Georgian Mathematical Journal Volume 9 (2002), Number 1, 53–56

CESÀRO MEANS OF TRIGONOMETRIC FOURIER SERIES

U. GOGINAVA

Abstract. L. Zhizhiashvili proved that if f ∈ H_{p}^{ω} for some p, 1 ≤ p ≤ ∞, and α ∈ (0, 1) , then the L^{p}-deviation of f from its Cesàro mean is O(n^{α}ω(1/n)) where ω(·) is a modulus of continuity. In this paper we show that this estimation is non-amplifiable for p = 1.

2000 Mathematics Subject Classification: 42A24.

Key words and phrases: Trigonometric Fourier series, Cesàro means.

1. Introduction

Let L^{p}, 1 ≤ p < ∞, denote the collection of all measurable, 2π-periodic functions defined on the [0, 2π) with the norm

∥f∥_{p} = (∫_{0}^{2π}|f (x)|^{p})^{1/p} < ∞,

we write L^{∞} instead of C, the space of continuous and 2π-periodic functions given on [0, 2π) with the norm ∥f∥_{∞} = ∥f∥_{C} = sup_{x∈[0,2π)} |f (x)|.

Let f ∈ L^{p}, 1 ≤ p ≤ ∞. The expression

ω (δ, f)_{p} = sup_{|h|≤δ} ∥f (· + h) − f (·)∥_{p}

is called the L^{p}-modulus of continuity. If ω (δ) is a modulus of continuity, then H_{p}^{ω} denotes the class of functions f ∈ L^{p} for which ω (δ, f)_{p} = O (ω (δ)) as δ → 0+. In particular, H_{p}^{ω} ≡ H_{p}^{δ^{β}} for ω(δ) = δ^{β} (β > 0) and H_{p}^{ω} ≡ H_{p}^{δ} for ω(δ) = δ.

The Cesàro (C, α)-means of trigonometric Fourier series are defined as follows:

σ_{n}^{α} (f, x) = \frac{1}{A_{n}^{α}} ∑_{k=0}^{n} A_{n−k}^{α} (a_{k} cos kx + b_{k} sin kx) ,

where a_{0}, a_{k}, b_{k}, k = 1, 2, …, are the Fourier coefficients and

A_{0}^{α} = 1, A_{n}^{α} = \frac{(α + 1) ⋯ (α + n)}{n!}, α ≠ −1, −2, −3, ….

It is well-known that ([1], Ch. 3)

c_{1}n^{α} < A_{n}^{α} < c_{2}n^{α}  (1)

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with some positive constants c_{1} and c_{2} depending on α.

For the uniform negative order Cesàro summability of trigonometric Fourier series the following result of Zygmund [2] is well-known: if f ∈ H_{∞}^{δ^{β}} and α ∈ (0, β), then the trigonometric Fourier series of the function f is uniformly (C, −α)-summable to f.

L. Zhizhiashvili ([3], Part. 1, Ch. 4) established approximate properties of Cesàro (C, −α)-means with α ∈ (−∞, 1) of trigonometric Fourier series. In particular, for α ∈ (0, 1) he proved the following assertion.

Theorem A. Let α ∈ (0, 1) and f ∈ H_{p}^{ω} for some p, 1 ≤ p ≤ ∞, then

∥f − σ_{n}^{−α} (f)∥_{p} = O(ω (1/n) n^{α}).  (2)

T. Akhobadze [4] has proved that this estimation is non-amplifiable for p = ∞. In the case p = 1 the question of the non-amplifiability of estimation (2) was open (see [3], p. 151). A complete answer to the question is given in this paper.

2. Formulation of the Main Results

Let \{l_{k} : k ≥1\} be a subsequence of natural numbers such that \overline{lim_{l→∞}}ω (1/l) l^{α} =

lim_{k→∞} ω (1/l_{k}) l_{k}^{α}.

Theorem 1. a) Let \frac{ω(δ)}{δ} ↑ ∞ as δ → 0+ and α ∈ (0, 1) . Then there exists a function f_{0} ∈ H_{∞}^{ω} such that

\overline{lim_{k→∞}} \frac{∥f_{0} − σ_{l_{k}}^{−α} (f_{0})∥_{1}}{ω (1/l_{k}) l_{k}^{α}} > 0.

b) Let α ∈ (0, 1) . Then there exists a function g_{0} ∈ H_{1}^{δ} such that

\overline{lim_{n→∞}} \frac{∥g_{0} − σ_{n}^{−α} (g_{0})∥_{1}}{n^{α−1}} > 0.

Theorem 2. Let α ∈ (0, 1) . Then there exists a function f ∈ H_{1}^{ω} such that

\overline{lim_{k→∞}} \frac{∥f − σ_{l_{k}}^{−α} (f)∥_{1}}{ω (1/l_{k}) l_{k}^{α}} > 0.

Corollary 1. Let α ∈ (0, 1) . For all trigonometric Fourier series of the class H_{1}^{ω} to be L^{1}-summable by the Cesàro (C, −α)-method it is necessary and sufficient that

lim_{n→∞} ω (1/n) n^{α} = 0.

Corollary 2. Let α ∈ (0, 1) . Then there exists a continuous function f for which ω (δ, f) = O (δ^{α}) and

\overline{lim_{n→∞}} ∥f − σ_{n}^{−α} (f)∥_{1} > 0.

3. Proofs

Proof of Theorem 1. a) We can define the sequence {n_{k} : k ≥1} ⊂ {l_{k} : k ≥1} satisfying the properties

ω (\frac{1}{n_{k+1}} ) ≤ \frac{1}{2}ω (\frac{1}{n_{k}} ) ,  (3)

∑_{j=1}^{k} ω (\frac{1}{n_{j}} ) n_{j} ≤ ω (\frac{1}{n_{k+1}} ) n_{k+1}.  (4)

Consider the function f_{0} defined by

f_{0} (x) = ∑_{k=1}^{∞} ω (\frac{1}{n_{k}} ) sin n_{k}x.

From (3) it is evident that f_{0} ∈ C. First we shall prove that f_{0} ∈ H_{∞}^{ω}. Let h ∈ [1/n_{s+1}, 1/n_{s}). Since \frac{ω(δ_{1})}{δ_{1}} ≤ 2\frac{ω(δ_{2})}{δ_{2}}, 0 < δ_{2} < δ_{1}, from (3) and (4) it follows that

|f_{0} (x + h) − f_{0} (x)| = 2 |∑_{k=1}^{∞} ω (\frac{1}{n_{k}} ) sin \frac{n_{k}h}{2} cos \frac{2x + h}{2}|

≤ ∑_{k=1}^{s} ω (\frac{1}{n_{k}} ) n_{k}h + 2 ∑_{k=s+1}^{∞} ω (\frac{1}{n_{k}} )

≤ 2ω (\frac{1}{n_{s}} ) n_{s}h + 4ω (\frac{1}{n_{s+1}} ) ≤ 4ω (h) + 4ω (h) = 8ω (h) .

Since b_{n_{k}} (f_{0}) = ω (\frac{1}{n_{k}} ) , we obtain

∥σ_{n_{k}}^{−α} (f_{0}) − f_{0}∥_{1} ≥ |∫_{0}^{2π}[σ_{n_{k}}^{−α} (f_{0}; x) − f_{0} (x)] sin n_{k}xdx|

≥ |∫_{0}^{2π}σ_{n_{k}}^{−α} (f_{0}; x) sin n_{k}xdx| − π |b_{n_{k}} (f_{0})|

= |\frac{1}{A_{n_{k}}^{−α}} ∑_{j=0}^{n_{k}} A_{n_{k−j}}^{−α} ∫_{0}^{2π}(a_{j} (f_{0}) cos jx + b_{j} (f_{0}) sin jx) sin n_{k}xdx|

−π |b_{n_{k}} (f_{0})| = \frac{π}{A_{n_{k}}^{−α}} |b_{n_{k}} (f_{0})| − π |b_{n_{k}} (f_{0})|

= \frac{π}{A_{n_{k}}^{−α}} ω(\frac{1}{n_{k}} ) − πω(\frac{1}{n_{k}} ).  (5)

b) Let

g_{0} (x) = ∑_{k=1}^{∞} \frac{sin kx}{k}.

It is well-known ([1], Ch. I) that g_{0} (x) is the function of bounded variation. According to the Hardy and Littlewood Theorem [5] we conclude that g_{0} (x) ∈ H_{1}^{δ}.

Since b_{n} (g_{0}) = \frac{1}{n}, we write

∥σ_{n}^{−α} (g_{0}) − g_{0}∥_{1} ≥ |∫_{0}^{2π}[σ_{n}^{−α} (g_{0}; x) − g_{0} (x)] sin nxdx|

≥ |∫_{0}^{2π}σ_{n}^{−α} (g_{0}; x) sin nxdx| − π |b_{n} (g_{0})|

= \frac{π}{A_{n}^{−α}} |b_{n} (g_{0})| − π |b_{n} (g_{0})| = \frac{π}{A_{n}^{−α}} \frac{1}{n} − \frac{π}{n}.  (6)

Owing to (1), (5), and (6) the proof of Theorem 1 complete.

The validity of Theorem 2 and Corollary 2 follows immediately from Theorem 1. As for Corollary 1, it follows from Theorem A and Theorem 1.

References

1. A. Zygmund, Trigonometric series, I. Cambridge University Press, Cambridge, 1959.

2. A. Zygmund, Sur la sommabilité des séries de Fourier des functions vérifiant la condition de Lipschitz. Bull. de Acad. Sci. Ser. Math. Astronom. Phys. 1925, 1–9.

3. L. Zhizhiashvili, Trigonometric Fourier series and their conjugates. (Russian) Tbilisi University Press, Tbilisi, 1993; English translation: Kluwer Acad. Publ., 1996.

4. T. Akhobadze, On the convergence and (C, −α)-summability of trigonometric Fourier series.(Russian) Soobshch. Akad. Nauk Gruz. SSR 128(1987), No. 2, 241–244.

5. G. H. Hardy and H. E. Littlewood, Some properties of fractional integral. Math. Z. 27(1928), 565–606.

(Received 18.01.2001)

Author’s address: Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia E-mail: z goginava@hotmail.com