JJ II

volume 7, issue 3, article 86, 2006.

Received 06 March, 2006;

accepted 08 March, 2006.

Communicated by:L. Leindler

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Journal of Inequalities in Pure and Applied Mathematics

CESÁRO MEANS OFN-MULTIPLE TRIGONOMETRIC FOURIER SERIES

USHANGI GOGINAVA

Department of Mechanics and Mathematics Tbilisi State University

Chavchavadze str. 1 Tbilisi 0128, Georgia

EMail:z_goginava@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 068-06

Cesáro Means ofN-multiple Trigonometric Fourier Series

Ushangi Goginava

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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 .

2000 Mathematics Subject Classification:42B08.

Key words: Trigonometric system, Cesáro means, Summability.

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

PN

i=1x²_i1/2

. Denote byC

[0,2]^N

the space of continuous on[0,2π]^N,2π-periodic rel- ative to each variable functions with the following norm

kfk_C = sup

x∈[0,2π]^N

|f(x)|

andL^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

<∞.

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For the casep=∞, byL^p

[0,2π]^N

we meanC

[0,2π]^N

.

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, . . . , xsi−1, xsi, xsi+1, . . . , xN). The expression we get by successive application of operators∆^{s1}(f, x, hs1), . . . ,∆^{sr}(f, x, h_sr)will be denoted by∆^B(f, x, h_s1, . . . , h_sr),i. e.

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

[0,2π]^N

.The expression ωB(δs1, . . . , δsr;f) := sup

|^h_si|^≤δ_si^,i=1,...,r

∆^B(f,·, hs1, . . . , hsr) p

is called a mixed or a particular modulus of continuity in the L^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^p norm 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 its N-dimensional Fourier series with respect to

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the trigonometric system is defined by

∞

X

i1=0

· · ·

∞

X

iN=0

2^−λ(i) X

B⊂M

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

N

Y

j∈B⁰

cosi_jx_j Y

k∈B

sini_kx_k, where

a^(B)_i

1,...,iN = 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 off andλ(i)is the number of those coordinates of the vectori:= (i₁, . . . , i_N)which are equal to zero.

LetSp1,...,pN (f, x)denote the(p1, . . . , pN)-th rectangular partial sums of the N-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, . . . , αN)-means ofN-multiple trigonometric Fourier se- ries defined by

σ_m^α1₁^,...,α_,...,m^N

N(f, x) =

N

Y

i=1

A^α_mⁱ

i

!⁻¹ _m

1

X

p1=0

· · ·

m_N

X

pN=0 N

Y

j=1

A^α_m^j_j−⁻¹_p

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

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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 trigonometric 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 uni- formly(C,−α)summable tof.

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

[0,2π]^N

,(1≤p≤ ∞). The following is proved.

Theorem A (Zhizhiashvili). Letf ∈L^p

[0,2π]^N

for somep∈[1,+∞]and α₁+· · ·+α_N <1, whereα_i ∈(0,1),i= 1,2, . . . , N. If

ω(δ, f)_p =o δ^{α1+···+αN} , then

σ_m^{−α1,...,−αN}

1,...,mN (f)−f

p →0 as m_i → ∞,i= 1, . . . , N.

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In casep=∞the sharpness of TheoremAhas been proved by Zhizhiashvili [2]. The following theorem shows that TheoremAcannot be improved in cases 1≤p < ∞. Moreover, we prove the following

Theorem 1 (for N = 1 see [1]). Let α₁ +· · ·+ α_N < 1 and α_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^−α_m,...,m^1,...,−αN(f₀)−f₀ 1 >0.

Proof. We can define the sequence{n_k: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 functionf0 defined by

f₀(x₁, . . . , x_N) :=

∞

X

j=1

f_j(x₁, . . . , x_N),

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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) ωi(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).

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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 ≥

σ_n^{−α1,...,−αN}

k,...,nk (f_k) 1

(7)

−

k−1

X

j=1

σ_n^{−α1,...,−αN}

k,...,nk (f_j)−f_j C

−

∞

X

j=k+1

σ_n^{−α1,...,−αN}

k,...,nk (f_j) C−

∞

X

j=k

kf_jk_C

=I−II −III −IV.

It is evident that

(8) σ^−α_n ^1,...,−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

.

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Since [2]

σ_n^−α_k,...,n^1,...,−α_k ^N(fj)−fj

C =O X

B⊂M

ωB

1 n_k, fj

C

n

P

s∈B

αs

k

!

and

ω_i 1

n_k, f_j

=O 1

n^α_j^1+···+αN nj

n_k

! ,

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+···+αN)!

=o(1) as k → ∞.

Since

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

N(f_k) = 0, for B ⊂M, B 6=M and

a^(M_i1,...,i⁾

N(f_k) =







n^−α_k ^{1−···−αN}, for i₁ =· · ·=i_N =n_k;

0, otherwise,

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from (1) we have σ_n^{−α1,...,−αN}

k,...,nk (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_n^{−α1,...,−αN}

k,...,n_k (f_k;x₁, . . . , x_N)

N

Y

i=1

sinn_kx_idx₁· · ·dx_N

=

1 A^−αnk¹

· · · 1 A^−αnk^N

nk

X

i1=0

· · ·

nk

X

iN=0 N

Y

j=1

A^−α_nk−¹_i⁻¹_j

× 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^−αn_k¹

· · · 1 A^−αnk^N

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

Combining (7) – (11) we complete the proof of Theorem1.

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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 Publishers, 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.