Lobachevskii Journal of Mathematics Vol. 12, 2003, 3–10 \begin{matrix}c \\ ◯ \end{matrix}S.S. Bhatia, Kulwinder Kaur, Babu Ram

S.S. Bhatia, Kulwinder Kaur, Babu Ram

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

(submitted by F. G. Avkhadiev)

Abstract. In this paper we study L^{1}–convergence of modified complex trigonometric sums introduced by Ram and Kumari [2] and obtain a necessary and sufficient condition for L^{1}-convergence of Fourier series under a new class K* of coefficients.

1. Introduction

It is well known that if a trigonometric series converges in L^{1}-metric to a function f ∈ L^{1}, then it is the Fourier series of the function f. Riesz [1,Vol.II, Ch. VIII § 22] gave a counter example showing that in a metric space L^{1} we cannot expect the converse of the above said result to hold true. This motivated the various authors to study L^{1}-convergence of the trigonometric series. During their investigations some authors introduced modified trigonometric sums as these sums approximate their limits better than the classical trigonometric series in the sense that they converge in L^{1}-metric to the sum of the trigonometric series whereas the classical series itself may not.

Let the partial sums of the complex trigonometric series

∑_{|n|<∞} c_{n}e^{int}, t ∈ T =\frac{R}{2πZ}

be denoted by S_{n}(c, t) = ∑_{|k|<n} c_{k}e^{ikt}.

If the trigonometric series is the Fourier of some f ∈ L^{1}(T), where f = lim_{n→∞} S_{n}(t) we shall write c_{n} = f̂(n) for all n, and S_{n}(c, t) = S_{n}(f, t) = S_{n}(f).

Key words and phrases. L^{1}-convergence of modified complex trigonometric sums, L^{1}-convergence of Fourier series, Dirichlet kernel.

2000 Mathematical Subject Classification. 42A20, 42A32.

Concerning the L^{1}-convergence of the Fourier series Goldberg and Stanojević [3] proved that if

{(f̂(n) − f̂(−n)) log n}  (1.1)

is a null sequence of bounded variation and for some 1 < p ≤ 2,

\frac{1}{n} ∑_{k=n}^{2n} k^{p} |Δˆ/ f(k)|^{p} = o(1)(n → ∞),  (1.2)

then ∥S_{n}(f, t) − f(t)∥ = o(1) (n → ∞), is equivalent to f̂(n) log |n| = o(1) (|n| → ∞).

In this paper instead of (1.1), a weaker condition will be assumed, i.e.,

\frac{1}{[λn]} ∑_{k=1}^{[λn]} (\frac{c_{k} − c_{−k}}{k} ) k log k = o(1), (n → ∞)

lim_{λ↓1} lim_{n→∞} ∑_{k=n}^{[λn]} △ (\frac{c_{k} − c_{−k}}{k} ) k log k = 0,

and (1.2) will be relaxed as follows

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

for some 1 < p ≤ 2.

Kumari and Ram [2] introduced modified cosine and sine sums as

f_{n}(x) = \frac{a_{0}}{2} + ∑_{k=1}^{n} ∑_{j=k}^{n} △(\frac{a_{j}}{j})k cos kx

g_{n}(x) = ∑_{k=1}^{n} ∑_{j=k}^{n} △(\frac{a_{j}}{j})k sin kx

and studied their L^{1}-convergence under different sets of conditions on the coefficients {a_{n}}. Also they deduced as corollaries the L^{1}-convergence of the trigonometric cosine and sine series.

The object of this paper is to study the L^{1}-convergence of the complex form of the above defined sums Kumari and Ram with coefficients belonging to a new class K* of sequence.

Let D_{n}(t), D̃_{n}(t) denote the Dirichlet kernel and conjugate Dirichlet kernel respectively and we shall use the following relations:

D′_{n}(t) = E′_{n}(t) + E′_{−n}(t) 2iD̃n^{′_{n}}(t) = E′_{n}(t) − E′_{−n}(t),

where E′_{n}(t) denotes the first differential of E_{n}(t) = ∑_{k=0}^{n} e^{int}.

The complex form of the above defined modified sums is

g_{n}(c, t) = S_{n}(c, t) + \frac{i}{n + 1} [c_{n+1}E′_{n}(t) − c_{−(n+1)}E′_{−n}(t)]  (1.3)

We introduce here a new class K^{∗}of sequence as follows. Definition. A null sequence of complex numbers belongs to the class K^{∗},

if for some 1 < p ≤ 2,

\frac{1}{[λn]} ∑_{k=1}^{[λn]} (\frac{c_{k} − c_{−k}}{k} ) k log k = o(1), (n → ∞)  (1.4)

lim_{λ↓1} lim_{n→∞} ∑_{k=n}^{[λn]} △ (\frac{c_{k} − c_{−k}}{k} ) k log k = 0,  (1.5)

lim_{λ↓1} lim_{n→∞} ∑_{k=n}^{[λn]} k^{p−1} |Δ (\frac{c_{k}}{k} )|^{p} = 0.  (1.6)

2 Lemmas The following lemmas will be useful for the proof of our result, of which the

first two are due to Sheng [47]:

Lemma 2.1. ∥D′_{n}(t)∥ = \frac{4}{π} (n log n) + O(n)

Lemma 2.2. ∥D̃′_{n}(t)∥ = O(n log n) Lemma 2.3. For each non-negative integer n, there holds

∥\frac{i}{n + 1} (c_{n+1}E′_{n}(t) − c_{−(n+1)}E′_{−n}(t))∥ = o(1), (n → ∞)

if and only if c_{n+1} log |n| = o(1), (|n| → ∞), where {c_{n}} is a complex null sequence.

This Lemma is just the slight modification of the following result of Sheng [47]. For each non-negative integer n, there holds

∥c_{n}E′_{n}(t) + c_{−n)}E′_{−n}(t)∥ = o(1), (n → ∞)

if and only if nc_{n} log |n| = o(1) (|n| → ∞), where {c_{n}} is a complex null sequence.

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

Theorem 3.1. Let {c_{n}} ∈ K^{∗}, then for f ∈ L^{1}(T),

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

∞).

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)),

6 S.S. BHATIA, KULWINDER KAUR, BABU RAM where V_{n}^{λ}(f, t) = \frac{1}{[λn] − n} ∑_{k=n+1}^{[λn]} S_{k}(f, t) is the truncated Cesàro means, and

σ_{n}(f, t) = \frac{1}{n + 1} ∑_{k=0}^{n} S_{k}(f, t). Also, V_{n}^{λ}(f, t)−S_{n}(f, t) = ∑_{|k|=n+1}^{[λn]} \frac{[λn] − |k| + 1}{[λn] − n}f̂(k)e^{ιkt}

With the use of (1.3) we obtain g_{n}(f, t) − V_{n}^{λ}(f, t)

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

Using summation by parts, we get

− ∑_{k=n+1}^{[λn]} \frac{[λn] − k + 1}{[λn] − n}c_{k}e^{ikt}

= i\{ ∑_{k=n+1}^{[λn]−1} \frac{[λn] − k}{[λn] − n}△ (\frac{c_{k}}{k} ) E′_{k}(t) + \frac{1}{[λn] − n} ∑_{k=n+1}^{[λn]} (\frac{c_{k}}{k} ) E′_{k}(t)

− (\frac{c_{n+1}}{n + 1}) E′_{n}(t)\}

Similarly, we have

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

= −i\{ ∑_{k=n+1}^{[λn]−1} \frac{[λn] − k}{[λn] − n}△ (\frac{c_{−k}}{k} ) E′_{−k}(t) + \frac{1}{[λn] − n} ∑_{k=n+1}^{[λn]} (\frac{c_{−k}}{k} ) E′_{−k}(t)

− (\frac{c_{−(n+1)}}{n + 1} ) E′_{−n}(t)\}

Therefore,

g_{n}(f, t) − V_{n}^{λ}(f, t)

= i ( ∑_{k=n+1}^{[λn]−1} \frac{[λn] − k}{[λn] − n}△ (\frac{c_{k}}{k} ) E′_{k}(t) − ∑_{k=n+1}^{[λn]−1} \frac{[λn] − k}{[λn] − n}△ (\frac{c_{−k}}{k} ) E′_{−k}(t))

+ i (\frac{1}{[λn] − n} ∑_{k=n+1}^{[λn]} (\frac{c_{k}}{k} ) E′_{k}(t) − \frac{1}{[λn] − n} ∑_{k=n+1}^{[λn]} (\frac{c_{−k}}{k} ) E′_{−k}(t))

Since E′_{n}(t) =2iD̃′_{n}(t) + E′_{−n}(t) Therefore g_{n}(f, t) − V_{n}^{λ}(f, t)

= i ( ∑_{k=n+1}^{[λn]−1} \frac{[λn] − k}{[λn] − n}△ (\frac{c_{k}}{k} ) 2iD̃′_{k}(t) + ∑_{k=n+1}^{[λn]−1} \frac{[λn] − k}{[λn] − n}△ (\frac{c_{k} − c_{−k}}{k} ) E′_{−k}(t))

+i (\frac{1}{[λn] − n} ∑_{k=n+1}^{[λn]} (\frac{c_{k}}{k} ) 2iD̃′_{k}(t) + \frac{1}{[λn] − n} ∑_{k=n+1}^{[λn]} (\frac{c_{k} − c_{−k}}{k} ) E′_{−k}(t))

(3.1) ∥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)∥

+ ∥\frac{2}{[λn] − n} ∑_{k=n+1}^{[λn]} (\frac{c_{k}}{k} ) D̃′_{k}(t) + \frac{1}{[λn] − n} ∑_{k=n+1}^{[λn]} (\frac{c_{k} − c_{−k}}{k} ) E′_{−k}(t)∥

+ ∥2 ∑_{k=n+1}^{[λn]} \frac{[λn] − k}{[λn] − n}△ (\frac{c_{k}}{k} ) D̃′_{k}(t) + ∑_{k=n+1}^{[λn]} \frac{[λn] − k}{[λn] − n}△ (\frac{c_{k} − c_{−k}}{k} ) E′_{−k}(t)∥

= \frac{[λn] + 1}{[λn] − n} ∥(σ_{[λn]}(f, t) − f(t)∥ + \frac{n + 1}{[λn] − n} ∥(σ_{n}(f, t) − f(t)∥

+I_{1} + I_{2} For the estimate of I_{1},we have

I_{1} = ∥\frac{2}{[λn] − n} ∑_{k=n+1}^{[λn]} (\frac{c_{k}}{k} ) D̃′_{k}(t) + \frac{1}{[λn] − n} ∑_{k=n+1}^{[λn]} (\frac{c_{k} − c_{−k}}{k} ) E′_{−k}(t)∥

≤ \frac{2}{[λn] − n} ∥∑_{k=n}^{[λn]} (\frac{c_{k}}{k} ) D̃′_{k}(t)∥ + \frac{1}{[λn] − n} ∥∑_{k=n}^{[λn]} (\frac{c_{k} − c_{−k}}{k} ) E′_{−k}(t)∥

= J_{11} + \frac{1}{[λn] − n} ∥∑_{k=n}^{[λn]} (\frac{c_{k} − c_{−k}}{k} ) E′_{−k}(t)∥

where

J_{11} = \frac{2}{[λn] − n} ∥∑_{k=n}^{[λn]} (\frac{c_{k}}{k} ) D̃′_{k}(t)∥

Since D̃′_{k}(t) = (k + 1)D_{k}(t) − (k + 1)K_{k}(t) (k + 1)K_{k}(t) = (k + 1)D_{k}(t) − D̃′_{k}(t),

where K_{k}(t) = \frac{1}{k + 1} ∑_{j=0}^{k} D_{j}(t) is the Fejér kernel

As K_{k}(t) ≥ 0 Therefore we have D̃′_{k}(t) ≤ (k + 1)D_{k}(t) This gives

J_{11} = \frac{2}{[λn] − n} ∥∑_{k=n}^{[λn]} (\frac{c_{k}}{k} ) D̃′_{k}(t)∥

≤ \frac{2}{[λn] − n} ∥∑_{k=n}^{[λn]}(k + 1) (\frac{c_{k}}{k} ) D_{k}(t)∥

≤ \frac{2([λn] + 1)}{[λn] − n} ∥∑_{k=n}^{[λn]} (\frac{c_{k}}{k} ) D_{k}(t)∥

Now

∫_{0}^{π} |∑_{k=n}^{[λn]} (\frac{c_{k}}{k} ) D_{k}(t)| dt

= ∫_{0}^{\frac{π}{[λn]}} |∑_{k=n}^{[λn]} (\frac{c_{k}}{k} ) D_{k}(t)| dt + ∫_{\frac{π}{[λn]}}^{π} |∑_{k=n}^{[λn]} (\frac{c_{k}}{k} ) D_{k}(t)| dt

= I_{n}+J_{n}

I_{n} ≤ C_{p} ∑_{k=n}^{[λn]} |(\frac{c_{k}}{k} )|

≤ C_{p} [λn] (\frac{1}{[λn]} ∑_{k=n}^{[λn]} |(\frac{c_{k}}{k} )|^{p})\frac{1}{p}

Therefore

J_{11} ≤ C_{p} [λn] (\frac{1}{[λn]} ∑_{k=n}^{[λn]} |(\frac{c_{k}}{k} )|^{p})\frac{1}{p}

where C_{p} is an absolute constant

J_{n} = ∫_{\frac{π}{[λn]}}^{π} |∑_{k=n}^{[λn]} (\frac{c_{k}}{k} ) D_{k}(t)| dt

= ∫_{\frac{π}{[λn]}}^{π} \frac{1}{sin \frac{t}{2}} |∑_{k=n}^{[λn]} (\frac{c_{k}}{k} ) sin(k + \frac{1}{2})t| dt

After applying Holder-inequality

J_{n} ≤ [ ∫_{\frac{π}{[λn]}}^{π} (\frac{1}{sin \frac{t}{2}} )^{p} dt]\frac{1}{p} [∫_{0}^{π} |∑_{k=n}^{[λn]} (\frac{c_{k}}{k} ) sin(k + \frac{1}{2})t| dt]\frac{1}{q}

and then the Hausdorff-Young inequality, we have

J_{n} ≤ C_{p} [λn]\frac{1}{q} (∑_{k=n}^{[λn]} |(\frac{c_{k}}{k} )|^{p})\frac{1}{p}

= C_{p} [λn] (\frac{1}{[λn]} ∑_{k=n}^{[λn]} |(\frac{c_{k}}{k} )|^{p})\frac{1}{p}

where C_{p} is an absolute constant depending on p, and \frac{1}{p} + \frac{1}{q} = 1

Also Lemma 1 and Lemma 2 imply that ∥E′_{−k}(t)∥ = O(k log k) Hence

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

+C_{2} \frac{[λn]}{[λn] − n} (\frac{1}{[λn]} ∑_{k=n}^{[λn]} |(\frac{c_{k} − c_{−k}}{k} )| k log k)

where C_{1}and C_{2} are absolute constants. Similarly the last term of (3.1) is

I_{2} = ∥2 ∑_{k=n+1}^{[λn]} \frac{[λn] − k}{[λn] − n}△ (\frac{c_{k}}{k} ) D̃′_{k}(t) + ∑_{k=n+1}^{[λn]} \frac{[λn] − k}{[λn] − n}△ (\frac{c_{k} − c_{−k}}{k} ) E′_{−k}(t)∥

≤ C_{3} ∑_{k=n}^{[λn]} |△ (\frac{c_{k} − c_{−k}}{k} )| k log k

+C_{4} (∑_{k=n}^{[λn]} k^{p−1} |(\frac{c_{k}}{k} )|^{p})^{\frac{1}{p}}

where C_{3} and C_{4} are absolute constants. Combining the above results, we get

∥g_{n}(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)∥

+C_{1} \frac{[λn]}{[λn] − n} (\frac{1}{[λn]} ∑_{k=1}^{[λn]} |(\frac{c_{k}}{k} )|^{p})^{\frac{1}{p}}

+C_{2} \frac{[λn]}{[λn] − n} (\frac{1}{[λn]} ∑_{k=n}^{[λn]} |(\frac{c_{k} − c_{−k}}{k} )| k log k)

+C_{3} ∑_{k=n}^{[λn]} |△ (\frac{c_{k} − c_{−k}}{k} )| k log k

(3.2) +C_{4} (∑_{k=n}^{[λn]} k^{p−1} |(\frac{c_{k}}{k} )|^{p})^{\frac{1}{p}}

Since {c_{n}} is a null sequence and λ > 1, we have

\frac{[λn]}{[λn] − n} ˜ \frac{λ}{λ − 1} as n → ∞, it follows that lim_{n→∞} \frac{[λn]}{[λn] − n}c_{n} = 0,

therefore the 3rd term of (3.2) tends to zero as n → ∞. Also as f ∈ L^{1}(T), it follows that ∥σ_{n}(f, t) − f(t)∥ = o(1), as n → ∞ . Taking lim sup of both sides of (3.2) and by condition (1.4), we have

lim_{n→∞} ∥g_{n}(f, t) − f(t)∥ ≤ C_{3} lim_{n→∞} ∑_{k=n}^{[λn]} |△ (\frac{c_{k} − c_{−k}}{k} )| k log k

+C_{4} lim_{n→∞} (∑_{k=1}^{[λn]} k^{p−1} |(\frac{c_{k}}{k} )|^{p})^{\frac{1}{p}}

Now by taking lim as λ → 1 and by conditions (1.5) and (1.6), we have

lim_{n→∞}^{−−−} ∥g_{n}(f, t) − f(t)∥ = 0

We further notice that ∥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)∥ + ∥\frac{i}{n + 1} (c_{n+1}E_{n}(t) − c_{−(n+1)}E_{−n}(t))∥

∥\frac{i}{n + 1} (c_{n+1}E_{n}(t) − c_{−(n+1)}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 (6.3.1)

and by Lemma 3, ∥\frac{i}{n + 1} (c_{n+1}E_{n}(t) − c_{−(n+1)}E_{−n}(t))∥ = o(1)(n → ∞),

if and only if c_{n+1} log |n| = o(1) (|n| → ∞). Therefore ∥S_{n}(f, t) − f(t)∥ = o(1)(|n| → ∞), if and only if c_{n+1} log |n| = o(1) (|n| → ∞), the assertion (ii) follows.

References

(1) Bary N.K, A Treatise on Trigonometric Series (1964) (London: Pergamon Press), Vol.II

(2) Ram B. and Kumari S., On L1-convergence of certain trigonometric sums, Indian J. Pure Appl. Math. 20 (1989) 908- 914.

(3) Goldberg R.R. and Stanojević .V., L1-convergence and Segal algebras of Fourier series, preprint (1980).

(4) Sheng S.Y., The extention of the theorem of ..V. Stanojević and V.B. Stanojević, Proc. Amer. Math. Soc. 110 (1990) 895-905.

School of Mathematics and Computer Applications, Thapar Institute of Engineering and Technology, Post Box No. 32, Patiala (Pb.)-147004. INDIA. Department of Mathematics, Maharshi Dyanand University, Rohtak.INDIA. E-mail address: mathkk@hotmail.com

Received October 14, 2002