M´oricz regarding theΛ-strong convergence of numerical sequences and Fourier series

Analysis

Volume 9, Number 2 (2016), 89–98 doi:10.7153/jca-09-10

ON Λ^r–STRONG CONVERGENCE OF NUMERICAL SEQUENCES AND FOURIER SERIES

P. K ´ORUS

Abstract. We prove theorems of interest about the recently given Λ^r-strong convergence. The main goal is to extend the results of F. M´oricz regarding theΛ-strong convergence of numerical sequences and Fourier series.

1. Introduction

Throughout this paper letΛ={λk:k=0,1,...}be a non-decreasing sequence of positive numbers tending to∞. The concept of Λ-strong convergence was introduced in [2]. We say, that a sequenceS={s_k:k=0,1,...} of complex numbers converges Λ-strongly to a complex number sif

n→∞lim 1 λn

∑

k=0|λk(s_k−s)−λk−1(s_k−1−s)|=0 with the agreementλ−1=s₋₁=0.

It is useful to note that Λ-strong convergence is an intermediate notion between bounded variation and ordinary convergence.

The following generalization was suggested recently in [1]. Throughout this paper, we assume thatr2 is an integer. A sequenceS={sk} of complex numbers is said to convergeΛ^r-strongly to a complex numbersif

n→∞lim 1 λn

∑

|λk(sk−s)−λk−r(sk−r−s)|=0 with the agreementλ−1=...=λ−r=s₋₁=...=s_−r=0.

It was seen that these Λ^r-convergence notions are intermediate notions between Λ-strong convergence and ordinary convergence. The following two basic results were introduced in [1] as Propositions 1 and 2, synthesized from [1, Lemmas 1 and 2].

LEMMA1. A sequence S converges Λ^r-strongly to a number s if and only if (i) S converges to s in the ordinary sense, and

Mathematics subject classification(2010): 40A05, 42A20.

Keywords and phrases: Λ-strong convergence, Λ^r-strong convergence, numerical sequence, Fourier series, Banach space.

c , Zagreb

Paper JCA-09-10 89

(ii) lim

n→∞

1 λn

∑

k=rλk−r|s_k−s_k−r|=0.

LEMMA2. A sequence S converges Λ^r-strongly to a number s if and only if σn:= 1

λn

∑

0knr|n−k

(λk−λk−r)s_k

converges to s in the ordinary sense and condition(ii)is satisfied.

2. Results on numerical sequences

Denote by c^r(Λ)the class of Λ^r-strong convergent sequencesS={sk} of com- plex numbers. Obviously,c^r(Λ)is a linear space. Let

Sc^r(Λ):=sup

n0

1 λn

∑

k=0|λks_k−λk−rs_k−r|, and consider the well-known norms

S∞:=sup

k0|sk|, Sbv:=

∑

k=0|sk−s_k−1|.

It is easy to see that._c^r_(Λ) is also a norm onc^r(Λ).

Moreover, one can easily obtain the inequality

∑

k=0|λks_k−λk−rs_k−r|r

∑

k=0|λks_k−λk−1s_k−1| and the equality

s_k= 1 λn

∑

0knr|n−k

(λks_k−λk−rs_k−r).

These together imply the following result.

PROPOSITION1. For every sequence S={sk} of complex numbers we have S∞Sc^r(Λ)rS_c(Λ)2rSbv.

As a consequence,bv⊂c(Λ)⊂c^r(Λ)⊂c.

It was seen in [2] thatc(Λ)endowed with the norm._c(Λ) is a Banach space. A similar results holds forc^r(Λ).

THEOREM1. The class c^r(Λ)endowed with the norm.c^r(Λ)is a Banach space.

Proof. With an analogous argument to the proof of [2, Theorem 1], we can get the required completeness ofc^r(Λ). The only needed modifications are

1 λn

∑

|λk(sk−s_k)−λk−r(s,k−r−s_k−r)|S−S∞1 λn

∑

(λk+λk−r)ε

and

1 λn

∑

k=0|λk(sjk−s_k)−λk−r(sj,k−r−s_k−r)|

1

λn

∑

k=0|λk(s_jk−s_k)−λk−r(s_j,k−r−s_,k−r)|

+ 1 λn

∑

|λk(sk−sk)−λk−r(s,k−r−s_k−r)|

Sj−Sc^r(Λ)+ε2ε for large enoughand j.

Now that we saw that c^r(Λ) is a Banach space, we show that it has a Schauder basis. In fact, putting

F^(j):= (0,0,...,0, j

1 ,0,0,...,0, j+r

1 ,0,0,...,0, j+2r

1 ,...), j=0,1,..., clearly eachF^(j)∈c^r(Λ).

THEOREM2. {F^(j): j=0,1,...} is a basis in c^r(Λ).

Proof. Existence. We will show that if S={s_k} is a Λ^r-strongly convergent sequence, then

m→∞limS−

∑

j=0(sj−s_j−r)F^(j)c^r(Λ)=0. (1) Since

S−

∑

j=0(sj−s_j−r)F^(j)

= (0,0,..., m

0 ,

m+1

s_m+1−s_m−r+1,s_m+2−s_m−r+2,...,

m+ar+b

s_m+ar+b−s_m−r+b,...),

where 0a, 1br, by definition, S−

∑

j=0(s_j−s_j−r)F^(j)c^r(Λ)

=sup

n1

1 λm+n

∑

1brbn

λm+b|(sm+b−s_m−r+b|

+^[n/r]

∑

a=1

∑

ar+bn1br

|λm+ar+b(s_m+ar+b−s_m−r+b)

−λm+ar−r+b(s_m+ar−r+b−s_m−r+b)|

sup

n1

1 λm+n

^[n/r]

a=0

∑ ∑

ar+bn1br

(λm+ar+b−λm+ar−r+b)|s_m+ar+b−s_m−r+b|

+^[n/r]

∑

a=0

∑

ar+bn1br

λm+ar−r+b|s_m+ar+b−s_m+ar−r+b|

r sup

j,k>m−r|sj−sk|+ sup

nm+1

1 λn

∑

k=m+1λk−r|sk−s_k−r|.

Applying Proposition1and Lemma1, respectively, results in (1) to be proved.

Uniqueness.It can be proved in basically the same way as it was seen in the proof of [2, Theorem 2].

3. Results on Fourier series: C-metric

Denote by C the Banach space of the 2π periodic complex-valued continuous functions endowed with the normfC:=maxt|f(t)|. Let

1 2a0+

∑

k=1(ak(f)coskt+b_k(f)sinkt) (2) be the Fourier series of a function f ∈Cwith the usual notations_k(f) =s_k(f,t)for the kth partial sum of the series (2). Denote byU,A, andS(Λ), respectively, the classes of functions f∈Cwhose Fourier series converges uniformly, converges absolutely and converges uniformlyΛ-strongly on[0,2π), endowed with the usual norms, see [2].

A function f ∈C belongs toS(Λ^r)if

n→∞lim

1 λn

∑

|λk(sk(f)−f)−λk−r(sk−r(f)−f)|

C

=0.

Set the norm

f_S(Λ^r₎:=sup

n0

1 λn

∑

k=0|λks_k(f)−λk−rs_k−r(f)|

_C,

which isfinite for every forS(Λ^r)since f_S(Λ^r₎fC+sup

n0

1 λn

∑

k=0|λk(sk(f)−f)−λk−r(sk−r(f)−f)|

C

. The norm inequalities corresponding to the ones in Proposition1are formulated below.

PROPOSITION2. For every function f ∈C we have fUf_S(Λ^r₎rf_S(Λ)2rfA. As a consequence, A⊂S(Λ)⊂S(Λ^r)⊂U .

The following results are the counterparts to Lemmas1and2and Theorems1and 2, respectively. We omit the details of the analogous proofs, except for Theorem4.

LEMMA3. A function f belongs to S(Λ^r)if and only if (iii) lim

k→∞sk(f)−fC=0, and (iv) lim

n→∞

1 λn

∑

k=rλk−r|sk(f)−s_k−r(f)|

C

=0.

LEMMA4. A function f belongs to S(Λ^r)if and only if (iii’) lim

n→∞σn(f)−fC=0 and condition(iv)is satisfied, where

σn(f) =σn(f,t):= 1 λn

∑

0knr|n−k

(λk−λk−r)s_k(f,t).

THEOREM3. The set S(Λ^r)endowed with the norm._S(Λ^r₎ is a Banach space.

THEOREM4. If f∈S(Λ^r), then

m→∞lims_m(f)−fS(Λ^r)=0. (3)

Proof. Since the sequence of partial sums of the Fourier series of the difference f−sm(f)is

(0,0,..., m

0 ,

m+1

s_m+1(f)−s_m(f),s_m+2(f)−s_m(f),...), then

s_m(f)−fS(Λ^r)

=sup

n1

1 λm+n

∑

1brbn

λm+b|(sm+b(f)−s_m(f)|

+^[n/r]

∑

a=1

∑

ar+bn1br

|λm+ar+b(s_m+ar+b(f)−s_m(f))

−λm+ar−r+b(sm+ar−r+b(f)−sm(f))|

C

sup

n1

1 λm+n

^[n/r]

∑

a=0

∑

ar+bn1br

(λm+ar+b−λm+ar−r+b)|sm+ar+b(f)−s_m(f)|

+^[n/r]

∑

a=0

∑

ar+bn1br

λm+ar−r+b|s_m+ar+b(f)−s_m+ar−r+b(f)|

_C

r sup

j,k>m−rs_j(f)−s_k(f)C+ sup

nm+1

1 λn

∑

k=m+1λk−r|s_k(f)−s_k−r(f)|

C, where 0a, 1br. Applying Proposition2and Lemma3, respectively, results in (3) to be proved.

In the following, our goal is to extend the well-known Denjoy–Luzin theorem presented below (see [3, p. 232]).

THEOREM5. (Theorem of Denjoy–Luzin) If

∑

k=1(a_kcoskt+b_ksinkt) (4)

converges absolutely for t belonging to a set A of positive measure, then

∑

k=1(|ak|+|bk|) converges.

This theorem was extended for Λ-strongly convergent trigonometric series by M´oricz in [2].

THEOREM6. If the nth partial sums sn(t) of the series(4)converge Λ-strongly for t belonging to a set A of positive measure or of second category, then

n→∞lim 1 λn

∑

k=1λk−1(|a_k|+|b_k|) =0. (5) Consequently, if f ∈C and the nth partial sums sn(f,t) of the Fourier series (2) converge uniformlyΛ-strongly to f(t) everywhere, then coefficients a_k=a_k(f) and b_k=b_k(f)satisfy(5).

First, we extend Theorem5for single sine and cosine series.

THEOREM7. If

∑

k=1|a_2k−1cos(2k−1)t+a_2kcos2kt| and

∑

k=1|a_2k−1sin(2k−1)t+a_2ksin 2kt| converge for t belonging to a set A of positive measure, then

∑

k=1|ak|converges.

Proof. We follow the proof of the Denjoy–Luzin theorem as in [3, pp. 232] with necessary modifications. We calculate

a_2k−1cos(2k−1)t+a_2kcos2kt

= (a2k−1+a_2kcost)cos(2k−1)t−(a2ksint)sin(2k−1)t and

a_2k−1sin(2k−1)t+a_2ksin 2kt

= (a_2k−1+a_2kcost)sin(2k−1)t+ (a2ksint)cos(2k−1)t, whence

a_2k−1cos(2k−1)t+a_2kcos2kt=ρk(t)cos((2k−1)t+f_k(t)) and

a_2k−1sin(2k−1)t+a_2ksin 2kt=ρk(t)sin((2k−1)t+f_k(t)) where

ρk(t) = a²_2k−1+a²_2k+2a_2k−1a_2kcost and f_k(t)is from

cosf_k(t) =a_2k−1+a_2kcost

ρk(t) , sinf_k(t) =a_2ksint ρk(t) . Now, we need that

ρk(t)C(|a_2k−1|+|a_2k|) (6)

is satisfied on a set E⊆A of positive measure where the constantCis independent of kandt. Inequality (6) can be obtained from

(1−C²)(a²_2k−1+a²_2k)2|a2k−1a_2k|(C²+|cost|), (7) since it implies

ρ_k²(t)a²_2k−1+a²_2k−2|a_2k−1a_2kcost|C²(|a_2k−1|+|a_2k|)². Hence we just need to defineC small enough so that the setE⊆Aon which

|cost|1−2C²

and consequently (7) and (6) hold is of positive measure. We setCand therebyE that way. SinceE⊆A, there is a setF⊆E of positive measure such that

∑

k=1αk(t) =

∑

k=1

(|a_2k−1cos(2k−1)t+a_2kcos2kt|+|a_2k−1sin(2k−1)t+a_2ksin 2kt|) is bounded onF, say by boundM. Hence we obtain the required estimation

∑

k=1(|a_2k−1|+|a_2k|) 1 C

∑

Fρk(t)

= 1 C

∑

Fρk(t)(cos²((2k−1)t+f_k(t))+sin²((2k−1)t+f_k(t)))dt

1

C

∑

Fρk(t)(|cos((2k−1)t+f_k(t))|+|sin((2k−1)t+f_k(t))|)dt

= 1 C

∑

Fαk(t)dtM

C|F|.

Second, we extend Theorem7forΛ²-strong convergent sine or cosine series.

THEOREM8. If

s¹_n(t) =

∑

k=1

a_kcoskt and s²_n(t) =

∑

k=1

a_ksinkt (8)

convergeΛ²-strongly for t belonging to a set A of positive measure, then

n→∞lim 1 λn

∑

k=1λk−1|ak|=0. (9)

Consequently, if f,g∈C has single Fourier series 1 2a₀+

∑

k=1

a_kcoskt and

∑

k=1

a_ksinkt , respectively, which partial sums converge uniformlyΛ²-strongly to f(t) and g(t) ev- erywhere, then coefficients a_k satisfy(9).

Proof. By Lemma1(in the second case Lemma3is used),Λ²-strong convergence implies for example, in the cosine case that

n→∞lim 1 λn

∑

k=2λk−2|s¹_k−s¹_k−2|=lim

n→∞

1 λn

∑

k=2λk−2|ak−1cos(k−1)t+a_kcoskt|=0, and the proof is analogous to the one of the previous theorem, Theorem7.

4. Results on Fourier series: L^p-metric

The results of Section 3 can be reformulated if we substitute L^p-metric for C- metric. Here and in the sequel 1p<∞. Along with the usual notations let us call a function f ∈L^pto be inS^p(Λ^r)if

n→∞lim 1 λn

∑

k=0|λk(s_k(f)−f)−λk−r(s_k−r(f)−f)|

p

=0,

and introduce the norm

fS^p(Λ^r):=sup

n0

1 λn

∑

k=0|λks_k(f)−λk−rs_k−r(f)|

p

, which isfinite for every forS^p(Λ^r).

The norm inequalities corresponding to the ones in Proposition2are the following.

PROPOSITION3. For every function f ∈L^p and r2 integer we have fU^pfS^p(Λ^r)rfS^p(Λ)2rfA.

As a consequence, A⊂S^p(Λ)⊂S^p(Λ^r)⊂U^p.

The next results are analogous to Lemmas3and4and Theorems3and4, respec- tively.

LEMMA5. A function f belongs to S^p(Λ^r)if and only if (v) lim

k→∞sk(f)−fp=0, and (vi) lim

n→∞

1 λn

∑

k=rλk−r|sk(f)−sk−r(f)|

_p=0.

LEMMA6. A function f belongs to S^p(Λ^r)if and only if (v’) lim

n→∞σn(f)−fp=0 and condition(vi)is satisfied.

THEOREM9. The set S^p(Λ^r)endowed with the norm._S^p_(Λ^r₎ is a Banach space.

THEOREM10. If f ∈S^p(Λ^r), then

m→∞limsm(f)−fS^p(Λ^r)=0.

Finally, we obtain the L^p-metric version of Theorem8.

THEOREM11. If the sums in(8)convergeΛ²-strongly in the L^p-metric restricted to a set of positive measure, then(9)holds true.

Consequently, if f,g∈L^p,1<p<∞, has single Fourier series 1 2a₀+

∑

k=1

a_kcoskt and

∑

k=1

a_ksinkt , respectively, then the partial sums of both series convergeΛ²-strongly to f(t) and g(t)in the L^p-metric if and only if coefficients a_ksatisfy(9).

Proof. Thefirst statement and the necessity part of the second statement is ob- tained in the same way as in the proof of Theorem7.

The sufficiency part of the second statement follows from two facts. First, by the theorem of M. Riesz [3, p. 266], (v) in Lemma5holds. Second, (vi) is also satisfied since

n→∞lim 1 λn

∑

k=2λk−2|s_k(f)−s_k−2(f)|

p

2 lim

n→∞

1 λn

∑

k=1λk−1|a_k|=0. PROBLEM. Can we prove similar statements to the above proved theorems about theΛ^r-strong convergence in the case r>2 as well?

R E F E R E N C E S

[1] P. K ´ORUS,OnΛ²-strong convergence of numerical sequences revisited, Acta Math. Hungar.,148, 1 (2016), 222–227.

[2] F. M ´ORICZ,OnΛ-strong convergence of numerical sequences and Fourier series, Acta Math. Hun- gar.,54, 3–4 (1989), 319–327.

[3] A. ZYGMUND,Trigonometric series, Vol. 1, Cambridge Univ. Press, 1959.

(Received February 22, 2016) P. K´orus

University of Szeged Department of Mathematics Juh´asz Gyula Faculty of Education Hattyas utca 10, H-6725 Szeged, Hungary e-mail:korpet@jgypk.u-szeged.hu

Journal of Classical Analysis www.ele-math.com jca@ele-math.com