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volume 1, issue 2, article 13, 2000.

Received 22 September, 1999;

accepted 7 March, 2000.

Communicated by:H.M. Srivastava

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Journal of Inequalities in Pure and Applied Mathematics

ON A BOJANIC-STANOJEVIC TYPE INEQUALITY AND ITS APPLICATIONS

Z. TOMOVSKI

Faculty of Mathematical and Natural Sciences P.O. Box 162

91000 Skopje MACEDONIA

EMail:[email protected]

2000c Victoria University ISSN (electronic): 1443-5756 006-99

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On a Bojani ´c–Stanojevi ´c Type Inequality and its Applications

Živorad Tomovski

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J. Ineq. Pure and Appl. Math. 1(2) Art. 13, 2000

http://jipam.vu.edu.au

Abstract

An extension of the Bojani´c–Stanojevi´c type inequality [1] is made by consid- ering ther-th derivate of the Dirichlet kernelD_k^(r)instead ofD_k. Namely, the following inequality is proved

n

X

k=1

αkD_k^(r)(x) 1

≤Mpn^r+1 1 n

n

X

k=1

|α_k|^p

!1/p

,

wherek · k₁is theL¹-norm,{α_k}is a sequence of real numbers,1< p≤ 2, r = 0,1,2, . . . andM_pis an absolute constant dependent only onp. As an application of this inequality, it is shown that the classF_pr is a subclass of BV ∩ C_r, whereF_pris the extension of the Fomin’s class,C_ris the extension of the Garrett–Stanojevi´c class [8] andBVis the class of all null sequences of bounded variation.

2000 Mathematics Subject Classification:26D15, 42A20

Key words: Bojani´c–Stanojevi´c inequality, Sidon–Fomin’s inequality, Bernstein’s in- equality.L¹-convergence cosine series.

1. Introduction

In 1939, Sidon [5] proved his namesake inequality, which is an upper estimate for the integral norm of a linear combination of trigonometric Dirichlet kernels expressed in terms of the coefficients. Since the estimate has many applications, for instance in L¹-convergence problems and summation methods with respect to trigonometric series, newer and newer improvements of the original inequality have been proved by several authors.

Fomin [2], by applying the linear method for summing of Fourier series, gave another proof of the inequality and thus it is known as Sidon-Fomin’s inequality. In addition, S. A. Telyakovskii in [7] has given an elegant proof of Sidon-Fomin’s inequality.

Lemma 1.1. (Sidon-Fomin). Let {α_k}ⁿ_k=0 be a sequence of real numbers such that |αk| ≤ 1for all k. Then there exists a positive constantM such that for anyn≥0,

(1.1)

n

X

k=0

α_kD_k(x) 1

≤M(n+ 1).

In [9] we extended this result and we gave two different proofs of the following lemma.

Lemma 1.2. [9]. Let{α_j}^k_j=0be a sequence of real numbers such that|α_k| ≤1 for allk. Then there exists a positive constantM, such that for anyn ≥0,

(1.2)

n

X

k=0

α_kD^(r)_k (x) 1

≤M(n+ 1)^r+1.

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However, Bojani´c and Stanojevi´c [1] proved the following more general inequality of (1.1).

Lemma 1.3. [1]. Let {α_i}ⁿ_i=0 be a sequence of real numbers. Then for any 1< p ≤2andn ≥0

(1.3)

n

X

k=0

α_kD_k(x) 1

≤M_p(n+ 1) 1 n+ 1

n

X

k=0

|α_k|^p

!1/p

,

where the constantM_pdepends only onp.

We note that this estimate is essentially contained (casep= 2) in Fomin [2].

Sidon-Fomin’s inequality is a special case of the Bojani´c-Stanojevi´c inequality, i.e., it can easily be deduced from Lemma1.3.

It is easy to see that the Bojani´c-Stanojevi´c inequality is not valid forp= 1.

Indeed, if α_n = 1 andα_k = 0 (k 6= n, k ∈ N) then the left side is of order logn/nwhile the right side is of order1/nasn→ ∞.

In order to prove our new results we need the following lemma.

Lemma 1.4. [10]. IfTn(x)is a trigonometrical polynomial of ordern, then kT_n^(r)k ≤n^rkT_nk.

This is S. Bernstein’s inequality in the L¹(0, π)-metric (see [10, Vol. 2, p.11]).

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2. Main Result

Now we will prove a counterpart of inequality (1.3) in the case where ther-th derivate of the Dirichlet’s kernelD_k^(r)is used instead ofD(x).

Theorem 2.1. Let{α_k}ⁿ_k=1 be a sequence of real numbers. Then for any 1 <

p≤2andr= 0,1,2, . . . , n∈Nthe following inequality holds:

(2.1)

n

X

k=1

αkD_k^(r)(x) 1

≤Mpn^r+1 1 n

n

X

k=1

|α_k|^p

!1/p

,

where the constantM_pdepends only onp.

Proof. Applying first the Bernstein inequality and then the Bojani´c-Stanojevi´c inequality, we have

n

X

k=1

αkD^(r)_k (x)

≤n^r

n

X

k=1

αkD^(r)_k (x)

≤Mpn^r+1 1 n

n

X

k=1

|αk|^p

!1/p

.

It is easy to see that the inequality (1.2) is a special case of the inequality (2.1), i.e. it can easily be deduced from Theorem2.1.

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3. Application

The problem of L¹-convergence via Fourier coefficients consists of finding the properties of Fourier coefficients such that the necessary and sufficient condition for kS_n−fk = o(1), n → ∞is given in the form a_nlgn = o(1), n → ∞.

HereS_ndenotes the partial sums of the cosine series a₀

2 +

∞

X

n=1

a_ncosnx .

The Sidon-Telyakovskii class S [7] is a classical example for which the condition a_nlgn = o(1), n → ∞ is equivalent to kS_n− fk = o(1), n → ∞.

Later Fomin [3] extended the Sidon-Telyakovskii class by defining a classF_p, p > 1of Fourier coefficients as follows: a sequence{ak}belongs toFp,p > 1 ifa_k →0ask → ∞and

(3.1)

∞

X

k=1

1 k

∞

X

i=k

|∆a_i|^p

!1/p

<∞.

We note that Fomin [3] has given an equivalent form of the condition (3.1).

Namely, he proved that{a_n} ∈ F_p,p >1iffP∞

s=12^s∆^(p)s <∞, where

∆^(p)_s = ( 1

2^s−1

2^s

X

k=2^s−1+1

|∆a_k|^p )1/p

.

Let BV denote the class of null sequence {a_n} of bounded variation, i.e.

P∞

n=1|∆an|<∞.

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The class C was defined by Garrett and Stanojevi´c [4] as follows: a null sequence of real numbers satisfy the conditionC if for everyε > 0there exists δ(ε)>0independent ofn, such that

δ

Z

0

∞

X

k=n

∆akDk(x)

dx < ε , for every n .

On the other hand, Stanojevi´c [6] proved the following inclusion between the classesF_p,C andBV.

Lemma 3.1. [6]. For all1< p≤2the following inclusion holds:F_p ⊂ BV∩C.

In [8] we defined an extensionC_r,r = 0,1,2, . . . ,of the Garrett-Stanojevi´c class. Namely, a null sequence {a_k}belongs to the classC_r, r = 0,1,2, . . .if for everyε >0there is aδ >0such that

δ

Z

0

∞

X

k=n

∆a_kD_k^(r)(x)

< ε , for all n .

Whenr= 0, we denoteC_r=C.

Denote by I_m the dyadic interval [2^m−1,2^m), for m ≥ 1. A null sequence {an}belongs to the classFpr,p > 1,r= 0,1,2, . . .if

∞

X

m=1

2^m(1/q+r) X

k∈Im

|∆a_k|^p

!1/p

<∞, where 1 p +1

q = 1. It is obvious thatFpr ⊂Fp. Forr= 0, we obtain the Fomin’s classFp.

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Theorem 3.2. For all 1 < p ≤ 2 and r = 0,1,2, . . . the following inclusion holdsFpr ⊂BV ∩Cr.

Proof. By Lemma3.1, it is clear thatF_pr ⊂BV. It suffices to show that

∞

X

k=n

∆akD_k^(r)(x)

=o(1), n → ∞.

Since

∞

X

m=1

2^m(1/q+r) X

k∈Im

|∆a_k|^p

!1/p

= 2

∞

X

m=1

(

2(m−1)[(r+1)p−1] X

k∈Im

|∆a_k|^p )1/p

,

we have

∞

X

k=1

k^(r+1)p−1|∆a_k|^p <∞.

Applying the Theorem2.1, we obtain

∞

X

k=n

∆akD^(r)_k (x)

≤Mp

∞

X

k=n

k^(r+1)p−1|∆ak|^p

!1/p

=o(1), n→ ∞.

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References

[1] R. BOJANI ´CANDC.V. STANOJEVI ´ˇ C, A class ofL¹-convergence, Trans.

Amer. Math. Soc., 269 (1982), 677–683.

[2] G.A. FOMIN, On linear method for summing Fourier series, Mat. Sb (Rus- sian), 66 (107), (1964), 144–152.

[3] G.A. FOMIN, A class of trigonometric series, Math. Zametki (Russian), 23 (1978), 117–124.

[4] J.W. GARRETT AND C.V. STANOJEVI ´ˇ C, Necessary and sufficient con- ditions forL¹convergence of trigonometric series, Proc. Amer. Math. Soc., 60 (1976), 68–72.

[5] S. SIDON, Hinreichende Bedingungen fur den Fouirier charakter einer Trigonometrischen Reihe, J. London, Math. Soc., 14 (1939), 158.

[6] ˇC.V. STANOJEVI ´C, Classes of L¹ convergence of Fourier series and Fourier Stiltjes series, Proc. Amer. Math. Soc., 82 (1981), 209–215.

[7] S.A. TELYAKOVSKII, On a sufficient condition of Sidon for the integra- bility of trigonometric series, Math. Zametki (Russian), (1973), 742–748.

[8] Ž. TOMOVSKI, An extension of the Garrett- Stanojevi´c class, Approx.

Theory Appl., 16(1) (2000), 46–51. [ONLINE] A corrected version is available in the RGMIA Research Report Collection, 3(4), Article 3, 2000.

URL:http://rgmia.vu.edu.au/v3n4.html

[9] Ž. TOMOVSKI, An extension of the Sidon-Fomin inequality and applica- tions, Math. Inequal. Appl., (to appear).

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[10] A. ZYGMUND, Trigonometric Series, Cambridge Univ. Press, 1959.