On L 1 -convergence of certain cosine sums ∗

On L ₁ -convergence of certain cosine sums ^∗

N. L. Braha and Xh. Z. Krasniqi

Abstract

In this paper a criterion forL1- convergence of a certain cosine sums with quasi semi-convex coefficients is obtained. Also a necessary and sufficient condition forL1-convergence of the cosine series is deduced as a corollary.

1 Introduction

It is well known that if a trigonometric series converges inL₁-metric to a function f ∈L₁, then it is the Fourier series of the function f. Riesz [2] gave a counter example showing that in a metric space L₁ we cannot expect the converse of the above said result to hold true. This motivated the various authors to study L1-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 inL1-metric to the sum of the trigonometric series whereas the classical series itself may not. In this contest we will introduce new modified cosine series given by relation

N_n(x) =− 1 2 sin^x₂2

n

X

k=1 n

X

j=k

(∆²a_j−1−∆²a_j) coskx+ a₁ 2 sin^x₂2, and for this modified cosine series we will prove L1-convergence, under condi- tions that coefficients (an) are quasi semi-convex.

2 Preliminaries

In what follows we will denote by

g(x) = a0

2 +

∞

X

k=1

akcoskx, (1)

∗Mathematics Subject Classifications: 42A20, 42A32.

Key words: cosine sums,L1-convergence, quasi semi-convex null sequences.

c

2009 Universiteti i Prishtines, Prishtine, Kosov¨e.

Submitted November, 2009. Published March, 2009.

55

with partial sums defined by

S_n(x) =a₀ 2 +

n

X

k=1

a_kcoskx, (2)

and

g(x) = lim

n→∞S_n(x). (3)

In the sequel we will mention some notations which are useful for the further work. First let us denote

Dn(t) =1 2 +

n

X

k=1

coskt= sin n+¹₂ t 2 sin^t₂ and

Den(t) =

n

X

k=1

coskt.

For all other notations see [11].

Definition 2.1 A sequence of scalars(an)is said to be semi-convex if an→0 asn→ ∞, and

∞

X

n=1

n|∆²a_n−1+ ∆²an|<∞,(a0= 0), (4)

where ∆²a_n= ∆a_n−∆a_n+1.

Definition 2.2 A sequence of scalars(an)is said to be quasi-convex if an→0 asn→ ∞, and

∞

X

n=1

n|∆²a_n−1|<∞,(a0= 0), (5)

Definition 2.3 A sequence of scalars (an) is said to be quasi semi-convex if an→0 asn→ ∞, and

∞

X

n=1

n|∆²an−1−∆²an|<∞,(a0= 0), (6)

where ∆²a_n= ∆a_n−∆a_n+1.

Kolmogorov in [5], proved the following theorem:

Theorem 2.4 If(an)is a quasi-convex null sequence, then for theL1-convergence of the cosine series (1), it is necessary and sufficient thatlim_n→∞an·logn= 0.

The case in which sequence (an) is convex, of this theorem was established by Young (see [10]). That is why, sometimes, this theorem is known as Young- Kolmogorov Theorem.

Remark 2.5 If (a_n) is a quasi-convex null scalar sequence, then it is quasi semi-convex scalars sequence too.

Bala and Ram in [1] have proved that Theorem 2.4 holds true for cosine series with semi-convex null sequences in the following form:

Theorem 2.6 If(an)is a semi-convex null sequence, then for the convergence of the cosine series (1) in the metric spaceL,it is necessary and sufficient that a_k−1logk= 0(1), k→ ∞.

Garret and Stanojevic in [3], have introduced modified cosine sums gn(x) =1

2

n

X

k=0

∆ak+

n

X

k=1 n

X

j=k

(∆aj) coskx. (7)

The same authors (see [4]), Ram in [8] and Singh and Sharma in [9] studied theL1-convergence of this cosine sum under different sets of conditions on the coefficients (an). Kumari and Ram in [7], introduced new modified cosine and sine sums as

fn(x) = a0

2 +

n

X

k=1 n

X

j=k

∆ aj

j

coskx (8)

and

Gn(x) =

n

X

k=1 n

X

j=k

∆ aj

j

sinkx, (9)

and have studied theirL₁-convergence under the condition that the coefficients (a_n) belong to different classes of sequences. Later one, Kulwinder in [6], intro- duced new modified sine sums as

K_n(x) = 1 2 sinx

n

X

k=1 n

X

j=k

(∆a_j−1−∆a_j+1) sinkx, (10)

and have studied theirL1-convergence under the condition that the coefficients (an) are semi-convex null.

3 Results

In this paper we introduce the following modified cosine sums Nn(x) =− 1

2 sin^x₂2 n

X

k=1 n

X

j=k

(∆²aj−1−∆²aj) coskx+ a1

2 sin^x₂2. (11)

The aim of this paper is to study the L1-convergence of this modified cosine sums with quasi semi-convex coefficients and to give necessary and sufficient condition forL1-convergence of the cosine series defined by relation (1).

Theorem 3.1 Let(an)a the quasi semi-convex null sequence, thenNn(x)con- verges tog(x) inL1 norm.

ProofWe have Sn(x) =a0

2 +

n

X

k=1

ak·coskx= 1 2 sin^x₂² ·

n

X

k=1

ak·coskx· 2 sinx

2 2

=− 1

2 sin^x₂² ·

n

X

k=1

ak[·cos (k+ 1)x−2 coskx+cos(k−1)x]

=− 1

2 sin^x₂²·

n

X

k=1

(a_k−1−2ak+ak+1)·coskx− a0cosx

2 sin^x₂²+ancos (n+ 1)x 2 sin^x₂² + a₁

2 sin^x₂2−a_n+1cosnx 2 sin^x₂2 ⇒ S_n(x) =− 1

2 sin^x₂2 ·

n

X

k=1

∆²a_k−1coskx− a₀cosx

2 sin^x₂2+a_ncos (n+ 1)x 2 sin^x₂2 + a1

2 sin^x₂² −an+1cosnx 2 sin^x₂² . Applying Abel’s transformation, we have

Sn(x) =− 1 2 sin^x₂²·

n−1

X

k=1

(∆²a_k−1−∆²ak)Dek(x) +∆²a_n−1·Den(x) 2 sin^x₂²

− a₀cosx

2 sin^x₂2 +a_ncos (n+ 1)x

2 sin^x₂2 + a₁

2 sin^x₂2 −a_n+1cosnx 2 sin^x₂2 .

SinceDe_n(x) is uniformly bounded on every segment [, π−],for every >0, g(x) = lim

n→∞S_n(x) =− 1 2 sin^x₂² ·

∞

X

k=1

(∆²a_k−1−∆²a_k)De_k(x) + a1

2 sin^x₂² Also

Nn(x) =− 1 2 sin^x₂²

n

X

k=1 n

X

j=k

(∆²a_j−1−∆²aj) coskx+ a1

2 sin^x₂² respectively

Nn(x) =− 1 2 sin^x₂²

n

X

k=1

∆²a_k−1coskx+∆²an·Den(x)

2 sin^x₂² + a1

2 sin^x₂².

Now applying Abel’s transformation we get the following relation:

N_n(x) =− 1 2 sin^x₂²

n−1

X

k=1

(∆²a_k−1−∆²a_k)De_k(x)+∆²a_n−1·Den(x)

2 sin^x₂² +∆²an·Den(x) 2 sin^x₂² + a1

2 sin^x₂² From above relation we will have:

g(x)−Nn(x) =− 1 2 sin^x₂²

∞

X

k=n+1

(∆²a_k−1−∆²ak)Dek(x)−∆²an−1·Den(x)

2 sin^x₂² −∆²an·Den(x) 2 sin^x₂² . Thus, we have

Z π

0

|g(x)−Nn(x)|dx→0, forn→ ∞,and definition 1.3.

Corollary 3.2 Let (an)be a quasi-convex null sequence, thenNn(x)converges tog(x)in L1 norm.

ProofProof of the corollary follows directly from Theorem 3.1 and Remark 2.5.

Corollary 3.3 If(an)is a quasi semi-convex null sequence of scalars, then the necessary and sufficient condition for L1-convergence of the cosine series (1) is lim_n→∞anlogn= 0.

ProofLet us start from this estimation:

||Sn(x)−g(x)||L₁ ≤ ||Sn(x)−Nn(x)||L₁+||Nn(x)−g(x)||L₁ =||Nn(x)−g(x)||L₁+

a_ncos (n+ 1)x

2 sin^x₂2 −a_n+1cosnx

2 sin^x₂2 −∆²a_n·De_n(x) 2 sin^x₂2

On the other hand

ancos (n+ 1)x

2 sin^x₂² −an+1cosnx

2 sin^x₂² −∆²an·Den(x) 2 sin^x₂²

= (12)

||Nn(x)−Sn(x)|| ≤ ||Nn(x)−g(x)||+||g(x)−Sn(x)||, and

∆²an =

∞

X

k=n

(∆²ak−∆²ak+1) =

∞

X

k=n

k

k(∆²ak−∆²ak+1)≤ 1

n

∞

X

k=n

(∆²ak−∆²ak+1) =o 1

n

.

Since

Z π

0

Den(x)

2 sin^x₂² =O(n), therefore

∆²a_n· Z π

0

Den(x)

2 sin^x₂² =o(1).

For the rest of the expression (11) we have this estimation:

Z π

0

a_ncos (n+ 1)x

2 sin^x₂2 −a_n+1cosnx 2 sin^x₂2

≤ Z π

0

a_n

cos (n+ 1)x

2 sin^x₂2 − cosnx 2 sin^x₂2

=

= Z π

0

an

Den(x)−1 2

dx∼(anlogn).

From Theorem 3.1 it follows that

||Nn(x)−g(x)||=o(1), n→ ∞.

Finally we get this estimation

n→∞lim Z π

0

|g(x)−S_n(x)|=o(1), if and only if

n→∞lim anlogn= 0, with which was proved corollary.

Corollary 3.4 If(an)is a quasi-convex null sequence of scalars, then the nec- essary and sufficient condition for L1-convergence of the cosine series (1) is lim_n→∞anlogn= 0.

References

[1] R. Bala and B.Ram, Trigonometric series with semi-convex coefficients, Tamang J. Math. 18(1) (1987), 75-84.

[2] Bary K. N., Trigonometric series, Moscow, (1961)(in Russian.)

[3] J. W. Garrett and C. V. Stanojevic, On integrability andL1- convergence of certain cosine sums, Notices, Amer. Math. Soc. 22(1975), A-166.

[4] J. W. Garrett and C. V. Stanojevic, OnL1- convergence of certain cosine sums, Proc. Amer. Math. Soc. 54(1976), 101-105.

[5] A.N. Kolmogorov, Sur l’ordere de grandeur des coefficients de la series de Fourier-Lebesque, Bull.Polon. Sci.Ser.Sci. Math.Astronom.Phys.(1923) 83- 86.

[6] Kulwinder Kaur, OnL1- convergence of modified sine sums, An electronic journal of Geography and Mathematics, Vol. 14, Issue 1, (2003), 1-6.

[7] Kumari Suresh and Ram Babu, L1-convergence modified cosine sum, In- dian J. Pure appl. Math. 19(11) (1988), 1101-1104.

[8] B. Ram, Convergence of certain cosine sums in the metric spaceL, Proc.

Amer. Math.Soc. 66(1977), 258-260.

[9] N. Singh and K.M.Sharma, Convergence of certain cosine sums in the met- ric spaceL, Proc. Amer. Math.Soc. 75(1978), 117-120.

[10] W.H.Young, On the Fourier series of bounded functions, Proc.London Math. Soc. 12(2)(1913), 41-70.

[11] A. Zygmund, Trigonometric series, Vol. 1, Cambridge University Press, Cambridge, 1959.

N. L. Braha

Department of Mathematics and Computer Sciences, Avenue ”Mother Theresa ” 5, Prishtin¨e, 10000, Kosova E-mail address: [email protected]

Xh. Z. Krasniqi

Department of Mathematics and Computer Sciences, Avenue ”Mother Theresa ” 5, Prishtin¨e, 10000, Kosova E-mail address:[email protected]