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A study on degree of approximation by Karamata summability method
Journal of Inequalities and Applications volume 2011, Article number: 85 (2011)
Abstract
Vuĉkoviĉ [Maths. Zeitchr. 89, 192 (1965)] and Kathal [Riv. Math. Univ. Parma, Italy 10, 33-38 (1969)] have studied summability of Fourier series by Karamata (Kλ) summability method. In present paper, for the first time, we study the degree of approximation of function f ∈ Lip (α,r) and f ∈ W(L r ,ξ(t)) by Kλ-summability means of its Fourier series and conjugate of function and by Kλ-summability means of its conjugate Fourier series and establish four quite new theorems.
MSC: primary 42B05; 42B08; 42A42; 42A30; 42A50.
1 Introduction
The method Kλwas first introduced by Karamata [1] and Lotosky [2] reintroduced the special case λ = 1. Only after the study of Agnew [3], an intensive study of these and similar cases took place. Vuĉkoviĉ [4] applied this method for summability of Fourier series. Kathal [5] extended the result of Vuĉkoviĉ [4]. Working in the same direction, Ojha [6], Tripathi and Lal [7] have studied Kλ-summability of Fourier series under different conditions. The degree of approximation of a function f ∈ Lip α by Cesàro and Nörlund means of the Fourier series has been studied by Alexits [8], Sahney and Goel [9], Chandra [10], Qureshi [11], Qureshi and Neha [12], Rhoades [13], etc. But nothing seems to have been done so far in the direction of present work. Therefore, in present paper, we establish two new theorems on degree of approximation of function f belonging to Lip (α,r) (r ≥ 1) and to weighted class W(L r , ξ (t))(r ≥ 1) by Kλ-means on its Fourier series and two other new theorems on degree of approximation of function , conjugate of a 2π-periodic function f belonging to Lip (α,r) (r > 1) and to weighted class W(L r ,ξ (t)) (r ≥ 1) by Kλ-means on its conjugate Fourier series.
2 Definitions and notations
Let us define, for n = 0, 1, 2,..., the numbers , for 0 ≤ m ≤ n, by
The numbers are known as the absolute value of stirling number of first kind
Let {s n } be the sequence of partial sums of an infinite series ∑u n , and let us write
to denote the n th Kλ-mean of order λ > 0. If as n → ∞, where s is a fixed finite number, then the sequence {s n } or the series ∑u n is said to be summable by Karamata method (Kλ) of order λ > 0 to the sum s, and we can write
Let f be a 2π-periodic function and integrable in the sense of Lebesgue. The Fourier series associated with f at a point x is defined by
with n th partial sums s n (f;x).
The conjugate series of Fourier series (2.4) is given by
with n th partial sums .
Throughout this paper, we will call (2.5) as conjugate Fourier series of function f.
L∞-norm of a function f: R → R is defined by
L r -norm is defined by
The degree of approximation of a function f: R → R by a trigonometric polynomial t n of degree n under sup norm || ||∞ is defined by
(Zygmund [14])
and E n (f) of a function f ∈ L r is given by
This method of approximation is called trigonometric Fourier approximation. A function f ∈ Lip α if
and
f ∈ Lip (α, r) for 0 ≤ x ≤ 2π, if
(definition 5.38 of McFadden [15]).
Given a positive increasing function ξ (t) and an integer r ≥ 1, f ∈ Lip (ξ(t), r), if
and that
f ∈ W (L r , ξ (t)) if
If β = 0, our newly defined weighted i.e. W (L r , ξ (t)) reduces to Lip (ξ (t), r), if ξ (t) = tαthen Lip (ξ (t), r) coincides with Lip (α, r) and if r → ∞ then Lip (α, r) reduces to Lip α.
We observe that
We write
3 The main results
3.1 Theorem 1
If a function f, 2π-periodic, belonging to Lip (α, r) then its degree of approximation by Kλ-summability means on its Fourier series is given by
where s n is Kλ-mean of Fourier series (2.4).
3.2 Theorem 2
If a function f, 2π-periodic, belonging to W (L r , ξ (t)) then its degree of approximation by Kλ-summability means on its Fourier series is given by
provided that ξ (t) satisfies the following conditions:
and
where δ is an arbitrary positive number such that s (1 - δ) - 1 > 0, , 1 ≤ r ≤ ∞, conditions (3.4) and (3.5) hold uniformly in x, s n is Kλ-mean of Fourier series (2.4).
3.3 Theorem 3
If a function , conjugate to a 2π-periodic function f, belonging to Lip(α,r) then its degree of approximation by Kλ-summability means on its conjugate Fourier series is given by
where is Kλ-mean of conjugate Fourier series (2.5) and
3.4 Theorem 4
If a function , conjugate to a 2π-periodic function f, belonging to W (L r ,ξ (t)) then its degree of approximation by Kλ-summability means on its conjugate Fourier series is given by
provided that ξ (t) satisfies the conditions (3.3)-(3.5) in which δ is an arbitrary positive number such that s (1 - δ) - 1 > 0, ,1≤r≤ ∞. Conditions (3.4) and (3.5) hold uniformly in x, is Kλ-mean of conjugate Fourier series (2.5) and
4 Lemmas
For the proof of our theorems, following lemmas are required.
4.1 Lemma 1
(Vuĉkoviĉ [14]). Let λ > 0 and , then
4.2 Lemma 2
Proof. For , sin nt ≤ nt and
Considering first part of (4.1) and using Lemma 1,
4.3 Lemma 3
Proof. For , sin nt ≤ nt and
Using Lemma 1,
□
4.4 Lemma 4
(McFadden [15]), Lemma 5.40) If f(x) belongs to Lip(α,r) on [0,π], then φ(t) belongs to Lip(α,r) on [0,π].
5 Proof of the theorems
5.1 Proof of Theorem 1
Following Titchmarsh [16] and using Riemann-Lebesgue theorem, the m th partial sum s m (x) of series (2.4) at t = x is given by
Therefore,
Now we consider,
Using Lemma 2,
Using Hölder's inequality and Lemma 4,
Since, for
Next we consider,
Using Hölder's inequality, (5.3) and Lemma 4,
Combining (5.1), (5.2) and (5.4),
This completes the proof of Theorem 1.
5.2 Proof of Theorem 2
Following the proof of Theorem 1,
We have
Hence, by Minkowiski's inequality,
Then f ∈ W (L r ,ξ(t))⇒ φ ∈ W(L r , ξ (t)).
Now we consider,
Using Lemma 2,
Using Hölder's inequality and the fact that φ (t) ∈ W (L r , ξ (t)),
Since sin t ≥ 2t/π,
Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,
Next we consider,
Using Hölder's inequality, |sin t| ≤ 1,sin t ≥ 2t/π, (5.3), conditions (3.3), (3.5) and second mean value theorem for integrals,
Putting
Now combining (5.5)-(5.7),
Now using L r -norm, we get
This completes the proof of Theorem 2.
5.3 Proof of Theorem 3
Following Lal [7], the m th partial sum of series (2.5) at t = x
Therefore,
We consider,
Using Lemma 3,
Now consider,
Using second mean value theorem for integrals,
Now we consider,
Since, for ,
Using Hölder's inequality and Lemma 4,
Next we consider,
Combining (5.9)-(5.12),
Since, for
Next we consider,
Using Hölder's inequality, (5.14) and Lemma 4,
Collecting (5.8), (5.13) and (5.15),
This completes the proof of Theorem 3.
5.4 Proof of Theorem 4
Following the calculations of Theorem 3,
Now,
Using Lemma 3,
Using Minkowiski's inequality, we have a fact that f ∈ W (L r , ξ (t))⇒ ψ ∈ W (L r , ξ (t)). Now we consider,
Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,
Now,
Since for ,
Hölder's inequality and the fact that ψ (t) ∈ W (L r , ξ (t)),
Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,
Next we consider,
Using Hölder's inequality and the fact that ψ (t) ∈ W (L r ,ξ (t)),
Since, sin t ≥ 2t/π,
Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,
Combining from (5.17) to (5.20),
Using Hölder's inequality, |sin t| ≤ 1, sin t ≥ 2t/π, conditions (3.3), (3.5) and second mean value theorem for integrals and the fact ψ (t) ∈ W (Lr,ξ (t)),
Putting
Combining from (5.16), (5.21) and (5.22)
Now using Lr-norm, we get
This completes the proof of Theorem 4.
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6 Competing interests
The authors declare that they have no competing interests.
7 Authors' contributions
HK framed the problems. HK and KS carried out the results and wrote the manuscripts. All the authors read and approved the final manuscripts.
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Nigam, H.K., Sharma, K. A study on degree of approximation by Karamata summability method. J Inequal Appl 2011, 85 (2011). https://doi.org/10.1186/1029-242X-2011-85
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DOI: https://doi.org/10.1186/1029-242X-2011-85