A Study on Degree of Approximation by (𝐸,1) Summability Means of the Fourier-Laguerre Expansion

Abstract

A very new theorem on the degree of approximation of the generating function by (𝐸,1) means of its Fourier-Laguerre series at the frontier point 𝑥=0 is obtained.

1. Introduction

Let ∑^∞_𝑛=0𝑢_𝑛 be an infinite series with the sequence of its 𝑛th partial sums {𝑠_𝑛}.

If 𝐸¹_𝑛=12^{𝑛𝑛𝑛𝑘=0}𝐶_𝑘𝑠_𝑛−𝑘→𝑠as𝑛→∞, (1.1) then we say that {𝑠_𝑛} is summable by (𝐸,1) means (see the study by Hardy [1]), and it is written as 𝑠_𝑛→𝑠(𝐸,1), where {𝑠_𝑛} is the sequence of 𝑛th partial sums of the series ∑^∞_𝑛=0𝑢_𝑛.

The Fourier-Laguerre expansion of a function 𝑓(𝑥)∈𝐿(0,∞) is given by 𝑓(𝑥)∼^∞_𝑛=0𝑎_𝑛𝐿_𝑛^(𝛼)(𝑥), (1.2) where 𝑎_𝑛𝑛={Γ(𝛼+1)}𝑛+𝛼^−1∞₀𝑒^−𝑦𝑦^𝛼𝑓(𝑦)𝐿_𝑛^(𝛼)(𝑦)𝑑𝑦 (1.3) and 𝐿_𝑛^(𝛼)(𝑥) denotes the 𝑛th Laguerre polynomial of order 𝛼>−1, defined by generating function ^∞_𝑛=0𝐿_𝑛^(𝛼)(𝑥)𝜔^𝑛=(1−𝜔)^−𝛼−1exp(−𝑥𝜔1−𝜔), (1.4) and existence of integral (1.3) is presumed.

We write 𝜙(𝑦)={Γ(𝛼+1)}⁻¹𝑒^−𝑦𝑦^𝛼{𝑓(𝑦)−𝑓(0)}. (1.5) Gupta [2] estimated the order of the function by Cesàro means of series (1.2) at the point 𝑥=0 after replacing the continuity condition in Szegö’s theorem [3] by a much lighter condition. He established the following theorem.

Theorem 1.1. If 𝐹(𝑡)=^𝑡₀|𝑓(𝑦)|𝑦1𝑑𝑦=𝑜{log(𝑡)}^1+𝑝,𝑡→0,−1<𝑝<∞,_∞1𝑒^−𝑦/2𝑦^{(3𝛼−3𝑘−1)/3}|𝑓(𝑦)|𝑑𝑦<∞, (1.6) then 𝜎^𝑘_𝑛(0)=𝑜(log𝑛)^𝑝+1 (1.7) provided that 𝑘>𝛼+1/2,𝛼>−1, with 𝜎^𝑘_𝑛(0) being the 𝑛th Cesàro mean of order 𝑘.

Denoting the harmonic means by {𝑡_𝑛}, Singh [4] estimated the order of function by harmonic means of series (1.2) at point 𝑥=0 by weaker conditions than those of Theorem 1.1. He proved the following theorem.

Theorem 1.2. For −5/6<𝛼<−1/2,𝑡_𝑛(0)−𝑓(0)=𝑜(log𝑛)^𝑝+1, (1.8) provided that ^𝛿_𝑡|𝜙(𝑦)|𝑦^𝛼+11𝑑𝑦=𝑜{log(𝑡)}^1+𝑝,𝑡→0,−1<𝑝<∞, (1.9)𝛿 is a fixed positive constant, ^𝑛_𝛿𝑒^𝑦/2𝑦^{−(2𝛼+3)/4}|𝜙(𝑦)|𝑑𝑦=𝑜{𝑛^{−(2𝛼+1)/4}(log𝑛)^1+𝑝},_∞𝑛𝑒^𝑦/2𝑦^−1/3|𝜙(𝑦)|𝑑𝑦=𝑜{(log𝑛)^𝑝+1},𝑛→∞. (1.10)

2. Main Theorem

The objects of present paper are as follows:(1)We prove our theorem for (𝐸,1) means which is entirely different from (𝐶,𝑘) and harmonic means.(2)We employ a condition which is weaker than condition (1.9) of Theorem 1.2.(3)In our theorem the range of 𝛼 is increased to −1<𝛼<−1/2, which is more useful for application.

In fact, we establish the following theorem.

Theorem 2.1. If 𝐸¹_𝑛=12^{𝑛𝑛𝑛𝑘=0}𝐶_𝑘𝑠_𝑘→∞as𝑛→∞, (2.1) then the degree of approximation of Fourier-Laguerre expansion (1.2) at the point 𝑥=0 by (𝐸,1) means 𝐸¹_𝑛 is given by 𝐸¹_𝑛(0)−𝑓(0)=𝑜{𝜉(𝑛)} (2.2) provided that Φ(𝑡)=^𝑡₀|𝜙(𝑦)|𝑑𝑦=𝑜{𝑡^𝛼+11𝜉(𝑡)},𝑡→0, (2.3)𝛿 is a fixed positive constant and −1<𝛼<−1/2, ^𝑛_𝛿𝑒^𝑦/2𝑦^{−(2𝛼+3)/4}|𝜙(𝑦)|𝑑𝑦=𝑜{𝑛^{−(2𝛼+1)/4}𝜉(𝑛)},_∞𝑛𝑒^𝑦/2𝑦^−1/3|𝜙(𝑦)|𝑑𝑦=𝑜{𝜉(𝑛)},𝑛→∞, (2.4)(2.5) where 𝜉(𝑡) is a positive monotonic increasing function of 𝑡 such that 𝜉(𝑛)→∞ as 𝑛→∞.

3. Lemmas

Lemma 3.1 (see the study by Szegö, 1959, [3, page 175]). Let 𝛼 be arbitrary and real, let 𝑐 and ∈ be fixed positive constants, and let 𝑛→∞. Then 𝐿_𝑛^(𝛼)(𝑥)=𝑂(𝑥^{−(2𝛼+1)/4}𝑛^{(2𝛼−1)/4})if𝑐𝑛≤𝑥≤∈,𝐿_𝑛^(𝛼)(𝑥)=𝑂(𝑛^𝛼)if∈0≤𝑥≤𝑛. (3.1)(3.2)

4. Proof of the Main Theorem

Since 𝐿_𝑛^(𝛼)𝛼(0)=,𝑛+𝛼 (4.1) therefore, 𝑠_𝑛(0)=^𝑛_𝑘=0𝑎_𝑘𝐿_𝑘^(𝛼)(0)={Γ(𝛼+1)}^−1∞₀𝑒^−𝑦𝑦^𝛼𝑓(𝑦)^𝑛_𝑘=0𝐿_𝑘^(𝛼)(𝑦)𝑑𝑦={Γ(𝛼+1)}^−1∞₀𝑒^−𝑦𝑦^𝛼𝑓(𝑦)𝐿_𝑛^(𝛼+1)(𝑦)𝑑𝑦. (4.2) Now, 𝐸¹_𝑛1(0)=2^{𝑛𝑛𝑛𝑘=0}𝐶_𝑘𝑠_𝑘=1(0)2^{𝑛𝑛𝑛𝑘=0}𝐶_𝑘{Γ(𝛼+1)}^−1∞₀𝑒^−𝑦𝑦^𝛼𝑓(𝑦)𝐿_𝑘^(𝛼+1)(𝑦)𝑑𝑦. (4.3) Using orthogonal property of Laguerre’s polynomial and (1.5), we have 𝐸¹_𝑛1(0)−𝑓(0)=2^{𝑛𝑛𝑛𝑘=0}𝐶_𝑘^∞₀𝜙(𝑦)𝐿_𝑘^(𝛼+1)(𝑦)𝑑𝑦=(₀^1/𝑛+^𝛿_1/𝑛+^𝑛_𝛿+^∞_𝑛)12^{𝑛𝑛𝑛𝑘=0}𝐶_𝑘𝜙(𝑦)𝐿_𝑘^(𝛼+1)(𝑦)𝑑𝑦=𝐼₁+𝐼₂+𝐼₃+𝐼₄(say). (4.4) Using orthogonal property and condition (3.2) (taking 𝛼+1 for 𝛼 and 𝛿 for ∈) of Lemma 3.1, we get 𝐼₁=12^{𝑛𝑛𝑛𝑘=0}𝐶_𝑘𝑂{𝑛^𝛼+1}₀^1/𝑛=1|𝜙(𝑦)|𝑑𝑦2^{𝑛𝑛𝑛𝑘=0}𝐶_𝑘𝑂{𝑛^𝛼+11}𝑜{𝑛^𝛼+1𝐼𝜉(𝑛)},₁1=𝑜(2^{𝑛𝑛𝑛𝑘=0}𝐶_𝑘𝜉(𝑛))=𝑜{𝜉(𝑛)}since^𝑛𝑛_𝑘=0𝐶_𝑘=2^𝑛. (4.5) Further, using orthogonal property and condition (3.1) (taking 𝛼+1 for 𝛼, 1 for 𝑐, and 𝛿 for ∈) of Lemma 3.1, we get 𝐼₂=12^{𝑛𝑛𝑛𝑘=0}𝐶_𝑘𝑂{𝑛^(2𝛼+1)/4}^𝛿_1/𝑛|𝜙(𝑦)|𝑦^{−(2𝛼+3)/4}𝑑𝑦. (4.6) Now, ^𝑛𝑛_𝑘=0𝐶_𝑘𝑛^(2𝛼+1)/4={^[𝑛/2]_𝑘=0+^𝑛_{𝑘=[𝑛/2]+1}}^𝑛𝐶_𝑘𝑛^(2𝛼+1)/4=(𝑛)^{(2𝛼+1)/4𝑛[𝑛/2]}_𝑘=0𝐶_𝑘+^𝑛𝐶_[𝑛/2](𝑛)^(2𝛼+5)/4≤(𝑛)^{(2𝛼+1)/4𝑛𝑛}_𝑘=0𝐶_𝑘+^𝑛𝐶_[𝑛/2](𝑛)^(2𝛼+5)/4,_{𝑛𝑛𝑘=0}𝐶_𝑘𝑛^(2𝛼+1)/4=(𝑛)^(2𝛼+1)/42^𝑛+^𝑛𝐶_[𝑛/2](𝑛)^(2𝛼+5)/4 (4.7) since 2^𝑛=^𝑛𝑛_𝑘=0𝐶_𝑘=^𝑛𝐶₀+^𝑛𝐶₁+^𝑛𝐶₂+⋯+^𝑛𝐶_[𝑛/2]+^𝑛𝐶_[𝑛/2]+1≥+⋯_𝑛𝐶_[𝑛/2]+^𝑛𝐶_[𝑛/2]+⋯+^𝑛𝐶_𝑛≥^𝑛𝐶_[𝑛/2]+^𝑛𝐶_[𝑛/2]+⋯+^𝑛𝐶_[𝑛/2]𝑛={[2]+1}^𝑛𝐶_[𝑛/2]≥𝑛2^𝑛𝐶_[𝑛/2]. (4.8) Therefore, 𝑛2^𝑛𝐶_[𝑛/2]≤2^𝑛. (4.9) By (4.7) and (4.9), we have, ^𝑛𝑛_𝑘=0𝐶_𝑘{𝑛^(2𝛼+1)/4}≤(𝑛)^(2𝛼+1)/42^𝑛+2(2^𝑛)(𝑛)^(2𝛼+1)/4=𝑂{(𝑛)^(2𝛼+1)/42^𝑛}. (4.10) Thus, 𝐼₂=𝑂{(𝑛)^(2𝛼+1)/4}^𝛿_1/𝑛𝑦^{−(2𝛼+3)/4}|𝜙(𝑦)|𝑑𝑦=𝑂{(𝑛)^(2𝛼+1)/4}[{𝑦^{−(2𝛼+3)/4}Φ(𝑦)}^𝛿_1/𝑛+^𝛿_1/𝑛(2𝛼+3)4𝑦^{−(2𝛼+7)/4}Φ(𝑦)𝑑𝑦]=𝑂{(𝑛)^(2𝛼+1)/4}[𝑦^{−(2𝛼+3)/4}𝑜{𝑦^𝛼+11𝜉(𝑦)}]+{^𝛿_1/𝑛𝑦^{−(2𝛼+7)/4}𝑜{𝑦^𝛼+11𝜉(𝑦)}𝑑𝑦}=𝑂{(𝑛)^(2𝛼+1)/4}[𝑜{𝑦^(2𝛼+1)/41𝜉(𝑦)}]^𝛿_1/𝑛+𝑜{^𝛿_1/𝑛𝑦^{(2𝛼−3)/4}1𝜉(𝑦)𝑑𝑦}=𝑂{(𝑛)^(2𝛼+1)/4}[𝑂(1)+𝑜{𝑛^{−(2𝛼+1)/4}𝜉(𝑛)}]+𝜉(𝑛)𝑜{^𝛿_1/𝑛𝑦^{(2𝛼−3)/4}𝑑𝑦}=𝑜{𝜉(𝑛)}+𝑜{(𝑛)^(2𝛼+1)/4𝜉(𝑛)}{^𝛿_1/𝑛𝑦^{(2𝛼−3)/4}𝑑𝑦}=𝑜{𝜉(𝑛)}+𝑜{𝑛^(2𝛼+1)/4𝑦𝜉(𝑛)}{^{(2𝛼−3)/4+1}}{(2𝛼−3)/4}+1^𝛿_1/𝑛=𝑜{𝜉(𝑛)}+𝑜{𝑛^(2𝛼+1)/4𝑦𝜉(𝑛)}{^(2𝛼+1)/4}(2𝛼+1)/4^𝛿_1/𝑛=𝑜{𝜉(𝑛)}+𝑜{𝑛^(2𝛼+1)/4𝜉(𝑛)}{𝑛^{−(2𝛼+1)/4}}=𝑜{𝜉(𝑛)}+𝑜{𝜉(𝑛)},𝐼₂=𝑜{𝜉(𝑛)}. (4.11)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10) Now, we consider 𝐼₃1=[(2^𝑛){^𝑛𝑛_𝑘=0𝐶_𝑘^𝑛_𝛿𝑒^𝑦/2𝑦^{−(2𝛼+3)/4}|𝜙(𝑦)|𝑒^−𝑦/2𝑦^(2𝛼+3)/4|𝐿_𝑛^(𝛼+1)1(𝑦)|𝑑𝑦}]=(2^𝑛){^𝑛𝑛_𝑘=0𝐶_𝑘𝑂{𝑛^(2𝛼+1)/4}^𝑛_𝛿𝑒^𝑦/2𝑦^{−(2𝛼+3)/4}𝐼|𝜙(𝑦)|𝑑𝑦},₃=𝑂{(𝑛)^(2𝛼+1)/4}𝑜{(𝑛)^{−(2𝛼+1)/4}𝜉(𝑛)},using𝐼(2.4),₃=𝑜{𝜉(𝑛)}. (4.12) Finally, 𝐼₄1=[(2^𝑛){^𝑛𝑛_𝑘=0𝐶_𝑘^∞_𝑛𝑒^𝑦/2𝑦^{−(3𝛼+5)/6}|𝜙(𝑦)|𝑒^−𝑦/2𝑦^(3𝛼+5)/6|𝐿_𝑛^(𝛼+1)1(𝑦)|𝑑𝑦}]=(2^𝑛){^𝑛𝑛_𝑘=0𝐶_𝑘𝑂{𝑘^(𝛼+1)/2}^∞_𝑛𝑒^𝑦/2𝑦^−1/3|𝜙(𝑦)|𝑦^(𝛼+1)/2𝐼𝑑𝑦},₄1=𝑂[(2^𝑛)(2^𝑛){𝑛^(𝛼+1)/2𝑛^{−(𝛼+1)/2}}𝑜{𝜉(𝑛)}],by𝐼(2.5)₄=𝑜{𝜉(𝑛)}. (4.13) Combining (4.4), (4.5), (4.11), (4.12), and (4.13), we get 𝐸¹_𝑛(0)−𝑓(0)=𝑜{𝜉(𝑛)}. (4.14) This completes the proof of the theorem.