Lobachevskii Journal of Mathematics Vol. 13, 2003, 39 – 43 \begin{matrix}c \\ ◯ \end{matrix}Kulwinder Kaur

Kulwinder Kaur

TAUBERIAN CONDITIONS FOR L^{1}-CONVERGENCE OF MODIFIED COMPLEX TRIGONOMETRIC SUMS

(submitted by F. G. Avkhadiev)

Abstract. An L^{1}–convergence property of the complex form g_{n}(c, t) = S_{n}(c, t)−[c_{n}E_{n}(t) + c_{−n}E_{−n}(t)] of the modified sums introduced by Garrett and Stanojević [3] is established and a necessary and sufficient condition for L^{1}-convergence of Fourier series is obtained.

1. Introduction.

Let S_{n}(f, t) and σ_{n}(f, t) denotes the n^{th} partial sum and n^{th} Cesàro means of the Fourier series ∑_{|n|<∞} c_{n}e^{ιnt}, t ∈ T = \frac{R}{2πZ} respectively. Define △f̂(n)as

follows: for n > 0, △f̂(n) = f̂(n)−f̂(n+1) and △f̂(−n) = f̂(−n)−f̂(−n−1). If the trigonometric series is the Fourier of some f ∈ L^{1}(T), we shall write c_{n} = f̂(n) for all n, and S_{n}(c, t) = S_{n}(f, t) = S_{n}(f). Also g_{n}(c, t) = g_{n}(f, t) = g_{n}(f).

Many authors have defined L^{1}-convergence classes in terms of the conditions on the sequences of the Fourier coefficients as there exists an integrable function on T whose Fourier series does not converge to itself in L^{1}-norm. An L^{1}-convergence class is a class of Fourier coefficients {f̂(n)}for which

|S_{n}(f, t) − f(t)| = o(1) (n → ∞),  (1.1)

if and only if f̂(n) log |n| = o(1) (|n| → ∞).

The following is the well-known L^{1}−convergence class for Fourier series:

lim_{λ↓1} lim_{n→∞} ∑_{|k|=n}^{[λn]} |k|^{p−1} |Δf̂(k)|^{p} = 0. (1 < p ≤ 2)  (1.2)

The above condition (1.2) is a Tauberian condition of Hardy-Karamata kind and is weaker than those considered by Fomin [2], Kolmogorov [4], Littlewood [5] and Telyakovskii [7]. Stanojevic [6] proved the following Tauberian Theorem for L^{1}−convergence of Fourier series of complex valued Lebesgue integrable functions on T = \frac{R}{2πZ} .

2000 Mathematical Subject Classification. 42A20, 42A32.

Theorem[3]. Let f ∼ ∑_{|n|<∞} c_{n}e^{ιnt} be a Fourier series of f ∈ L^{1}(T), whose

coefficients satisfy

\frac{1}{n} ∑_{k=1}^{n} |f̂(k) − f̂(−k)| log k = o(1) (n → ∞),  (1.3)

lim_{λ↓1} lim_{n→∞} ∑_{k=n}^{[λn]} |△ (f̂(k) − f̂(−k))| log k = 0.  (1.4)

If for some 1 < p ≤ 2,

lim_{λ↓1} lim_{n→∞} ∑_{k=n}^{[λn]} k^{p−1} |Δf̂(k)|^{p} = 0,  (1.5)

then |S_{n}(f) − f| = o(1) (n → ∞), if and only if

|f̂(n)E_{n}(t) + f̂(−n)E_{−n}(t)| = o(1) (n → ∞)  (1.6)

where E_{n}(t) = ∑_{k=0}^{n} e^{ιnt}.

Sequences satisfying conditions (1.3) and (1.4) are called asymptotically even. In the case of even or odd coefficients, condition (1.6) is equivalent with f̂(n) log n = o(1) (n → ∞).

The object of this paper is to study the L^{1}-convergence of the complex form

g_{n}(c, t) = S_{n}(c, t) − [f̂(n)E_{n}(t) + f̂(−n)E_{−n}(t)]  (1.7)

of the modified sums introduced by Garrett and Stanojević [3] and to obtain the above mentioned theorem of

Stanojević [6] without the notion of asymptotic evenness.

2. Lemma.

We shall use the following Lemma for the proof of our result:

Lemma[1]. For each non-negative integer n, there holds

|f̂(n)E_{n}(t) + f̂(−n)E_{−n}(t)| = o(1) (n → ∞)

if and only if

f̂(n) log |n| = o(1) (|n| → ∞),

where {f̂(n)} is a complex null sequence.

3. Main Result The main result of this paper is the following theorem:

Theorem. Let f ∼ ∑_{|n|<∞} c_{n}e^{ιnt} be a Fourier series of f̂ ∈ L^{1}(T).

If for some 1 < p ≤ 2,

lim_{λ↓1} lim_{n→∞} ∑_{|k|=n}^{[λn]} |k|^{p−1} |Δf̂(k)|^{p} = 0,  (3.1)

then (i) |g_{n}(f, t) − f(t)| = o(1) (n → ∞). (ii) |S_{n}(f, t) − f(t)| = o(1) (n → ∞), if and only if

f̂(n) log |n| = o(1) (|n| → ∞).

Here and in the sequel, [.] means the greatest integral part and |.|denotes L^{1}(T)-norm:

|f| = \frac{1}{π} ∫_{0}^{π} |f(t)| dt

We draw three corollaries of the above theorem: Corollary 1. If f ∈ L^{1}(T) and for some 1 < p ≤ 2, the limit

lim_{n→∞} \frac{1}{n} ∑_{k=1}^{n} |k|^{p} |Δf̂(k)|^{p}  (3.2)

exists and is finite, then we have both part (i) and (ii) of the theorem. Corollary 2. If f ∈ L^{1}(T) and for some 1 < p ≤ 2,

∑_{|k|=1}^{∞} |k|^{p−1} |Δf̂(k)|^{p} < ∞,  (3.3)

then we have both part (i) and (ii) of the theorem.

It is well known that (3.3) implies the existence of (3.2), and the latter implies (3.1).

Corollary 3. If f ∈ L^{1}(T) and for some 1 < p ≤ 2,

lim_{λ↓1} lim_{n→∞} ∑_{k=|n|}^{[λn]} (\frac{[λn] − k}{[λn] − n})^{p} |k|^{p−1} |Δf̂(k)|^{p} = 0,  (3.4)

then

lim_{λ↓1} lim_{n→∞} |g_{n}(f, t) − f(t)| = o(1) (|n| → ∞),

and |S_{n}(f, t) − f(t)| = o(1) (|n| → ∞), if and only if

f̂(n) log |n| = o(1) (|n| → ∞).

Also corollary 1 extends for coefficient sequences satisfying (3.4) instead of (3.1).

Proof of Theorem.

Let λ > 1 and n > 1, then we have V_{n}^{λ}(f, t) − f(t) = \frac{[λn]+1}{[λn]−n} (σ_{[λn]}(f, t) − f(t)) − \frac{n+1}{[λn]−n} (σ_{n}(f, t) − f(t))

42 KULWINDER KAUR

where V_{n}^{λ}(f, t) = \frac{1}{[λn]−n} ∑_{k=n+1}^{[λn]} S_{k}(f, t) is the generalized de Ia Vallée-Poussin means.

And σ_{n}(f, t) = \frac{1}{n+1} ∑_{k=0}^{n} S_{k}(f, t)

Since |σ_{n}(f, t) − f(t)| = o(1) (|n| → ∞), then it follows that |V_{n}^{λ}(f, t) − f(t)| = o(1) (|n| → ∞). Consequently it is sufficient to prove that

lim_{λ↓1} lim_{|n|→∞} |V_{n}^{λ}(f, t) − g_{n}(f, t)| = 0.  (3.5)

Elementary calculation gives

V_{n}^{λ}(f, t) − S_{n}(f, t) = ∑_{|k|=n+1}^{[λn]} \frac{[λn] − |k| + 1}{[λn] − n}f̂(k)e^{ιkt}

By (1.7), we have

\begin{matrix}V_{n}^{λ}(f, t) − g_{n}(f, t) = ∑_{|k|=n+1}^{[λn]} \frac{[λn] − |k| + 1}{[λn] − n}f̂(k)e^{ιkt} \\ +f̂(n)E_{n}(t) + f̂(−n)E_{−n}(t) \end{matrix}  (3.6)

By using summation by parts, we get

∑_{k=n+1}^{[λn]} \frac{[λn]−k+1}{[λn]−n} f̂(k)e^{ιkt}

= ∑_{k=n+1}^{[λn]−1} △ (\frac{[λn]−k+1}{[λn]−n} f̂(k)) E_{k}(t) + \frac{1}{[λn]−n} f̂([λn])E_{[λn]}(t) − f̂(n + 1)E_{n}(t)

= ∑_{k=n+1}^{[λn]−1} \frac{[λn]−k}{[λn]−n}△f̂(k)E_{k}(t) + \frac{1}{[λn]−n} f̂(k)E_{k}(t)

+\frac{1}{[λn]−n} f̂([λn])E_{[λn]}(t) − f̂(n + 1)E_{n}(t)

= ∑_{k=n}^{[λn]} \frac{[λn]−k}{[λn]−n}△f̂(k)E_{k}(t) + \frac{1}{[λn]−n} ∑_{k=n+1}^{[λn]} f̂(k)E_{k}(t) − f̂(n)E_{n}(t)

Similarly

∑_{k=n+1}^{[λn]} \frac{[λn]−k+1}{[λn]−n} f̂(−k)e^{−ιkt}

= ∑_{k=n}^{[λn]} \frac{[λn]−k}{[λn]−n}△f̂(−k)E_{−k}(t) + \frac{1}{[λn]−n} ∑_{k=n+1}^{[λn]} f̂(−k)E_{−k}(t) − f̂(−n)E_{−n}(t)

Therefore V_{n}^{λ}(f, t) − g_{n}(f, t)

= ∑_{|k|=n}^{[λn]} \frac{[λn]−|k|}{[λn]−n}△f̂(k)E_{k}(t) + \frac{1}{[λn]−n} ∑_{|k|=n+1}^{[λn]} f̂(k)E_{k}(t)

Note that

|V_{n}^{λ}(f, t) − g_{n}(f, t)| = \frac{1}{π} \{∫_{0}^{\frac{π}{n}} + ∫_{\frac{π}{n}}^{π}\} |V_{n}^{λ}(f, t) − g_{n}(f, t)| dt

= I_{1} + I_{2}

For the first integral, we have the following estimate from (3.5)

I_{1} ≤ \frac{1}{n} ∑_{|k|=n+1}^{[λn]} \frac{[λn]−|k|+1}{[λn]−n} |f̂(k)| ≤ \frac{1}{n} ∑_{|k|=n+1}^{[λn]} |f̂(k)| = o(1) (|n| → ∞),

since {f̂(n)} = o(1) (|n| → ∞). Therefore (3.2) holds if and only if

lim_{λ↓1} lim_{|n|→∞} ∫_{\frac{π}{n}}^{π} |V_{n}^{λ}(f, t) − g_{n}(f, t)| dt = 0.

To estimate I_{2}

V_{n}^{λ}(f, t) − g_{n}(f, t) = ∑_{|k|=n}^{[λn]} \frac{[λn]−|k|}{[λn]−n}△f̂(k)E_{k}(t) + \frac{1}{[λn]−n} ∑_{|k|=n+1}^{[λn]} f̂(k)E_{k}(t)

= I_{n1} + I_{n2}

After applying Hölder-inequality and then the Hausdorff-Young inequality to I_{n2}, we have

I_{n2} ≤ C_{p} (\frac{n}{[λn]−n})^{\frac{1}{q}} (\frac{1}{[λn]−n} ∑_{|k|=n+1}^{[λn]} |f̂(k)|^{p})^{\frac{1}{p}}

Similarly

I_{n1} ≤ B_{p} ( ∑_{|k|=n}^{[λn]} |k|^{p−1} |△f̂(k)|^{p})^{\frac{1}{p}}

where C_{p} and B_{p} are absolute constants depending on p, and \frac{1}{p} + \frac{1}{q} = 1. Since{f̂(n)} is a null sequence and λ > 1, we have

lim_{λ↓1} lim_{|n|→∞}I_{n2} = 0.

Hence

\begin{matrix}|g_{n}(f, t) − f(t)| ≤ |g_{n}(f, t) − V_{n}^{λ}(f, t)| + |V_{n}^{λ}(f, t) − f(t)| \\ ≤ \frac{[λn] + 1}{[λn] − n} |(σ_{[λn]}(f, t) − f(t)| + \frac{n + 1}{[λn] − n} |(σ_{n}(f, t) − f(t)| \\ +B_{p} ( ∑_{|k|=n}^{[λn]} |k|^{p−1} |△f̂(k)|^{p})^{\frac{1}{p}} \end{matrix}  (3.7)

Also as f ∈ L^{1}(T), it follows that |σ_{n}(f, t) − f(t)| = o(1) (|n| → ∞). Taking lim sup of both sides of (3.6), we have

lim_{n→∞} |g_{n}(f, t) − f(t)| ≤ B_{p} lim_{n→∞} ( ∑_{|k|=n}^{[λn]} |k|^{p−1} |△f̂(k)|^{p})^{\frac{1}{p}} .

By taking lim as λ ↓ 1 and by condition (3.1), we obtain

lim_{λ↓1} lim_{|n|→∞} |g_{n}(f, t) − f(t)| = 0.

(ii) |S_{n}(f, t) − f(t)| ≤ |S_{n}(f, t) − g_{n}(f, t)| + |g_{n}(f, t) − f(t)| = |g_{n}(f, t) − f(t)| + |f̂(n)E_{n}(t) + f̂(−n)E_{−n}(t)| ,

|f̂(n)E_{n}(t) + f̂(−n)E_{−n}(t)| = |g_{n}(f, t) − S_{n}(f, t)| ≤ |g_{n}(f, t) − f(t)| + |S_{n}(f, t) − f(t)| Since |g_{n}(f, t) − f(t)| = o(1), (|n| → ∞) by (i) and by Lemma,

|f̂(n)E_{n}(t) + f̂(−n)E_{−n}(t)| = o(1) (n → ∞),

if and only if f̂(n) log |n| = o(1) (|n| → ∞).

Therefore |S_{n}(f, t) − f(t)| = o(1) (|n| → ∞), if and only if f̂(n) log |n| = o(1) (|n| → ∞).

This proves part (ii)

References

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(2) G.A. Fomin, A class of trigonometric series, Mat. Zametki 23 (1978), 213-222.

(3) J.W. Garrett and Č.V. Stanojević, On L^{1}- convergence of certain cosine sums, Proc. Amer. Math. Soc. 54(1976), 101-105.

(4) A.N. Kolmogorov, Sur I’, Ordre des coefficients de Ia serie de Fourier-Lebesgue, Bull. Acad. Polon. Ser. Sci. Math. Astronom Phys. (1923), 83-86.

(5) J.E. Littlewood, The convergence of Abel’s Theorem on power series, Pro. London. Math. Soc., 9(1911), 434-448.

(6) Č.V. Stanojević, Tauberian conditions L^{1}-convergence of Fourier series, Trans. Amer. Math. Soc.271(1982), 237-244.

(7) S.A. Telyakosvskii, On conditions of integrability of multiple trigonometric series, Trudy Mat. Inst.Steklov. 164(1983), 180-188 (Russian).

(8) A. Zygmund. Trigonometric Series, Cambridge University Press, 1959.

Govt. Polytechnic Institute, Bathinda.15100 Punjab. INDIA E-mail address: mathkk@hotmail.com