Error Bound of Periodic Signals in the Hölder Metric

Abstract

We obtain two theorems to determine the error bound between input periodic signals and processed output signals, whenever signals belong to 𝐻_𝜔-space and as a processor we have taken (𝐶,1)(𝐸,1)-mean and generalized an early result of Lal and Yadav in (2001).

1. Introduction

Chandra [1] was first to extend Prössdorf’s [2] result to find the degree of approximation of a continuous function using the Nörlund transform. Later on, Mohapatra and Chandra [3] obtained a number of interesting results on the degree of approximation in the Hölder metric using matrix transforms, which generalize all the previous results based on Cesàro and Nörlund transforms. In 1992, Singh [4] introduced 𝐻_𝜔-space in place of 𝐻_𝛼-space and obtained several results on the degree of approximation of functions and deduced many previous results based on 𝐻_𝛼-spaces. In 1996, Das et al. [5] used 𝐻_(𝛼,𝑝)-space in place of 𝐻_𝛼-space and obtained degree of approximation of functions and generalized the results of Mohapatra and Chandra [3]. In 2000, Mittal and Rhoades [6] also obtained the degree of approximation of functions in a normed space and generalized the results of Singh [4] by removing the hypothesis of monotonicity of the rows of the matrix. Singh and Soni [7], and Mittal et al. [8] used the technique of approximation of functions in measuring the errors in the input signals and the processed output signals.

2. Definitions and Notations

Let the transforms

A:𝜆_𝑛=^𝑛_𝑘=1𝑎_𝑛𝑘𝑠_𝑘, (2.1)B:𝜏_𝑛=^𝑛_𝑘=1𝑏_𝑛𝑘𝑠_𝑘, (2.2) be two regular methods of summability. Then, the A transform of the B transform of a sequence {𝑠_𝑛} is given by 𝑡_𝑛=^𝑛_𝑝=1𝑎_𝑛𝑝𝜏_𝑝=^𝑛𝑛_𝑝=1𝑘=1𝑎_𝑛𝑝𝑏_𝑝𝑘𝑠_𝑘, (2.3)the sequence {𝑠_𝑛} is said to be summable 𝑡_𝑛 to the sum 𝑠, if Lim_𝑛→∞𝑡_𝑛=𝑠. (2.4) Let 𝑠(𝑡)∈𝐶_2𝜋 be a 2𝜋-periodic analog signal whose Fourier trigonometric expansion be given by 1𝑠(𝑡)∼2𝑎₀+^∞_𝑛=1(𝑎_𝑛cos𝑛𝑡+𝑏_𝑛sin𝑛𝑡)≡^∞_𝑛=0𝐴_𝑛(𝑡), (2.5) and let {𝑠_𝑛(𝑡)} be the sequence of partial sums of (2.5).

Let the (𝐸,1) and (𝐶,1) transforms for the sequence {𝑠_𝑛} be defined by 𝐸¹_𝑛=12^𝑛𝑛_𝑘=0𝑛𝑘𝑠_𝑘𝜎(𝑡),_𝑛=1𝑛+1^𝑛_𝑘=0𝑠_𝑘(𝑡), (2.6)(2.7) respectively.

The product (𝐶,1)(𝐸,1)-transform is expressed as the (𝐶,1)-transform of (𝐸,1)-transform of {𝑠_𝑛} and is given by sequence-to-sequence transformation (see, e.g., [9]): 𝑡_𝑛1(𝑠;𝑡)=𝑛+1^𝑛_𝑘=0𝐸¹_𝑘. (2.8) The sequence {𝑠_𝑛} is said to be summable (𝐶,1)(𝐸,1) to the sum 𝑠, if Lim_𝑛→∞𝑡_𝑛(𝑠;𝑡)=𝑠. (2.9)

2.1. Regularity Condition of (𝐶,1)(𝐸,1)-Method

𝑡_𝑛1(𝑠;𝑡)=𝑛+1^𝑛_𝑘∑₌₀𝐸¹_𝑘=1𝑛+1^𝑛_𝑘∑₌₀{12^𝑘𝑘_𝜐∑₌₀𝑘𝜐𝑠_𝑘}=∑^∞_𝑘=0𝐶_𝑛,𝑘𝑠_𝑘, (2.10) where𝐶_𝑛,𝑘=12𝑛+1^𝑘−𝑘_𝜐=0𝑘𝜐,𝑘≤𝑛0,𝑘>𝑛. (2.11)Now,

(i)∑^∞_𝑘=0|𝐶_𝑛,𝑘|=∑^𝑛_𝑘=0|(1/(𝑛+1))2^−𝑘∑^𝑘_𝜐=0(^𝑘_𝜐)|=1,(ii)𝐶_𝑛,𝑘=(1/(𝑛+1))(1)→0, as 𝑛→∞, for fixed 𝑘,(iii)∑^∞_𝑘=0𝐶_𝑛,𝑘=1, thus, (𝐶,1)(𝐸,1)-method is regular.

Singh [4] defined the space 𝐻_𝜔 by

𝐻_𝜔={𝑠(𝑡)∈𝐶_2𝜋∶|𝑠(𝑡₁)−𝑠(𝑡₂)|≤𝐾𝜔(|𝑡₁−𝑡₂|)}, (2.12) and the norm ‖⋅‖_𝜔∗ by ‖𝑠‖_𝜔∗=‖𝑠‖_𝑐+Sup_𝑡1,𝑡2{Δ^𝜔∗𝑠(𝑡₁,𝑡₂)}, (2.13) where ‖𝑠‖_𝑐=Sup_{0≤𝑡≤2𝜋}Δ|𝑠(𝑡)|,_𝜔∗𝑠(𝑡₁,𝑡₂)=|𝑠(𝑡₁)−𝑠(𝑡₂)|𝜔^∗(|𝑡₁−𝑡₂|),𝑡₁≠𝑡₂, (2.14)and choosing Δ⁰𝑠(𝑡₁,𝑡₂)=0, 𝜔(𝑡) and 𝜔^∗(𝑡) being increasing signals of 𝑡. If 𝜔(|𝑡₁−𝑡₂|)≤𝐴|𝑡₁−𝑡₂|^𝛼 and 𝜔^∗(|𝑡₁−𝑡₂|)≤𝐾|𝑡₁−𝑡₂|^𝛽, 0≤𝛽<𝛼≤1, 𝐴 and 𝐾 being positive constants, then the space 𝐻_𝛼={𝑠(𝑡)∈𝐶_2𝜋∶|𝑠(𝑡₁)−𝑠(𝑡₂)|≤𝐾|𝑡₁−𝑡₂|^𝛼,0<𝛼≤1} (2.15) is Banach space [2] and the metric induced by the norm ‖⋅‖_𝛼 on 𝐻_𝛼 is said to be Hölder metric.

We write 𝜙_𝑡1(𝑡)=𝑠(𝑡₁+𝑡)+𝑠(𝑡₁−𝑡)−2𝑠(𝑡₁𝐾),_𝑛𝑡(𝑡)=sin(𝑛+1)2^𝑛_𝑘=0𝑛𝑘1sin(𝑘+2)𝑡. (2.16)(2.17)

3. Known Result

Lal and Yadav [10] established the following theorem to estimate the error between the input signal 𝑠(𝑡) and the signal obtained after passing through the (𝐶,1)(𝐸,1)-transform.

Theorem A. If a function 𝑠∶𝑅→𝑅 is 2𝜋-periodic and belonging to class 𝐿𝑖𝑝𝛼, 0<𝛼≤1, then the degree of approximation by (𝐶,1)(𝐸,1) means of its Fourier series is given by

‖𝑡_𝑛(𝑠;𝑡₁)−𝑠(𝑡₁)‖_∞=𝑂(𝑛^−𝛼),0<𝛼<1𝑂(log𝑛𝑛),𝛼=1. (3.1)

4. Main Result

The object of this paper is to generalize the above result under much more general assumptions. We will measure the error between the input signal 𝑠(𝑡) and the processed output signal 𝑡_𝑛(𝑠;𝑡)=(1/(𝑛+1))∑^𝑛_𝑘=1𝐸¹_𝑘(𝑡), by establishing the following theorems.

Theorem 4.1. Let 𝜔(𝑡) defined in (2.12) be such that ^𝜋_𝑡𝜔(𝑢)𝑢²𝑑𝑢=𝑂{𝐻(𝑡)},𝐻(𝑡)≥0,_𝑡0𝐻(𝑢)𝑑𝑢=𝑂{𝑡𝐻(𝑡)},as𝑡→0⁺, (4.1)(4.2) then, for 0≤𝛽<𝜂≤1 and 𝑠∈𝐻_𝜔, we have ‖𝑡_𝑛(𝑠;𝑡₁)−𝑠‖_𝜔∗=𝑂{((𝑛+1)⁻¹𝜋𝐻(𝑛+1))^{1−𝛽/𝜂}}. (4.3)

Theorem 4.2. Let 𝜔(𝑡) defined in (2.12) and for 0≤𝛽<𝜂≤1 and 𝑠∈𝐻_𝜔, we have ‖𝑡_𝑛(𝑠;𝑡₁)−𝑠‖_𝜔∗𝜋=𝑂{(𝜔(𝑛+1))^{1−𝛽/𝜂}+((𝑛+1)^−1𝑛+1_𝑘=11𝜔(𝑘+1))^{1−𝛽/𝜂}}. (4.4)

5. Lemmas

We will use following lemmas.

Lemma 5.1. Let 𝜙_𝑡1(𝑡) be defined in (2.16), then for 𝑠∈𝐻_𝜔, we have |𝜙_𝑡1(𝑡)−𝜙_𝑡2(𝑡)|≤4𝐾𝜔(|𝑡₁−𝑡₂|),|𝜙_𝑡1(𝑡)−𝜙_𝑡2(𝑡)|≤4𝐾𝜔(|𝑡|). (5.1)(5.2) It is easy to verify.

Lemma 5.2. Let 𝐾_𝑛(𝑡) be defined in (2.17), then 𝐾_𝑛2(𝑡)≤𝐶(^𝑛+1𝑡)cos^𝑛(𝑡2𝑡)sin(𝑛+1)(2), (5.3) where “𝐶" is an absolute constant, not necessarily the same at each occurrence.

Proof. 𝐾_𝑛1(𝑡)=sin(𝑡/2)𝐼.𝑃.{^𝑛_𝑘=0𝑛𝑘𝑒^{𝑖(𝑘+1/2)𝑡}}=1sin(𝑡/2)𝐼.𝑃.{𝑒^𝑖𝑡/2(1+𝑒^𝑖𝑡)^𝑛}=1sin(𝑡/2)𝐼.𝑃.{2^𝑛cos^𝑛(𝑡2)𝑒^{𝑖(𝑛+1)𝑡/2}}2≤𝐶(^𝑛+1𝑡)cos^𝑛(𝑡2𝑡)sin(𝑛+1)(𝑛). (5.4)

Lemma 5.3. ^𝑛_𝑘=0(1𝑡)cos^𝑘(𝑡2𝑡)sin(𝑘+1)(2𝐶)≤(𝑡²𝑡)(1−cos(𝑛+1)(2)cos^𝑛+1(𝑡2)). (5.5)

Proof. ^𝑛_𝑘=0(1𝑡)cos^𝑘(𝑡2𝑡)sin(𝑘+1)(2)=^𝑛_𝑘=0(1𝑡)𝐼.𝑃.{𝑒^{𝑖(𝑘+1)𝑡/2}cos^𝑘(𝑡21)}=(𝑡)𝐼.𝑃.{𝑒^𝑖𝑡/2(1−𝑒^{𝑖(𝑛+1)𝑡/2}cos^𝑛+1(𝑡/2)1−𝑒^𝑖𝑡/2𝐶cos(𝑡/2))}≤(𝑡²𝑡)𝐼.𝑃.{𝑖−𝑖cos(𝑛+1)(2)cos^𝑛+1(𝑡2𝑡)+sin(𝑛+1)(2)cos^𝑛+1(𝑡2𝐶)}=(𝑡²𝑡)(1−cos(𝑛+1)(2)cos^𝑛+1(𝑡2)). (5.6)

Lemma 5.4 (see [9]). For 0≤𝑡≤1/𝑛+1, then 𝑡1−cos(𝑛+1)(2)cos^𝑛+1(𝑡2)=𝑂{(𝑛+1)²𝑡²}. (5.7)

Lemma 5.5 (see [6]). If 𝜔(𝑡) satisfies conditions (4.1) and (4.2), then ^𝑢₀𝑡⁻¹𝜔(𝑡)𝑑𝑡=𝑂(𝑢𝐻(𝑢)),𝑢→0⁺. (5.8)

6. Proof of Theorem 4.1

Proof of Theorem 4.1. Following Zygmund [11], we have 𝑠_𝑛(𝑡₁1)−𝑠=2𝜋^𝜋₀𝜙_𝑡1(𝑡)1sin(𝑡/2)sin(𝑛+2)𝑡𝑑𝑡. (6.1)From (2.6) and (2.16), we have 𝐸¹_𝑛(𝑡₁2)−𝑠=^−𝑛2𝜋^𝜋₀𝜙_𝑡1(𝑡)𝐾_𝑛(𝑡)𝑑𝑡. (6.2)Using Lemma 5.2, we have 𝐸¹_𝑛(𝑡₁2)−𝑠≤𝐶^−(𝑛+1)𝜋^𝜋₀𝜙_𝑡1(𝑡)𝑡2^𝑛+1cos^𝑛(𝑡2𝑡)sin(𝑛+1)(2)𝑑𝑡. (6.3)Now from (2.8), the (𝐶,1)-transform of (𝐸,1)-transform is given by |𝑡_𝑛(𝑠;𝑡₁𝐶)−𝑠|≤𝑛+1^𝜋₀|𝜙_𝑡1(𝑡)|𝑡|^𝑛_𝑘=0cos^𝑘(𝑡2𝑡)sin(𝑘+1)(2)|𝑑𝑡. (6.4) Setting 𝐸_𝑛(𝑡₁)=|𝑡_𝑛(𝑠;𝑡₁)−𝑠(𝑡₁𝐶)|≤𝑛+1^𝜋₀|𝜙_𝑡1(𝑡)|𝑡|^𝑛_𝑘=0cos^𝑘(𝑡2𝑡)sin(𝑘+1)(2𝐸)|𝑑𝑡,_𝑛(𝑡₁,𝑡₂)=|𝐸_𝑛(𝑡₁)−𝐸_𝑛(𝑡₂𝐶)|≤𝑛+1^𝜋₀|𝜙_𝑡1(𝑡)−𝜙_𝑡2(𝑡)|𝑡|^𝑛_𝑘=0cos^𝑘(𝑡2𝑡)sin(𝑘+1)(21)|𝑑𝑡=𝑂(𝑛+1)(₀^𝜋/𝑛+1+^𝜋_𝜋/𝑛+1)=𝐼₁+𝐼₂,say, (6.5) now using (4.1), (4.2), (5.2), and Lemma 5.5, we get 𝐼₁1=𝑂(1)𝑛+1₀^𝜋/𝑛+1𝑡⁻¹𝜔(𝑡)𝑑𝑡=𝑂{(𝑛+1)⁻¹𝜋𝐻(𝑛+1)}. (6.6) Again using (5.2), (4.1), and Lemma 5.3, we have 𝐼₂1=𝑂(1)𝑛+1^𝜋_𝜋/𝑛+1𝑡⁻²𝑡𝜔(𝑡)|1−cos(𝑛+1)(2)cos^𝑛+1(𝑡21)|𝑑𝑡=𝑂(1)𝑛+1^𝜋_𝜋/𝑛+1𝑡⁻²𝜔(𝑡)𝑑𝑡=𝑂{(𝑛+1)⁻¹𝜋𝐻(𝑛+1)}. (6.7) Now from (5.1), Lemmas 5.3 and 5.4, we have 𝐼₁1=𝑂(1)𝑛+1₀^𝜋/𝑛+1𝜔(|𝑡₁−𝑡₂|)𝑡²𝑡|1−cos(𝑛+1)(2)cos^𝑛+1(𝑡2)|𝑑𝑡=𝑂(1)𝜔(|𝑡₁−𝑡₂|)𝑛+1₀^𝜋/𝑛+1𝑡⁻²(𝑛+1)²𝑡²𝑑𝑡=𝑂{𝜔(|𝑡₁−𝑡₂𝐼|)},₂=𝑂(1)𝜔(|𝑡₁−𝑡₂|)𝑛+1^𝜋_{𝜋/(𝑛+1)}𝑡⁻²𝑑𝑡=𝑂{𝜔(|𝑡₁−𝑡₂|)}. (6.8)(6.9) Now noting that 𝐼_𝑟=𝐼_𝑟^{1−𝛽/𝜂}𝐼_𝑟^𝛽/𝜂,𝑟=1,2, (6.10) we have, from (6.6) and (6.8), 𝐼₁=𝑂{(𝜔(|𝑡₁−𝑡₂|))^𝛽/𝜂((𝑛+1)⁻¹𝜋𝐻(𝑛+1))^{1−𝛽/𝜂}}, (6.11) and from (6.7) and (6.9), we have 𝐼₂=𝑂{(𝜔(|𝑡₁−𝑡₂|))^𝛽/𝜂((𝑛+1)⁻¹𝜋𝐻(𝑛+1))^{1−𝛽/𝜂}}. (6.12) Thus, from (2.13), (6.11) and (6.12), we have sup_𝑡1,𝑡2Δ^𝜔∗|𝐸_𝑛(𝑡₁,𝑡₂)|=sup_𝑡1,𝑡2|𝐸_𝑛(𝑡₁)−𝐸_𝑛(𝑡₂)|𝜔^∗(|𝑡₁−𝑡₂|)=𝑂{(𝜔(|𝑡₁−𝑡₂|))^𝛽/𝜂(𝜔^∗(|𝑡₁−𝑡₂|))⁻¹((𝑛+1)⁻¹𝜋𝐻(𝑛+1))^{1−𝛽/𝜂}}. (6.13) It is to be noted from (6.6) and (6.7), ‖𝐸_𝑛(𝑡₁)‖_𝑐=max_{0≤𝑡1≤2𝜋}|𝑡_𝑛(𝑠;𝑡₁)−𝑠|=𝑂{(𝑛+1)⁻¹𝜋𝐻(𝑛+1)}. (6.14) Combining (6.13) and (6.14), we get ‖𝑡_𝑛(𝑠;𝑡₁)−𝑠‖_𝜔∗=𝑂{((𝑛+1)⁻¹𝜋𝐻(𝑛+1))^{1−𝛽/𝜂}}. (6.15)This completes the proof of Theorem 4.1.

Proof of Theorem 4.2. Follows analogously as the proof of Theorem 4.1 with slight changes, so we omit details.

7. Applications

The following results can easily be derived from the Theorem 4.1. If we put 𝜔^∗(|𝑡₁−𝑡₂|)≤𝐾|𝑡₁−𝑡₂|^𝛽, 𝜔(|𝑡₁−𝑡₂|)≤𝐴|𝑡₁−𝑡₂|^𝛼 and replace 𝜂 by 𝛼 and set 𝑢𝐻(𝑢)=^𝛼−11,0<𝛼<1log(𝑢),𝛼=1, (7.1) then we get Corollary 7.1.

Corollary 7.1. If 𝑠∈𝐻_𝛼, 0≤𝛽<𝛼≤1, then ‖𝑡_𝑛(𝑠;𝑡₁)−𝑠‖_𝛽=𝑂(𝑛+1)^𝛽−𝛼,0<𝛼<1𝑂(log(𝑛+1))(𝑛+1)^1−𝛽,𝛼=1. (7.2) If we put 𝛽=0, then from above corollary, we have Corollary 7.2.

Corollary 7.2. If 𝑠∈Lip𝛼, 0<𝛼≤1, then ‖𝑡_𝑛(𝑠;𝑡₁)−𝑠‖=𝑂(𝑛^−𝛼),0<𝛼<1𝑂(log𝑛𝑛),𝛼=1. (7.3) Hence Theorem 3 is particular case of Theorem 4.1.

Acknowledgment

The authors are highly grateful to the referee for his valuable comments and suggestions for the improvement and the better presentation of the paper.