α1+···+αN} , then

σ_m^−α_1,...,m^1,...,−αN

N (f)−f

p →0 as m_i → ∞,i= 1, . . . , N.

In case p = ∞ the sharpness of Theorem A has been proved by Zhizhiashvili [2]. The following theorem shows that Theorem A cannot be improved in cases1≤p <∞. Moreover, we prove the following

Theorem 1 (forN = 1see [1]). Letα₁+· · ·+α_N <1andα_i ∈ (0,1),i= 1,2, . . . , N,then there exists the functionf₀ ∈C

[0,2π]^N

for which

(2) ω(δ, f₀)_C =O δ^{α1+···+αN} and

m→∞lim

σ_m,...,m^{−α1,...,−αN} (f₀)−f₀ 1 >0.

Proof. We can define the sequence{nk :k ≤1} satisfying the properties (3)

∞

X

j=k+1

1

n^α_j^1+···+αN =O

1 n^α_k^1+···+αN

,

(4)

k−1

X

j=1

n^1−(α_j ^{1+···+αN)} =O

n^1−(α_k ^{1+···+αN)}

,

(5) nk−1

n_k < 1 k. Consider the functionf₀defined by

f₀(x₁, . . . , x_N) :=

∞

X

j=1

f_j(x₁, . . . , x_N), where

f_j(x₁, . . . , x_N) := 1 n^α_j^1+···+αN

N

Y

i=1

sinn_jx_i.

From (3) it is easy to show thatf₀ ∈C

[0,2π]^N

.First we shall prove that (6) ω_i(δ, f)_C =O δ^{α1+···+αN}

, i= 1, . . . , N.

Let _n¹

k ≤δ < _n¹

k−1. Then from (3) and (4) we can write that

|f₀(x₁, . . . , x_i−1, x_i+δ, x_i+1, . . . , x_N)−f₀(x₁, . . . , x_i−1, x_i, x_i+1, . . . , x_N)|

≤

∞

X

j=1

1

n^α_j^1+···+αN |sinn_j(x_i+δ)−sinn_jx_i|

≤

k−1

X

j=1

1

n^α_j^1+···+αN |sinnj(xi+δ)−sinnjxi|+ 2

∞

X

j=k

1 n^α_j^1+···+αN

≤

k−1

X

j=1

n_jδ

n^α_j^1+···+αN +O

1 n^α_k^1+···+αN

=O

δn^1−(α_k−1 ^{1+···+αN)} +O

1 n^α_k^1+···+αN

=O δ^{α1+···αN} , which proves (6).

Since

ω(δ, f)_C ≤

N

X

i=1

ω_i(δ, f)_C, we obtain the proof of estimation (2).

Next we shall prove thatσ_n^{−α1,...,−αN}

k,...,nk (f₀)diverge in the metric ofL¹

[0,2π]^N

.It is clear that

σ^−α_n ^1,...,−αN

k,...,nk (f₀)−f₀ 1

(7)

≥

σ^−α_n ^1,...,−αN

k,...,nk (f_k) 1

−

k−1

X

j=1

σ^−α_nk,...,n^1,...,−α_k ^N(f_j)−f_j C −

∞

X

j=k+1

σ^−α_nk,...,n^1,...,−α_k ^N(f_j) C −

∞

X

j=k

kf_jk_C

=I−II−III−IV.

It is evident that

(8) σ_n^{−α1,...,−αN}

k,...,nk (f_j) = 0, j =k+ 1, k+ 2, . . . . Using (3) forIV we have

(9) IV ≤

∞

X

j=k

1

n^α_j^1+···+αN =O

1 n^α_k^1+···+αN

.

Since [2]

σ_n^{−α1,...,−αN}

k,...,nk (f_j)−f_j

C =O X

B⊂M

ω_B 1

n_k, f_j

C

n

P

s∈B

αs

k

!

and

ω_i 1

nk

, f_j

=O 1

n^α_j^1+···+αN n_j nk

! ,

from (4) and (5) we get

II =O 1

n^1−(α_k ^{1+···+αN)}

k−1

X

j=1

n^1−(α_j ^{1+···+αN)}

! (10)

=O 1

n^1−(α_k ^{1+···+αN)}

k−2

X

j=1

n^1−(α_j ^{1+···+αN)}+n^1−(α_k−1 ^{1+···+αN)} n^1−(α_k ^{1+···+αN)}

!

=O n^1−(α_k−1 ^{1+···+αN)} n^1−(α_k ^{1+···+αN)}

!

=O 1

k

^1−(α1+···+α_N)!

=o(1) as k→ ∞.

Since

a^(B)_i

1,...,iN(f_k) = 0, for B ⊂M, B 6=M and

a^(M)_i

1,...,iN(f_k) =







n^−α_k ^{1−···−αN}, for i1 =· · ·=iN =nk;

0, otherwise,

from (1) we have σ_n^{−α1,...,−αN}

k,...,n_k (f_k) 1

(11)

= Z 2π

0

· · · Z 2π

0

σ_n^{−α1,...,−αN}

k,...,nk (f_k;x₁, . . . , x_N)

dx₁· · ·dx_N

≥

Z 2π 0

· · · Z 2π

0

σ^−α_nk,...,n^1,...,−α_k ^N(f_k;x₁, . . . , x_N)

N

Y

i=1

sinn_kx_idx₁· · ·dx_N

=

1 A^−αn_k¹

· · · 1 A^−αnk^N

nk

X

i1=0

· · ·

nk

X

iN=0 N

Y

j=1

A^−α_n ¹⁻¹

k−ij

× Z 2π

0

· · · Z 2π

0

S_i1,...,iN (f_k;x₁, . . . , x_N)

N

Y

i=1

sinn_kx_idx₁· · ·dx_N

=π^N 1 A^−αn_k¹

· · · 1 A^−αnk^N

a^(M_n ⁾

k,...,nk(f_k)

=π^N 1 A^−αnk¹

· · · 1 A^−αn_k^N

n^−α_k ^{1−···−αN} ≥c(α₁, . . . , α_N)>0.

Combining (7) – (11) we complete the proof of Theorem 1.

REFERENCES

[1] U. GOGINAVA, Cesáro means of trigonometric Fourier series, Georg. Math. J., 9 (2002), 53–56.

[2] L.V. ZHIZHIASHVILI, Trigonometric Fourier Series and their Conjugates, Kluwer Academic Pub- lishers, Dobrecht, Boston, London, 1996.

[3] A. ZYGMUND, Sur la sommabilite des series de Fourier des functions verfiant la condition de Lipshitz, Bull. de Acad. Sci. Ser. Math. Astronom. Phys., (1925), 1–9.

[4] A. ZYGMUND, Trigonometric Series, Vol. 1, Cambridge Univ. Press, 1959.