On the Approximation of Generalized Lipschitz Function by Euler Means of Conjugate Series of Fourier Series

Abstract

Approximation theory is a very important field which has various applications in pure and applied mathematics. The present study deals with a new theorem on the approximation of functions of Lipschitz class by using Euler’s mean of conjugate series of Fourier series. In this paper, the degree of approximation by using Euler’s means of conjugate of functions belonging to Lip(𝜉(𝑡),𝑝) class has been obtained. Lip𝛼 and Lip(𝛼,𝑝) classes are the particular cases of Lip(𝜉(𝑡),𝑝) class. The main result of this paper generalizes some well-known results in this direction.

1. Introduction and Definitions

Let𝑓be periodic with period2𝜋and integrable in the sense of Lebesgue. The Fourier series associated with𝑓at the point𝑥is given by 𝑓(𝑥)∼12𝑎₀+^∞_𝑛=1𝑎_𝑛cos𝑛𝑥+𝑏_𝑛sin𝑛𝑥=^∞_𝑛=0𝐴_𝑛(𝑥) (1) with partial sums𝑠_𝑛(𝑓;𝑥). The conjugate series of (1) is given by 𝐵_𝑛(𝑥)=^∞_𝑛=1𝑎_𝑛sin𝑛𝑥−𝑏_𝑛cos𝑛𝑥 (2) with partial sums𝑠_𝑛̃(𝑓;𝑥). Throughout this paper, we call (2) as conjugate series of Fourier series of function𝑓. If𝑓is Lebesgue integrable, then 𝑓(𝑥)1=−2𝜋^𝜋₀𝜓(𝑡)𝑡cot21𝑑𝑡=−2𝜋lim_𝜀→0^𝜋_𝜀𝜓(𝑡)𝑡cot2𝑑𝑡 (3) exists for almost all𝑥(Hardy [1], page 131).𝑓(𝑥)is called the conjugate function of𝑓(𝑥).

A function𝑓∈Lip𝛼if 𝑓(𝑥+𝑡)−𝑓(𝑥)=𝑂|𝑡|^𝛼for0<𝛼≤1. (4)

𝑓∈Lip(𝛼,𝑝),𝑝>1consider that if ₀^2𝜋𝑓(𝑥+𝑡)−𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝=𝑂|𝑡|^𝛼,0<𝛼≤1,𝑝≥1 (5)

(Definition5.38of Chandra [2]).

Given a positive increasing function𝜉(𝑡),𝑓∈Lip(𝜉(𝑡),𝑝), 𝑓(𝑥+𝑡)−𝑓(₀^2𝜋𝑥)^𝑝𝑑𝑥^1/𝑝≤𝑀𝜉(𝑡)𝑡^−1/𝑝,𝑝>1, (6) where𝑀is a positive number independent of𝑥and𝑡.

In case𝜉(𝑡)=𝑡^𝛼, thenLip(𝜉(𝑡),𝑝)coincides withLip(𝛼,𝑝). If𝑝→∞inLip(𝛼,𝑝), then it coincides withLip𝛼.

𝐿_∞-norm of a function𝑓∶𝑅→𝑅is defined by 𝑓_∞=sup𝑓(𝑥)∶𝑥∈𝑅. (7)

𝐿_𝑝-norm is defined by 𝑓_𝑝=₀^2𝜋𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝,𝑝≥1. (8)

The degree of approximation of a function𝑓∶𝑅→𝑅by a trigonometric polynomial𝑡_𝑛of order𝑛under sup norm ‖‖_∞ is defined by ([1], page 114-115]) 𝑡_𝑛−𝑓_∞=sup𝑡_𝑛(𝑥)−𝑓(𝑥)∶𝑥∈𝑅, (9) and𝐸_𝑛(𝑓)of a function𝑓∈𝐿_𝑝is given by 𝐸_𝑛𝑓=min𝑡_𝑛−𝑓_𝑝. (10)

Let{𝑆_𝑛}be the sequence of partial sums of the given series∑^∞_𝑛=0𝑢_𝑛. Then, for𝑞>0, the Euler (𝐸,𝑞) means of{𝑆_𝑛}are defined to be 𝑊_𝑛=1+𝑞^𝑛−𝑛_𝑘=0𝑛𝑘𝑞^𝑛−𝑘𝑆_𝑘. (11)

The series is said to be Euler (𝐸,𝑞) summable to𝑆provided that the sequence{𝑊_𝑛}converges to𝑆as𝑛→+∞.

We write 𝜓(𝑡)()(),𝜎=𝑓𝑥+𝑡−𝑓𝑥−𝑡_𝑛𝑓;𝑥=1+𝑞^𝑛−𝑛_𝑘=0𝑛𝑘𝑞^𝑛−𝑘𝑆_𝑘,𝑆(𝑡)=^𝑛_𝑘=0𝑛𝑘𝑞^𝑛−𝑘1cos𝑘+2𝑅(𝑡)𝑡𝑡,=sin2+𝑛tan⁻¹sin𝑡.𝑞+cos𝑡 (12)

2. Main Theorem

Hardy [1] established a theorem on (𝐶,𝛼), (𝛼>0) summability of the series. Harmonic summability is weaker than (𝐶,𝛼) summability. Iyengar [3] proved a theorem on harmonic summability of a Fourier series. The result of Iyengar [3] has been generalized by several researchers like Siddiqi [4], Pati [5], Lal and Kushwaha [6], and Rajagopal [7], for Nörlund means.

Alexits [8] proved the following theorem concerning the degree of approximation of a function𝑓∈Lip𝛼by the (𝐶,𝛿) means of its Fourier series.

Theorem A. If a periodic function𝑓∈Lip𝛼,0<𝛼≤1, then the degree of approximation of the (𝐶,𝛿) means of its Fourier series for0<𝛼<𝛿≤1is given by max_{0≤𝑥≤2𝜋}𝑓(𝑥)−𝜎^𝛿_𝑛(𝑥1)=𝑂𝑛^𝛼 (13) and for0<𝛼≤𝛿≤1is given by max_{0≤𝑥≤2𝜋}𝑓(𝑥)−𝜎^𝛿_𝑛(𝑥)=𝑂log𝑛𝑛^𝛼, (14) where𝜎^𝛿_𝑛(𝑥)are the (𝐶,𝛿) means of the partial sums of (2).

Later on, Hölland et al. [9] extended Theorem A to functions belonging to𝐶∗[0,2𝜋], the class of2𝜋-periodic continuous functions on [0,2𝜋], using Nörlund means of Fourier series. Their theorem is as follows.

Theorem B. If𝑤(𝑡)is the modulus of continuity of𝑓∈𝐶∗[0,2𝜋], then the degree of approximation of𝑓by the Nörlund means of the Fourier series for f is given by 𝐸_𝑛=max_{0≤𝑡≤2𝜋}𝑓(𝑡)−𝑇_𝑛(𝑡1)=𝑂𝑝_𝑛^𝑛_𝑘=1𝑝_𝑘𝑤(1/𝑘)𝑘, (15) where𝑇_𝑛are the(𝑁,𝑝_𝑛)means of Fourier series of𝑓.

Hölland et al. [9] have shown that Theorem B reduces to Theorem A if we deal with Cesàro means of order𝛿and consider a function𝑓∈Lip𝛼,0<𝛼≤1. Working in same direction we prove the following theorem.

Theorem 1. If𝑓∶𝑅→𝑅is a2𝜋periodic, Lebesgue integrable and belonging toLip(𝜉(𝑡),𝑝)for>1and if ^√1/𝑛₀𝑡𝜉(𝑡)𝑡^1/𝑝𝑝𝑑𝑡^1/𝑝𝜉1=𝑂√𝑛_{𝜋√1/𝑛},𝜉(𝑡)𝑡^1/𝑝+2𝑝𝑑𝑡^1/𝑝𝜉1=𝑂√𝑛𝑛 (16)(17) conditions (16) and (17) hold uniformly in𝑥, then degree of approximation of𝑓(𝑥), conjugate of𝑓∈Lip{𝜉(𝑡),𝑝}, by Euler (𝐸,𝑞) mean 𝜎_𝑛𝑓;𝑥=1+𝑞^𝑛−𝑛_𝑘=0𝑛𝑘𝑞^𝑛−𝑘𝑆_𝑘, (18) of the conjugate series (2) is given by 𝜎_𝑛(𝑓;𝑥)−𝑓(𝑥)_𝑝𝜉1=𝑂√𝑛(𝑛)^1/2𝑝. (19)

In order to prove our theorem, we need the following lemma.

Lemma 2. If0<𝑡≤𝜋, then 1+𝑞^−𝑛1+𝑞²+2𝑞cos𝑡^𝑛/2≤𝑒^{−2𝑞𝑡2𝑛/{𝜋(1+𝑞)}2}. (20)

Proof. We have 1+𝑞⁻²1+𝑞²+2𝑞cos𝑡=1−4𝑞sin²(𝑡/2)1+𝑞²≤1−4𝑞𝑡²𝜋²1+𝑞²≤𝑒^{−4𝑞𝑡2/{𝜋(1+𝑞)}2}, (21) since𝑒^𝑥(1−𝑥)<1when0<𝑥<1. Therefore, 1+𝑞^−𝑛1+𝑞²+2𝑞cos𝑡^𝑛/2≤𝑒^{−2𝑞𝑡2𝑛/{𝜋(1+𝑞)}2}. (22)

Proof of Theorem 1. The𝑘thpartial sum of the conjugate series of the Fourier series (2) is given by 𝑆_𝑘1𝑓;𝑥=−2𝜋^𝜋₀𝑡cot2𝜓(𝑡)+1𝑑𝑡2𝜋^𝜋₀cos(𝑘+1/2)𝑡()𝜓(𝑡)𝑆sin𝑡/2𝑑𝑡_𝑘−−1𝑓;𝑥2𝜋^𝜋₀𝑡cot2𝜓(𝑡)=1𝑑𝑡2𝜋^𝜋₀cos(𝑘+1/2)𝑡()𝜓(𝑡)sin𝑡/2𝑑𝑡. (23) Taking Euler (𝐸,𝑞) means, we get 1+𝑞^𝑛−𝑛_𝑘=0𝑛𝑘𝑞^𝑛−𝑘𝑆_𝑘−𝑓(𝑥=1𝑓;𝑥)𝑑𝑡𝜋1+𝑞^𝑛𝜋₀𝜓(𝑡)()×sin𝑡/2_𝑛𝑘=0𝑛𝑘𝑞^𝑛−𝑘1cos𝑘+2𝑡𝜎_𝑛−𝑓(𝑥)=1𝑓;𝑥𝜋1+𝑞^𝑛𝜋₀𝜓(𝑡)()𝑆(𝑡)=1sin𝑡/2𝑑𝑡𝜋1+𝑞^{𝑛√1/𝑛}₀+^𝜋_√1/𝑛𝜓(𝑡)()𝑆(𝑡)=𝐾sin𝑡/2𝑑𝑡₁(𝑥)+𝐾₂(𝑥),𝑆(𝑡)1+𝑞^𝑛≤11+𝑞^𝑛×^𝑛_𝑘=0𝑛𝑘𝑞^𝑛−𝑘𝑒^{𝑖(𝑘+1/2)𝑡}=𝑞+𝑒^𝑖𝑡𝑛1+𝑞^𝑛=1+𝑞²+2𝑞cos𝑡^𝑛/21+𝑞^𝑛≤𝑒^{−2𝑞𝑡2𝑛/{𝜋(1+𝑞)}2}, (24)(25) using Lemma 2.
Clearly, 𝜓()(𝑥≤𝑓()(+𝑓()(.𝑥+𝑡−𝜓)𝑢+𝑥+𝑡−𝑓𝑢+𝑥)𝑢+𝑥−𝑓𝑢−𝑥−𝑡) (26) Hence, by Minkowski’s inequality, ∫₀^2𝜋𝜓(𝑥+𝑡)−𝜓(𝑥)^𝑝𝑑𝑥^1/𝑝≤₀^2𝜋𝑓(𝑢+𝑥+𝑡)−𝑓(𝑢+𝑥)^𝑝𝑑𝑥^1/𝑝+₀^2𝜋𝑓(𝑢+𝑥)−𝑓(𝑢−𝑥−𝑡)^𝑝𝑑𝑥^1/𝑝=𝑂(𝜉(𝑡)). (27) Then,𝑓∈Lip(𝜉(𝑡),𝑝)⇒𝜓∈Lip(𝜉(𝑡),𝑝).
Using Hölder’s inequality,𝜓(𝑡)∈Lip(𝜉(𝑡),𝑝), condition (16),sin𝑡≥(2𝑡/𝜋), lemma, and second mean value theorem for integrals, we have 𝐾₁(𝑥)=𝑂^√1/𝑛₀𝜓(𝑡)^𝑝𝑑𝑡^1/𝑝×^√1/𝑛₀1+𝑞^−𝑛𝑆(𝑡)sin(𝑡/2)^𝑝′𝑑𝑡^1/𝑝′, (28) where 𝑝^′=𝑝𝐾𝑝−1¹(𝑥)=𝑂^√1/𝑛₀𝜉(𝑡)𝑡^1/𝑝𝑝𝑑𝑡^1/𝑝×^√1/𝑛₀𝑒^{−2𝑞𝑡2𝑛/{𝜋(1+𝑞)}2}|sin(𝑡/2)|^𝑝′𝑑𝑡^1/𝑝′𝜉1=𝑂√𝑛^√1/𝑛₀𝑡^−𝑝′𝑑𝑡^1/𝑝′𝜉1=𝑂√𝑛(𝑛)^1/2𝑝. (29) Now, 𝐾₂(𝑥)=𝑂^{𝜋√1/𝑛}𝜓(𝑡)sin(𝑡/2)1+𝑞^−𝑛1+𝑞²+2𝑞cos𝑡^𝑛/2|𝑅(𝑡)|𝑑𝑡=𝑂^𝜋_√1/𝑛𝜓(𝑡)sin(𝑡/2)1+𝑞^−𝑛×1+𝑞²+2𝑞cos𝑡^𝑛/2𝑑𝑡=𝑂^𝜋_√1/𝑛𝜓(𝑡)()𝑒sin𝑡/2^{−2𝑞𝑡2𝑛/{𝜋(1+𝑞)}2}1𝑑𝑡=𝑂𝑛^𝜋_√1/𝑛𝜓(𝑡)()×𝜕sin𝑡/2𝑒𝜕𝑡^{−2𝑞𝑡2𝑛/{𝜋(1+𝑞)}2}𝑑𝑡. (30) Using Hölder’s inequality,𝜓(𝑡)∈Lip(𝜉(𝑡),𝑝), and condition (17), we have 𝐾₂(𝑥1)=𝑂𝑛^𝜋_√1/𝑛𝜉(𝑡)𝑡^1/𝑝+2𝑝𝑑𝑡^1/𝑝×^𝜋_√1/𝑛𝜕𝑒𝜕𝑡^{−2𝑞𝑡2𝑛/{𝜋(1+𝑞)}2𝑝′}𝑑𝑡^1/𝑝′𝜉1=𝑂√𝑛(𝑛)^1/2𝑝. (31) Combining (24) with (31), we have 𝜎_𝑛(𝑓;𝑥)−𝑓(𝑥)_𝑝𝜉1=𝑂√𝑛(𝑛)^1/2𝑝, (32) which completes the proof of the theorem.

3. Corollaries

The following corollaries may be derived from our theorem.

Corollary 3. If𝜉(𝑡)=𝑡^𝛼, then the degree of approximation of a function𝑓(𝑥), conjugate of𝑓∈Lip(𝛼,𝑝),1/𝑝<𝛼<1, by Euler’s means(𝐸,𝑞)of the conjugate series of the Fourier series (2) is given by 𝜎_𝑛𝑓;𝑥−𝑓(𝑥)_𝑝1=𝑂𝑛^{(𝛼𝑝−1)/2𝑝}. (33)

Corollary 4. If𝑝→∞in Corollary 3, then, for0<𝛼<1, 𝜎_𝑛𝑓;𝑥−𝑓(𝑥)_∞1=𝑂𝑛^𝛼/2. (34)