PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 107–113 DOI: https://doi.org/10.2298/PIM1716107Y

A NEW THEOREM ON ABSOLUTE MATRIX SUMMABILITY OF FOURIER SERIES

Şebnem Yildiz

Abstract. We generalize a main theorem dealing with absolute weighted mean summability of Fourier series to the |A, p_{n} |_{k} summability factors of Fourier series under weaker conditions. Also some new and known results are obtained.

1. Introduction

Let ∑ a_{n} be a given infinite series with partial sums (s_{n}). By u_{n}^{α} and t_{n}^{α} we denote the nth Cesàro means of order α, with α > −1, of the sequence (s_{n}) and (na_{n}), respectively, that is (see [6])

u_{n}^{α} = \frac{1}{A_{n}^{α}} ∑_{v=0}^{n} A_{n−v}^{α−1} s_{v} and t_{n}^{α} = \frac{1}{A_{n}^{α}} ∑_{v=0}^{n} A_{n−v}^{α−1} va_{v},

where

A_{n}^{α} = \frac{(α + 1)(α + 2)...(α + n)}{n!} = O(n^{α}), A_{−n}^{α} = 0 for n > 0.

The series ∑ a_{n} is said to be summable |C, α|_{k}, k ⩾ 1, if (see [8, 10])

∑_{n=1}^{∞} n^{k−1} |u_{n}^{α} − u_{n−1}^{α} |^{k} = ∑_{n=1}^{∞} \frac{1}{n}|t_{n}^{α}|^{k} < ∞.

If we take α = 1, then |C, α|_{k} summability reduces to |C, 1|_{k} summability. Let (p_{n}) be a sequence of positive real numbers such that

P_{n} = ∑_{v=0}^{n} p_{v} → ∞ as n → ∞, (P_{−i} = p_{−i} = 0, i ⩾ 1).

2010 Mathematics Subject Classification: 26D15; 42A24; 40F05; 40G99.

Key words and phrases: summability factors, absolute matrix summability, Fourier series, infinite series, Hölder inequality, Minkowski inequality.

Communicated by Gradimir Milovanović.

The sequence-to-sequence transformation t_{n} = \frac{1}{P_{n}} ∑ _{v=0}^{n} p_{v}s_{v} defines the sequence (t_{n}) of the Riesz mean or simply the (N̄, p_{n}) mean of the sequence (s_{n}) generated by the sequence of coefficients (p_{n}) (see [9]).

The series ∑ a_{n} is said to be summable |N̄, p_{n}|_{k}, k ⩾ 1, if (see [1])

∑_{n=1}^{∞} ( \frac{P_{n}}{p_{n}} )^{k−1} |t_{n} − t_{n−1} |^{k} < ∞.

In the special case when p_{n} = 1 for all values of n (resp. k = 1), |N̄, p_{n}|_{k} summability is the same as |C, 1|_{k} (resp. |N̄, p_{n}|) summability.

2. Known Results

The following theorems are dealing with |N̄, p_{n}|_{k} summability factors of infinite series.

Theorem 2.1. [2] Let (p_{n}) be a sequence of positive numbers such that

(2.1)  P_{n} = O(np_{n}) as n → ∞.

Let (X_{n}) be a positive monotonic nondecreasing sequence. If the sequences (X_{n}), (λ_{n}) and (p_{n}) satisfy the conditions

λ_{m}X_{m} = O(1) as m → ∞, (2.2)

∑_{n=1}^{m} nX_{n}|Δ^{2}λ_{n}| = O(1) as m → ∞, (2.3)

∑_{n=1}^{m} \frac{p_{n}}{P_{n}} |t_{n}|^{k} = O(X_{m}) as m → ∞, (2.4)

then the series ∑ a_{n}λ_{n} is summable |N̄, p_{n}|_{k}, k ⩾ 1.

Theorem 2.2. [4] Let (X_{n}) be a positive monotonic nondecreasing sequence. If the sequences (X_{n}), (λ_{n}), and (p_{n}) satisfy the conditions (2.1)–(2.3) and

(2.5)  ∑_{n=1}^{m} \frac{p_{n}}{P_{n}} \frac{|t_{n}|^{k}}{X_{n}^{k−1}} = O(X_{m})  as m → ∞,

then the series ∑ a_{n}λ_{n} is summable |N̄, p_{n}|_{k}, k ⩾ 1.

Remark 2.1. It should be noted that condition (2.5) is reduced to the condition (2.4), when k = 1. When k > 1, condition (2.5) is weaker than condition (2.4) but the converse is not true (see [4] for details).

3. An application of absolute matrix summability to Fourier series

Let A = (a_{nv}) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence s = (s_{n}) to As = (A_{n}(s)), where A_{n}(s) = ∑ _{v=0}^{n} a_{nv} s_{v}, n = 0, 1, … The series ∑ a_{n} is said to be summable |A|_{k}, k ⩾ 1, if (see [13])

∑_{n=1}^{∞} n^{k−1} |Δ̄ A_{n}(s)|^{k} < ∞,

and it is said to be summable |A, p_{n}|_{k}, k ⩾ 1, if (see [12])

∑_{n=1}^{∞} ( \frac{P_{n}}{p_{n}} )^{k−1} |Δ̄ A_{n}(s)|^{k} < ∞.

where Δ̄ A_{n}(s) = A_{n}(s) − A_{n−1} (s).

If we take p_{n} = 1 for all n, then |A, p_{n}|_{k} summability is the same as |A|_{k} summability. Also, if we take a_{nv} = \frac{p_{v}}{P_{n}} , then |A, p_{n}|_{k} summability is the same as |N̄, p_{n}|_{k} summability. For any sequence (λ_{n}) we write Δ^{2}λ_{n} = Δλ_{n} − Δλ_{n+1} and Δλ_{n} = λ_{n} − λ_{n+1} . A sequence (λ_{n}) is said to be of bounded variation, denoted by (λ_{n}) ∈ BV, if ∑ _{n=1}^{∞} |Δλ_{n}| < ∞. Let f(t) be a periodic function with period 2π, and Lebesgue integrable over (−π, π). Write

f(x) ∼ \frac{1}{2}a_{0} + ∑_{n=1}^{∞} (a_{n} cosnx + b_{n} sinnx) = ∑_{n=0}^{∞} C_{n}(x),

ϕ(t) = \frac{1}{2}[f(x + t) + f(x − t)], and ϕ_{α} (t) = \frac{α}{t^{α}} ∫_{0}^{t}(t − u)^{α−1} ϕ(u) du (α > 0).

It is well known that if ϕ(t) ∈ BV(0, π), then t_{n}(x) = O(1), where t_{n}(x) is the (C, 1) mean of the sequence (nC_{n}(x)) (see [7]).

Many works have been done dealing with absolute summability factors of Fourier series (see [3–5, 11]). Among them, in [4], Bor has proved the following theorem dealing with the Fourier series.

Theorem 3.1. If ϕ_{1}(t) ∈ BV(0, π), (X_{n}) is a positive monotonic nondecreasing sequence, the sequences (p_{n}), (λ_{n}) satisfy conditions (2.1)–(2.3) and

∑_{n=1}^{m} \frac{p_{n}}{P_{n}} \frac{|t_{n}(x)|^{k}}{X_{n}^{k−1}} = O(X_{m}) as m → ∞,

then the series ∑ C_{n}(x)λ_{n} is summable |N̄, p_{n}|_{k}, k ⩾ 1.

If we take p_{n} = 1 for all values of n, then we obtain a new result dealing with |C, 1|_{k} summability factors of Fourier series.

4. Main Results

We generalize Theorem 3.1 for |A, p_{n}|_{k} summability factors of Fourier series. Before stating the main theorem, we must first introduce some further notations.

With a normal matrix A = (a_{nv}), we associate two lower semimatrices Ā = (ā_{nv}) and Â = (â_{nv}) where ā_{nv} = ∑ _{i=v}^{n} a_{ni}, n, v = 0, 1, … and â_{00} = ā_{00} = a_{00}, â_{nv} = ā_{nv} −ā_{n−1,v} , n = 1, 2, … We note that Ā and Â are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. So, we have

(4.1)  A_{n}(s) = ∑_{v=0}^{n} a_{nv} s_{v} = ∑_{v=0}^{n} ā_{nv} a_{v} and Δ̄ A_{n}(s) = ∑_{v=0}^{n} â_{nv} a_{v}.

Theorem 4.1. Let k ⩾ 1 and A = (a_{nv}) be a positive normal matrix such that

ā_{n0} = 1, n = 0, 1, …, a_{n−1,v} ⩾ a_{nv} , for n ⩾ v + 1,
a_{nn} = O(p_{n}/P_{n}), â_{n,v+1} = O(v|Δ_{v}(â_{nv} |).

If all the conditions of Theorem 3.1 are satisfied, then the series ∑ C_{n}(x)λ_{n} is summable |A, p_{n}|_{k}, k ⩾ 1.

If we take a_{nv} = \frac{p_{v}}{P_{n}} , then we get Theorem 3.1. We need the following lemma for the proof of our theorem.

Lemma 4.1. [2] Under the conditions of Theorem 2.2 we have

nX_{n}|Δλ_{n}| = O(1) as n → ∞, and ∑_{n=1}^{∞} X_{n}|Δλ_{n}| < ∞.

5. Proof of Theorem 4.1

Let (I_{n}(x)) denote the A-transform of the series ∑ _{n=1}^{∞} C_{n}(x)λ_{n}. Then, by (4.1), we have Δ̄ I_{n}(x) = ∑ _{v=1}^{n} â_{nv} C_{v}(x)λ_{v}. Applying Abel’s transformation to this sum, we get

Δ̄ I_{n}(x) = ∑_{v=1}^{n} â_{nv} C_{v}(x)λ_{v} \frac{v}{v} = ∑_{v=1}^{n−1} Δ_{v}( \frac{â_{nv} λ_{v}}{v} ) ∑_{r=1}^{v} rC_{r}(x) + \frac{â_{nn} λ_{n}}{n} ∑_{r=1}^{n} rC_{r}(x)

= ∑_{v=1}^{n−1} Δ_{v}( \frac{â_{nv} λ_{v}}{v} ) (v + 1)t_{v}(x) + â_{nn} λ_{n} \frac{n + 1}{n} t_{n}(x)

= ∑_{v=1}^{n−1} Δ_{v}(â_{nv})λ_{v}t_{v}(x)\frac{v + 1}{v} + ∑_{v=1}^{n−1} â_{n,v+1} Δλ_{v}t_{v}(x)\frac{v + 1}{v}

+ ∑_{v=1}^{n−1} â_{n,v+1} λ_{v+1} \frac{t_{v}(x)}{v} + a_{nn} λ_{n}t_{n}(x)\frac{n + 1}{n}

= I_{n,1} (x) + I_{n,2} (x) + I_{n,3} (x) + I_{n,4} (x).

To complete the proof of Theorem 4.1, by Minkowski’s inequality, it is sufficient to show that

∑_{n=1}^{∞} ( \frac{P_{n}}{p_{n}} )^{k−1} |I_{n,r} (x)|^{k} < ∞, for r = 1, 2, 3, 4.

First, by applying Hölder’s inequality with indices k and k′, where k > 1 and \frac{1}{k} + \frac{1}{k′} = 1, we have

∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} |I_{n,1} (x)|^{k} ⩽ ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} \{ ∑_{v=1}^{n−1} |\frac{v + 1}{v} ||Δ_{v}(â_{nv})||λ_{v}||t_{v}(x)|\}^{k}

= O(1) ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} ∑_{v=1}^{n−1} |Δ_{v}(â_{nv})||λ_{v}|^{k}|t_{v}(x)|^{k}

× { ∑_{v=1}^{n−1} |Δ_{v}(â_{nv})|}^{k−1}

= O(1) ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} a_{nn}^{k−1} \{ ∑_{v=1}^{n−1} |Δ_{v}(â_{nv})||λ_{v}|^{k}|t_{v}(x)|^{k}\}

= O(1) ∑_{v=1}^{m} |λ_{v}|^{k−1} |λ_{v}||t_{v}(x)|^{k} ∑_{n=v+1}^{m+1} |Δ_{v}(â_{nv})|

= O(1) ∑_{v=1}^{m} \frac{1}{X_{v}^{k−1}} |λ_{v}||t_{v}(x)|^{k} \frac{p_{v}}{P_{v}}

= O(1) ∑_{v=1}^{m−1} Δ|λ_{v}| ∑_{r=1}^{v} \frac{p_{r}}{P_{r}} \frac{|t_{r}(x)|^{k}}{X_{r}^{k−1}} + O(1)|λ_{m}| ∑_{v=1}^{m} \frac{p_{v}}{P_{v}} \frac{|t_{v}(x)|^{k}}{X_{v}^{k−1}}

= O(1) ∑_{v=1}^{m−1} |Δλ_{v}|X_{v} + O(1)|λ_{m}|X_{m} = O(1) as m → ∞,

by virtue of the hypotheses of Theorem 4.1 and Lemma 4.1. Now, using Hölder’s inequality we have

∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} |I_{n,2} (x)|^{k} ⩽ ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} \{ ∑_{v=1}^{n−1} |\frac{v + 1}{v} ||â_{n,v+1} ||Δλ_{v}||t_{v}(x)|\}^{k}

= O(1) ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} \{ ∑_{v=1}^{n−1} |â_{n,v+1} ||Δλ_{v}||t_{v}(x)|\}^{k}

= O(1) ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} ∑_{v=1}^{n−1} (v|Δλ_{v}|)^{k}|Δ_{v}(â_{nv})||t_{v}(x)|^{k}

× { ∑_{v=1}^{n−1} |Δ_{v}(â_{nv})|}^{k−1}

= O(1) ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} a_{nn}^{k−1} ∑_{v=1}^{n−1} (v|Δλ_{v}|)^{k}|Δ_{v}(â_{nv})||t_{v}(x)|^{k}

= O(1) ∑_{v=1}^{m} (v|Δλ_{v}|)^{k−1} (v|Δλ_{v}|)|t_{v}(x)|^{k} ∑_{n=v+1}^{m+1} |Δ_{v}(â_{nv})|

= O(1) ∑_{v=1}^{m} \frac{p_{v}}{P_{v}} \frac{1}{X_{v}^{k−1}} |t_{v}(x)|^{k}(v|Δλ_{v}|)

= O(1) ∑_{v=1}^{m−1} Δ(v|Δλ_{v}|) ∑_{r=1}^{v} \frac{p_{r}}{P_{r}} \frac{1}{X_{r}^{k−1}} |t_{r}(x)|^{k}

+ O(1)m|Δλ_{m}| ∑_{v=1}^{m} \frac{p_{v}}{P_{v}} \frac{1}{X_{v}^{k−1}} |t_{v}(x)|^{k}

= O(1) ∑_{v=1}^{m−1} |Δ(v|Δλ_{v}|)|X_{v} + O(1)m|Δλ_{m}|X_{m}

= O(1) ∑_{v=1}^{m−1} vX_{v}|Δ^{2}λ_{v}| + O(1) ∑_{v=1}^{m−1} X_{v}|Δλ_{v}| + O(1)m|Δλ_{m}|X_{m} = O(1)

as m → ∞, by virtue of the hypotheses of Theorem 4.1 and Lemma 4.1. Again, we have that

∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} |I_{n,3} (x)|^{k} = ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} | ∑_{v=1}^{n−1} â_{n,v+1} λ_{v+1} \frac{t_{v}(x)}{v} |^{k}

⩽ ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} \{ ∑_{v=1}^{n−1} |â_{n,v+1} ||λ_{v+1} |\frac{|t_{v}(x)|}{v}\}^{k}

= O(1) ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} \{ ∑_{v=1}^{n−1} |Δ_{v}(â_{nv})||λ_{v+1} ||t_{v}(x)|\}^{k}

= O(1) ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} ∑_{v=1}^{n−1} |Δ_{v}(â_{nv})||λ_{v+1} |^{k}|t_{v}(x)|^{k}

× { ∑_{v=1}^{n−1} |Δ_{v}(â_{nv})|}^{k−1}

= O(1) ∑_{n=2}^{m+1} ( \frac{P_{n}}{p_{n}} )^{k−1} a_{nn}^{k−1} ∑_{v=1}^{n−1} |Δ_{v}(â_{nv})||λ_{v+1} |^{k}|t_{v}(x)|^{k}

= O(1) ∑_{v=1}^{m} |λ_{v+1} |^{k}|t_{v}(x)|^{k} ∑_{n=v+1}^{m+1} |Δ_{v}(â_{nv})|

= O(1) ∑_{v=1}^{m} \frac{p_{v}}{P_{v}}|t_{v}(x)|^{k}|λ_{v+1} |^{k−1} |λ_{v+1} |

= O(1) ∑_{v=1}^{m} \frac{1}{X_{v}^{k−1}} |λ_{v+1} ||t_{v}(x)|^{k} \frac{p_{v}}{P_{v}} = O(1) as m → ∞,

by virtue of the hypotheses of Theorem 4.1 and Lemma 4.1. Finally, as in T_{n,1} , we have that

∑_{n=1}^{m} ( \frac{P_{n}}{p_{n}} )^{k−1} |I_{n,4} (x)|^{k} = O(1) ∑_{n=1}^{m} ( \frac{P_{n}}{p_{n}} )^{k−1} a_{nn}^{k} |λ_{n}|^{k}|t_{n}(x)|^{k}

= O(1) ∑_{n=1}^{m} \frac{p_{n}}{P_{n}} |λ_{n}|^{k−1} |λ_{n}||t_{n}(x)|^{k}

= O(1) ∑_{n=1}^{m} \frac{1}{X_{n}^{k−1}} |λ_{n}||t_{n}(x)|^{k} \frac{p_{n}}{P_{n}} = O(1) as m → ∞,

by virtue of hypotheses of the Theorem 4.1 and Lemma 4.1. This completes the proof of Theorem 4.1.

If we take a_{nv} = \frac{p_{v}}{P_{n}} in Theorem 4.1, then we get Theorem 3.1 and if we take p_{n} = 1 for all values of n in Theorem 4.1, then we get a new result dealing with the |A|_{k} summability method. Also, if we take a_{nv} = \frac{p_{v}}{P_{n}} and p_{n} = 1 for all values of n in Theorem 4.1, then we get a result concerning the |C, 1|_{k} summability methods.

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Department of Mathematics (Received 16 07 2016) Ahi Evran University Kırşehir Turkey

   sebnemyildiz@ahievran.edu.tr,
   sebnem.yildiz82@gmail.com