Lobachevskii Journal of Mathematics Vol. 13, 2003, 39 – 43

©Kulwinder Kaur

Kulwinder Kaur
TAUBERIAN CONDITIONS FOR L1-CONVERGENCE OF MODIFIED COMPLEX TRIGONOMETRIC SUMS
(submitted by F. G. Avkhadiev)

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2000 Mathematical Subject Classification. 42A20, 42A32.

ABSTRACT. An L1–convergence property of the complex form gn(c,t) = Sn(c,t) −cnEn(t) + c−nE−n(t) of the modified sums introduced by Garrett and Stanojević [3] is established and a necessary and sufficient condition for L1-convergence of Fourier series is obtained.

1. Introduction.

Let Sn(f,t) and σn (f,t) denotes the nth partial sum and nth Cesàro means of the Fourier series ∑ n<∞cneιnt,t ∈ T = 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 ∈ L1(T), we shall write cn = f̂(n) for all n, and Sn(c,t) = Sn(f,t) = Sn(f). Also gn(c,t) = gn(f,t) = gn(f).

Many authors have defined L1 -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 L1 -norm. An L1 -convergence class is a class of Fourier coefficients {f̂(n)}for which

Sn(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 L1 −convergence class for Fourier series:

lim λ↓1 lim n →∞___ ∑ k=nλn kp−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 L1 −convergence of Fourier series of complex valued Lebesgue integrable functions on T = R 2πZ.

Theorem[3]. Let f ∼∑ n<∞cneιnt be a Fourier series of f ∈ L1(T), whose coefficients satisfy

1 n ∑ k=1n f̂(k) −f̂(−k) log k = o(1)(n →∞), (1.3)
lim λ↓1 lim n →∞___ ∑ k=nλn △f̂(k) −f̂(−k) log k = 0. (1.4)

If for some 1 < p ≤ 2,

lim λ↓1 lim n →∞___ ∑ k=nλnkp−1 Δf̂(k) p = 0, (1.5)

then Sn(f) − f = o(1)(n →∞), if and only if

f̂(n)En(t) + f̂(−n)E−n(t) = o(1)(n →∞) (1.6)

where En(t) = ∑ k=0neιnt.

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 f ̂ (n) log n = o(1) (n →∞).

The object of this paper is to study the L1 -convergence of the complex form

gn(c,t) = Sn(c,t) −f̂(n)En(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)En(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<∞cneιnt be a Fourier series of f̂ ∈ L1(T).

If for some 1 < p ≤ 2,

lim λ↓1 lim n →∞___ ∑ k=nλn kp−1 Δf̂(k) p = 0, (3.1)

then

(i) gn(f,t) − f(t) = o(1)(n →∞).

(ii) Sn(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 L1 (T)-norm:

f = 1 π ∫ 0π f(t) dt

We draw three corollaries of the above theorem:

Corollary 1. If f ∈ L1(T) and for some 1 < p ≤ 2, the limit

lim n→∞1 n ∑ k=1n kp Δf̂(k) p (3.2)

exists and is finite, then we have both part (i) and (ii) of the theorem.

Corollary 2. If f ∈ L1(T) and for some 1 < p ≤ 2,

∑ k=1∞kp−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 ∈ L1(T) and for some 1 < p ≤ 2,

lim λ↓1 lim n →∞___ ∑ k=nλn λn − k λn − np kp−1 Δf̂(k) p = 0, (3.4)

then

lim λ↓1 lim n →∞___ gn(f,t) − f(t) = o(1)(n →∞),

and Sn(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 nλ(f,t) − f(t) = λn+1 λn−n σλn(f,t) − f(t) − n+1 λn−n σn(f,t) − f(t)

where V nλ(f,t) = 1 λn−n ∑ k=n+1λnS k(f,t) is the generalized de Ia Vallée-Poussin means.

And σn(f,t) = 1 n+1 ∑ k=0nS k(f,t)

Since σn(f,t) − f(t) = o(1)(n →∞), then it follows that

V nλ(f,t) − f(t) = o(1)(n →∞).

Consequently it is sufficient to prove that

lim λ↓1 lim n →∞___ V nλ(f,t) − g n(f,t) = 0. (3.5)

Elementary calculation gives

V nλ(f,t) − S n(f,t) = ∑ k=n+1λn λn −k + 1 λn − n f̂(k)eιkt

By (1.7), we have

V nλ(f,t)−g n(f,t) = ∑ k=n+1λn λn −k + 1 λn − n f̂(k)eιkt+f̂(n)E n(t)+f̂(−n)E−n(t) (3.6)

By using summation by parts, we get

∑ k=n+1λnλn−k+1 λn−n f̂(k)eιkt

=∑ k=n+1λn−1△λn−k+1 λn−n f̂(k) Ek(t) + 1 λn−nf̂(λn)Eλn(t) −f̂(n + 1)En(t)

=∑ k=n+1λn−1 λn−k λn−n△f̂(k)Ek(t) + 1 λn−nf̂(k)Ek(t)

+ 1 λn−nf̂(λn)Eλn(t) −f̂(n + 1)En(t)

=∑ k=nλnλn−k λn−n△f̂(k)Ek(t) + 1 λn−n ∑ k=n+1λnf̂(k)E k(t) −f̂(n)En(t)

Similarly

∑ k=n+1λnλn−k+1 λn−n f̂(−k)e−ιkt

=∑ k=nλnλn−k λn−n△f̂(−k)E−k(t) + 1 λn−n ∑ k=n+1λnf̂(−k)E −k(t) −f̂(−n)E−n(t)

Therefore

V nλ(f,t) − g n(f,t)

=∑ k=nλnλn−k λn−n △f̂(k)Ek(t) + 1 λn−n ∑ k=n+1λnf̂(k)E k(t)

Note that

V nλ(f,t) − g n(f,t) = 1 π ∫ 0π n + ∫ π n π V nλ(f,t) − g n(f,t) dt

= I1 + I2

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

I1 ≤ 1 n ∑ k=n+1λnλn−k+1 λn−n f̂(k) ≤ 1 n ∑ k=n+1λn f̂(k) = o(1)(n →∞),

since {f̂(n)} = o(1)(n →∞).

Therefore (3.2) holds if and only if

limλ↓1lim n →∞___ ∫ π n π V nλ(f,t) − g n(f,t) dt = 0.

To estimate I2

V nλ(f,t) − g n(f,t) = ∑ k=nλnλn−k λn−n △f̂(k)Ek(t) + 1 λn−n ∑ k=n+1λnf̂(k)E k(t)

= In1 + In2

After applying Ho¨lder-inequality and then the Hausdorff-Young inequality to In2 , we have

In2 ≤ Cp n λn−n 1 q 1 λn−n ∑ k=n+1λn f̂(k) p 1 p

Similarly

In1 ≤ Bp ∑ k=nλn kp−1 △f̂(k) p 1 p

where Cp and Bp are absolute constants depending on p, and 1 p + 1 q = 1.

Since{f̂(n)} is a null sequence and λ > 1, we have

limλ↓1lim n →∞___In2 = 0.

Hence

gn(f,t) − f(t) ≤gn(f,t) − V nλ(f,t) + V nλ(f,t) − f(t) ≤ λn + 1 λn − n σλn(f,t) − f(t + n + 1 λn − n σn(f,t) − f(t + Bp ∑ k=nλn kp−1 △f̂(k) p 1 p (3.7)

Also as f ∈ L1(T), it follows that σn(f,t) − f(t) = o(1)(n →∞).

Taking lim sup of both sides of (3.6), we have

limn →∞___ gn(f,t) − f(t) ≤ Bp lim n →∞___ ∑ k=nλn kp−1 △f̂(k) p 1 p .

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

lim λ↓1 lim n →∞___ gn(f,t) − f(t) = 0.

(ii) Sn (f,t) − f(t) ≤Sn(f,t) − gn(f,t) + gn(f,t) − f(t)

= gn(f,t) − f(t) + f̂(n)En(t) + f̂(−n)E−n(t) ,

f̂(n)En(t) + f̂(−n)E−n(t) = gn(f,t) − Sn(f,t)

≤gn(f,t) − f(t) + Sn(f,t) − f(t)

Since gn(f,t) − f(t) = o(1), (n →∞) by (i) and by Lemma,

f̂(n)En(t) + f̂(−n)E−n(t) = o(1)(n →∞),

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

Therefore Sn(f,t) − f(t) = o(1)(n →∞), if and only if f̂(n) log n = o(1) (n →∞).

This proves part (ii)

References

  1. W.O. Bray and Č.V. Stanojević, Tauberian L1-convergence class of Fourier series, Trans. Amer. Math.Soc. 275(1983), 59-69
  2. G.A. Fomin, A class of trigonometric series, Mat. Zametki 23 (1978), 213-222.
  3. J.W. Garrett and Č.V. Stanojević, On L1- 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 L1-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

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