On the rate of summability of Fourier series by means of the method of the Voronoi-N¨orlund

Stanis̷law Stoi´ nski

On the rate of summability of Fourier series by means of the method of the Voronoi-N¨orlund

Abstract. In this note we present the theorem on the rate of pointwise summability of Fourier series by means of generalized Voronoi-N¨orlund sums.

2000 Mathematics Subject Classification: 42A10.

Key words and phrases: Fourier series, generalized sum, rate of summability, Lebes- gue point.

1. Preliminaries. Let us denote by 𝑀 = [𝑐𝑚𝑘], 𝑚 = 0, 1, 2, ...; 𝑘 = 0, 1, 2, ..., 𝑚, the infinite triangular matrix, where 𝑐𝑚0 > 0 for 𝑚 = 0, 1, 2, ..., 𝑐𝑚𝑘 ≥ 0 for 𝑚 = 1, 2, ...; 𝑘 = 1, 2, ..., 𝑚. Write 𝐶^𝑚 = ∑𝑚

𝑘=0𝑐𝑚𝑘. For the num- ber series ∑∞

𝑛=0𝑎𝑛 with partial sums 𝐴𝑛 we make the sequence (𝑤𝑚)^∞_𝑚=0, where 𝑤𝑚= 1

𝐶𝑚

∑𝑚 𝑘=0

𝑐𝑚𝑚−𝑘𝐴𝑘

If 𝑤𝑚→ 𝐴 ∈ ℝ, then the number 𝐴 is called the generalized Voronoi-N¨orlund sum of the series ∑∞

𝑛=0𝑎𝑛. In the particular case, when 𝑐𝑚𝑘 = 𝑐𝑘 for 𝑚 = 0, 1, 2, ...;

𝑘 = 0, 1, 2, ..., 𝑚, we have the method of a generalized summation which is linear.

This summation method is regular if and only if 𝑐𝑚/𝐶𝑚→ 0 ([2], [6]).

Let 𝐿^𝑝_2𝜋, 𝑝≥ 1 and 𝐶^2𝜋 be the spaces composed of all 2𝜋-periodic real-valued functions Lebesgue-integrable on the interval [−𝜋, 𝜋] with 𝑝-th power and all 2𝜋- periodic real-valued functions continuous on [−𝜋, 𝜋], respectively. In the case 𝑝 = 1 we shall write 𝐿2𝜋 instead of 𝐿¹_2𝜋. Let us introduce in these spaces the usual norms

𝑓 _𝐿𝑝 =( ∫ ^𝜋

−𝜋∣𝑓(𝑡)∣^𝑝𝑑𝑡)¹_𝑝 , 𝑓

𝐶= max{∣𝑓(𝑡)∣ : −𝜋 ≤ 𝑡 ≤ 𝜋}.

In [4] there were applicated the method of a generalized summation of the Voronoi-N¨orlund to the Fourier series of 𝑓 ∈ 𝐿^2𝜋 putting 𝑐𝑚𝑘 = ( 𝑚

𝑚−𝑘

)𝑠^𝑚−𝑘, 𝑚 = 0, 1, 2, ...; 𝑘 = 0, 1, 2, ..., 𝑚 for 𝑠 ∈ (0, 1]. Let 𝑆^𝑛[𝑓 ](𝑥) denote the 𝑛-th partial sum of the Fourier series

𝑆[𝑓 ](𝑥) = 𝑎0

2 +

∑∞ 𝑚=1

(𝑎𝑚cos 𝑚𝑥 + 𝑏𝑚sin 𝑚𝑥)

of 𝑓 . Linearity and regularity of the method will be maintained if a sequence of mean arithmetic forms

(𝑉 𝑁 )𝑛[𝑓 ](𝑠, 𝑥) = 1 𝑛 + 1

∑𝑛 𝑚=0

1 (1 + 𝑠)^𝑚

∑𝑚 𝑘=0

(𝑚 𝑘

)

𝑠^𝑘𝑆𝑘[𝑓 ](𝑥) =

∫ 𝜋

−𝜋

𝑓 (𝑥 + 𝑡)𝒦^𝑛(𝑠, 𝑡)𝑑𝑡,

where

𝒦^𝑛(𝑠, 𝑡) = 1−(√

1+2𝑠 cos 𝑡+𝑠² 1+𝑠

)𝑛+1

cos(

(𝑛 + 1)arc tg_{1+𝑠 cos 𝑡}^{𝑠 sin 𝑡} )

4𝜋(𝑛 + 1)_1+𝑠^𝑠 (sin₂^𝑡)² for 𝑡∕= 0, is taken into consideration.

Let us write for 𝑓 ∈ 𝐿^2𝜋

𝜑𝑥(𝑡) = 𝑓 (𝑥 + 𝑡)− 2𝑓(𝑥) + 𝑓(𝑥 − 𝑡), Φ𝑥(ℎ) =

∫ ℎ

0 ∣𝜑^𝑥(𝑡)∣𝑑𝑡 (ℎ≥ 0) and let us introduce the following pointwise characteristic

𝜔𝑥(𝛿) = 𝜔𝑥(𝛿; 𝑓 ) = sup

0<ℎ≤𝛿

{1 ℎΦ𝑥(ℎ)

}

(see e.g. [1] and [5]),

˜

𝜔𝑥(𝛿) =𝜔˜𝑥(𝛿; 𝑓 ) = 1 𝛿Φ𝑥(𝛿)

(see e.g. [3]). If lim𝛿→0⁺𝜔˜𝑥(𝛿; 𝑓 ) = 0, then 𝑥 is called a Lebesgue point (L-point) of the second order of 𝑓 . For every L-point 𝑥 of 𝑓 we have 𝜔𝑥(𝛿)→ 0 as 𝛿 → 0⁺. If, in particular 𝑓 ∈ 𝐶^2𝜋, then

(1) 𝜔𝑥(𝛿; 𝑓 )≤ 𝜔²(𝛿; 𝑓 )𝐶.

uniformly in 𝑥 ∈ [−𝜋, 𝜋], where 𝜔²(𝛿; 𝑓 )𝑐 = sup_{∣ℎ∣≤𝛿} 𝜑(ℎ) _𝐶 is its modulus of smoothness. If 𝑓 ∈ 𝐿^𝑝2𝜋, 𝑝 ≥ 1, and 𝜔²(𝛿; 𝑓 )𝐿^𝑝 = sup_{∣ℎ∣≤𝛿} 𝜑(ℎ) _𝐿𝑝 is its integral modulus of smoothness, then

(2)

˜𝜔(𝛿; 𝑓) _𝐿𝑝≤ 𝜔²(𝛿; 𝑓 )𝐿^𝑝.

2. Main results. Now, we present estimates for the rate of poitwise conver- gence of (𝑉 𝑁 )𝑛[𝑓 ](𝑠, 𝑥) at the L-points of 𝑓 .

Let us write

Δ(𝑠, 𝑡) = 1 + 2𝑠 cos 𝑡 + 𝑠²= (1 + 𝑠)²− 4𝑠 sin² 𝑡 2, 𝛿𝑛= 𝛿𝑛(𝑠) = 2(1 + 𝑠)

√

(𝑛 + 1)𝑠 +(

(𝑛 + 1)𝑠(1 + 𝑠))² for 𝑠∈ (0, 1], 𝑛 = 1, 2, ...

Theorem 2.1 If 𝑓 ∈ 𝐿^2𝜋 and 𝑠∈ (0, 1], then at every real 𝑥

∣(𝑉 𝑁)^𝑛[𝑓 ](𝑠, 𝑥)− 𝑓(𝑥)∣ ≤ (1 + 𝑠) 2(𝑛 + 1)𝑠

(𝜔˜𝑥(𝜋) + 2𝜋

∫ 𝜋 𝛿𝑛(𝑠)

˜ 𝜔𝑥(𝑡)

𝑡² 𝑑𝑡) , where 𝑛 is such large that 𝛿𝑛(𝑠)≤ ^𝜋2.

Proof We can write

∣(𝑉 𝑁)^𝑛[𝑓 ](𝑠, 𝑥)− 𝑓(𝑥)∣ ≤( ∫ ^{𝛿𝑛(𝑠)}

0

+

∫ 𝜋 𝛿𝑛(𝑠)

)∣𝜑^𝑥(𝑡)∣𝒦^𝑛(𝑠, 𝑡)𝑑𝑡 = 𝑃𝑛(𝑠, 𝑥) + 𝑄𝑛(𝑠, 𝑥).

Using the estimates

1−( √1 + 2𝑠 cos 𝑡 + 𝑠² 1 + 𝑠

)𝑛+1

≤ 𝑐 sin²( 𝑡 2

) for 𝑐 = 2(𝑛 + 1)𝑠 (1 + 𝑠)² and Δ(𝑠, 𝑡)≤ (1 + 𝑠)², we obtain

𝑃𝑛(𝑠, 𝑥) ≤ (1 + 𝑠)𝜋 16(𝑛 + 1)𝑠

∫ 𝛿𝑛(𝑠) 0 ∣𝜑^𝑥(𝑡)∣

𝑐𝑡²+ 8 sin² (

1

2(𝑛 + 1)arc tg_{1+𝑠 cos 𝑡}^{𝑠 sin 𝑡} )

𝑡² 𝑑𝑡

≤ (1 + 𝑠)𝜋

16(𝑛 + 1)𝑠(𝑐 + 2(𝑛 + 1)²𝑠²)

∫ 𝛿𝑛(𝑠)

0 ∣𝜑^𝑥(𝑡)∣𝑑𝑡 and

𝑄𝑛(𝑠, 𝑥)≤ 1 + 𝑠 2𝜋(𝑛 + 1)𝑠

∫ 𝜋 𝛿𝑛(𝑠)

∣𝜑^𝑥(𝑡)∣

sin²(₂^𝑡)𝑑𝑡≤ 𝜋(1 + 𝑠) 2(𝑛 + 1)𝑠

∫ 𝜋 𝛿𝑛(𝑠)

𝑑Φ𝑥(𝑡) 𝑡² . Integration by parts, gives

𝑄𝑛(𝑠, 𝑥)≤ 𝜋(1 + 𝑠) 2(𝑛 + 1)𝑠

( Φ𝑥(𝜋)

𝜋² −Φ𝑥(𝛿𝑛) 𝛿_𝑛² + 2

∫ 𝜋 𝛿𝑛(𝑠)

Φ𝑥(𝑡) 𝑡³

)𝑑𝑡.

Hence we obtain

∣(𝑉 𝑁)^𝑛[𝑓 ](𝑠, 𝑥)− 𝑓(𝑥)∣ ≤ Φ^𝑥(𝛿𝑛)( (1 + 𝑠) 4(𝑛 + 1)𝑠𝜋( 𝑐

4+1

2(𝑛 + 1)²𝑠²)

− 𝜋(1 + 𝑠) 2(𝑛 + 1)𝑠𝛿²_𝑛

)

+ (1 + 𝑠)

2𝜋(𝑛 + 1)𝑠Φ𝑥(𝜋) +𝜋(1 + 𝑠) (𝑛 + 1)𝑠

∫ 𝜋 𝛿𝑛(𝑠)

Φ𝑥(𝑡) 𝑡³ 𝑑𝑡.

In consideration of 𝛿𝑛 we have our result. _■

Corollary 2.2 If 𝑓 ∈ 𝑋, where 𝑋 = 𝐶^2𝜋 or 𝑋 = 𝐿^𝑝_2𝜋, 𝑝≥ 1, then for 𝑠 ∈ (0, 1]

(3) (𝑉 𝑁)𝑛[𝑓 ](𝑠,⋅) − 𝑓(⋅) _𝑋 ≤ 1 + 𝑠 2(𝑛 + 1)𝑠

(𝜔2(𝜋; 𝑓∣^𝑋+ 2𝜋

∫ 𝜋 𝛿𝑛(𝑠)

𝜔2(𝑡; 𝑓 )𝑋

𝑡² 𝑑𝑡) , where 𝑛 is such large that 𝛿𝑛(𝑠)≤ 𝜋/2.

Proof By the condition (1) it follows the estimate (3) for 𝑋 = 𝐶2𝜋. Appling the Minkowski inequality, we have

(𝑉 𝑁)𝑛[𝑓 ](𝑠;⋅) − 𝑓(⋅)

𝐿^𝑝≤ 1 + 𝑠 2(𝑛 + 1)𝑠

(

˜𝜔⋅(𝜋; 𝑓 )

𝐿^𝑝+ 2𝜋

∫ 𝜋 𝛿𝑛(𝑠)

˜ 𝜔_⋅(𝑡; 𝑓 )

𝑡² 𝑑𝑡

𝐿^𝑝

) . Using the generalized Minkowski inequality and the condition (2), we obtain

∫ 𝜋 𝛿𝑛(𝑠)

˜ 𝜔_⋅(𝑡; 𝑓 )

𝑡² 𝑑𝑡 _𝐿𝑝

≤

∫ 𝜋 𝛿𝑛(𝑠)

𝜔2(𝑡; 𝑓 )𝐿^𝑝

𝑡² 𝑑𝑡.

Hence it follows the estimate (3) for 𝑋 = 𝐿^𝑝_2𝜋. _■ Corollary 2.3 If 𝑓 ∈ 𝐿^2𝜋, then for 𝑠∈ (0, 1]

𝑛→∞lim(𝑉 𝑁 )𝑛[𝑓 ](𝑠; 𝑥) = 𝑓 (𝑥) at every L-point 𝑥 of 𝑓 .

Proof It is known ([5]) that almost all real numbers are the L-points of 𝑓 ∈ 𝐿^2𝜋. Let 𝑥∈ ℝ be the L-point of 𝑓. For an arbitrary 𝜀 > 0 there exists Δ ∈ (0, 𝜋/2] such that 𝜔𝑥(𝛿; 𝑓 ) < 𝑠𝜀/(√

5𝜋) for 0 < 𝛿 < Δ. Since for 𝑠∈ (0, 1] we have 𝛿^𝑛(𝑠)→ 0 with 𝑛→ ∞, so there exists 𝑛⁰∈ ℕ such that 𝛿^𝑛0 = 𝛿𝑛0(𝑠) < Δ. Let us write

∫ 𝜋 𝛿_𝑛0

˜ 𝜔𝑥(𝑡)

𝑡² 𝑑𝑡 =

∫ Δ

𝛿_𝑛0

+

∫ 𝜋

Δ. The following estimations are true

∫ Δ

𝛿_𝑛0

˜ 𝜔𝑥(𝑡)

𝑡² 𝑑𝑡≤ 𝑠𝜀

√5𝜋 ( 1

𝛿𝑛0

− 1 Δ

)

and ∫ 𝜋

Δ

˜ 𝜔𝑥(𝑡)

𝑡² 𝑑𝑡≤ 𝜔𝑥(𝜋) Δ . Hence we have

∣(𝑉 𝑁)^𝑛0[𝑓 ](𝑠, 𝑥)− 𝑓(𝑥)∣ < 𝑊 (𝑥, 𝑛⁰, 𝑠) + 𝜀 2, where

𝑊 (𝑥, 𝑛0, 𝑠) = 1 + 𝑠 2(𝑛0+ 1)𝑠

((2𝜋 Δ + 1

)

𝜔𝑥(𝜋)− 2 𝑠𝜀

√5Δ )

.

Taking the natural number 𝑛1 ≥ 𝑛⁰ such that 𝑊 (𝑥, 𝑛1, 𝑠) < ^𝜀₂, we obtain our

result. _■

Acknowledgement. I would like to express my gratitude to the Reviewer for his invaluable critical remarks and suggestions.

References

[1] S. Aljanˇci´c, R. Bojanic and M. Tomi´c, On the degree of convergence of Fej´er-Lebesgue sums, L’Enseignement Mathematique, Geneve, Tome XV (1969), 21–28.

[2] G. H. Hardy, Divergent series, Oxford 1949.

[3] W. ̷Lenski, On the rate of pointwise strong (𝐶, 𝛼) summability of Fourier series, Colloquia Math. Soc. J´anos Bolyai, 58 Approx. Theory, Kecskem´et (Hungary) (1990), 453–486.

[4] S. Stoi´nski, Approximation of periodic functions by means of certain linear operators with regard to Hausdorff metric, Funct. Approximatio Comment. Math. 1 (1974), 123–131.

[5] R. Taberski, Aproksymacja funkcji wielomianami trygonometrycznymi , Wyd. Nauk. UAM, Pozna´n 1979.

[6] K. Zeller, Theorie der Limitierungsverfahren , Springer Verlag, Berlin-G¨ottingen-Heidelberg 1958.

Stanis̷law Stoi´nski

Faculty of Mathematics and Computer Science, Adam Mickiewicz University Umultowska 87, 61-614 Pozna´n, Poland

E-mail: stoi@amu.edu.pl

(Received: 17.12.2008)