Investigating the Degree of Approximation of Fourier Series Through Product Means

1. Introduction

In mathematical analysis, the degree of approximation is crucial for understanding how infinite series behave. This concept goes beyond standard convergence theory, providing a more detailed understanding of series that might diverge or converge conditionally. Through this method, mathematicians can establish meaningful limits for such series, uncovering inherent patterns and order in what would otherwise seem complex or unsolvable expressions.

A vibrant area where these ideas find resonance is in the study of the Fourier series. These series provide elegant representations of periodic functions as infinite sums of $sines$ and $cosines$, bridging the gap between pure mathematical theory and practical applications. From solving differential equations to signal processing and acoustics to quantum mechanics and electrical engineering, Fourier analysis plays a central role. However, the classical theory of the Fourier series is not without its limitations. Functions with discontinuities or irregular behavior often resist uniform convergence, presenting challenges that call for more nuanced analysis methods.

This is where the interplay between Banach summability and the Fourier series becomes both natural and fruitful. The framework of Banach summability offers powerful tools to investigate the convergence properties of Fourier series, especially in cases where classical convergence fails. By broadening the notion of limits, Banach summability methods allow for a deeper understanding of the behavior of series and open pathways to new applications in both theoretical and applied contexts.

The origins of summability theory can be traced back to the seminal work of Godfrey Harold Hardy [1], whose famous evergreen book “Divergent Series" (1970) laid the foundation for a rigorous treatment of divergent series. Hardy’s work inspired future generations of mathematicians, including Stefan Banach [2], Salomon Bochner [3], Ram Chandran [4], and Shyam Lal Singh [5], among others. Of particular significance is the contribution of Stefan Banach [6], whose introduction of Banach limits and summability revolutionized the analysis of convergence and divergence in infinite series.

Building upon these foundations, researchers have developed more sophisticated summability methods tailored to specific mathematical contexts. For instance, S.K. Paikray et al. [6] introduced the notion of absolute indexed summability factors using quasi-monotone sequences. In contrast, R.K. Jati et al. [7] employed absolute indexed matrix summability to study infinite series in a more generalized setting. Further contributions by J.K. Mishra & M. Mishra [8], G.D. Dikshit [9], L. McFadden [10], and T. Pati [11] have advanced the field by extending various methods of absolute summability and exploring their applications to Fourier series. H.K. Nigam [12] found the way to the degree of approximation of product means. E.C. Titchmarch [13] contributed to the development of trigonometric theory, and A. Zygmund [14] revealed the development of trigonometric series.

This convergence of ideas from classical analysis to modern summability techniques underscores the evolving nature of mathematical inquiry. By integrating the tools of Banach summability with the rich structure of Fourier analysis, we not only deepen our theoretical understanding but also enhance our ability to address practical challenges in science and engineering.

2. Definitions

Definition 1.

Let $\left(a_n\right)$ be a sequence of real or complex numbers. The Euler mean of order 1 is denoted $(E,1)$ and defined by

$$(E,1)={\displaystyle \frac{1}{2^n}}\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)a_k$$

(1)

Further, if ${lim}_{n\to \infty }(E,1)=s$, a finite number, then we say $a_n$ is Euler summable of order 1 to s and written $\Sigma a_n=s(E,1).$

Definition 2.

Let $\sum a_n$ be a sequence of real or complex numbers and $s_n$ denotes its $n^{th}$ partial sums. For two sequences $\left\{p_n\right\}$ and $\left\{q_n\right\}$, define $\left\{t_n\right\}$ by

$$t_n={\displaystyle \frac{1}{r_n}}\sum _{v=0}^n{p}_{n-v}{q}_v{s}_v$$

(2)

where $r_n=p_0{q}_n+p_1{q}_{n-1}+...+p_n{q}_0.$ If ${lim}_{n\to \infty }{t}_n=S$, a finite number, then we say $\sum a_n$ is said to be $(N,p,q)$ summable to s and written $\Sigma a_n=s(N,p,q).$

Further, if the $(E,1)$ transform of the $(N,p,q)$ transform $\left\{s_n\right\}$ is defined by ${\tau }_n=(E,1).(N,p.q)$, then

$${\tau }_n=\left({\displaystyle \frac{1}{2^n}}\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)a_k\right)\left({\displaystyle \frac{1}{r_n}}\sum _{v=0}^n{p}_{n-v}{q}_v{S}_v\right)$$

(3)

If ${\tau }_n\to s$ when $n\to \infty ,$ then we say $\sum a_n$ is said to be $(E,1).(N,p,q)$ summable to a finite number s.

Definition 3.

When we express $f\left(n\right)=O\left(g\right(n\left)\right)$, it signifies that there exists a positive constant C and a threshold value $n_0$ such that for all n exceeding $n_0$ the relationship holds true:

$$\left|f\right(n\left)\right|\le Cg\left(n\right),$$

(4)

where big $`O^{\prime }$ notation stands for an upper limit on the growth of a function.

Example 1.

Let $f\left(n\right)=5n^2+3n-3$. We can say that $f\left(n\right)=O\left(n^2\right)$. For sufficiently large values of n, the expression $3n^2+2n+1$ is bounded above by $C{n}^2$ for some constant C. In the context of asymptotic notation, the small ’o’ notation, denoted $f\left(n\right)=o\left(g\right(n\left)\right)$, signifies that $f\left(n\right)$ grows strictly slower than $g\left(n\right)$ as $n\to \infty .$ . Formally, this is defined by the limit:

$$\underset{n\to \infty }{lim}{\displaystyle \frac{f\left(n\right)}{g\left(n\right)}}=0.$$

(5)

Example 2.

If $f\left(n\right)=n$ and $g\left(n\right)=n^2$, then $f\left(n\right)=o\left(g\right(n\left)\right)$ because n becomes insignificant compared to $n^2$ as n increases.

Let $f\left(t\right)$ be a periodic function Lebesgue integrable in $(-\pi ,\pi )$. Then the series

$$f\left(x\right)={\displaystyle \frac{a_0}{2}}+\sum _{n=1}^{\infty }(a_{n}cosnx+b_{n}sinnx)=\sum _{n=0}^{\infty }{A}_n\left(x\right)$$

(6)

is referred to as the Fourier series of $f\left(t\right)$, where

$$\begin{array}{cc}\hfill a_0& ={\displaystyle \frac{1}{\pi }}{\int }_{-\pi }^{\pi }f\left(t\right)dt\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill a_n& ={\displaystyle \frac{1}{\pi }}{\int }_{-\pi }^{\pi }f\left(t\right)cosntdt\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill b_n& ={\displaystyle \frac{1}{\pi }}{\int }_{-\pi }^{\pi }f\left(t\right)sinntdt\hfill \end{array}$$

(9)

Let $\left\{s_n\right\}$ be the $n^{th}$ partial sum of $\sum a_n$. Then $L_{\infty }$ norm of a function $R\to R$ is defined by

$${\left|\right|f\left|\right|}_{\infty }=sup\left\{\right|f\left(x\right):x\in R\left|\right\}$$

(10)

The $L_v$-norm is defined by

$${\left|\right|f\left|\right|}_v={\left\{{\int }_0^{2\pi }{\left|f\left(x\right)\right|}^{v}dx\right\}}^{{\displaystyle \frac{1}{v}}}$$

(11)

The degree of approximation $f:R⟶R$ defined by polynomial $p_n\left(x\right)$ of degree n under norm ${\left|\right|.\left|\right|}_{\infty }$ is defined by

$$\left|\right|p_n-{f\left|\right|}_{\infty }=sup\left\{\right|p_n\left(x\right)-f\left(x\right)|:x\in R\}$$

(12)

Also, the degree of approximation of a function $f\in L_v$ is defined by

$$E_n\left(f\right)=inf\left\{\right||p_n\left(x\right)-{f\left|\right|}_v:x\in R\}.$$

(13)

A function f is said to be in the class $Lip\alpha$ if

$$|f(x+p)-f\left(x\right)|=O\left(\right|p{|}^{\alpha })\mathrm{where}0<\alpha <1.$$

(14)

We use the following notation throughout this paper:

$$\varphi \left(t\right)=f(x+t)+f(x-t)-2f\left(x\right).$$

(15)

$$K_n\left(t\right)={\displaystyle \frac{1}{2^{n+1}\pi }}\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left\{{\displaystyle \frac{1}{r_n}}\sum _{v=0}^k{p}_{n-v}{q}_v{\displaystyle \frac{sin(v+\frac{1}{2})t}{sin\frac{t}{2}}}\right\}$$

(16)

3. Known Result

Theorem 1.

If f be a $2\pi$ periodic function of $Lip\alpha$, the degree of approximation by the product $(E.q)(N.P_n)$ summability means on its Fourier series (6) is given by $\left|\right|{\tau }_n-f{\left|\right|}_{\infty }=O\left\{{\displaystyle \frac{1}{{(n+1)}^{\alpha }}}\right\}$ where $0<\alpha <1$ and ${\tau }_n$ is defined by (3).

Theorem 2.

If f is a periodic function of period $2\pi$ and class of $Lip\alpha$, then the degree of approximation by product Euler & Cesaro summability of its Fourier series (6) is given by $O\left\{{\displaystyle \frac{1}{{(n+1)}^{\alpha }}}\right\}$ where $0<\alpha <1$.

4. Principal theorem

In this paper, we have proved the degree of approximation by product mean $(E,1)(N,p,q)$ of the Fourier series of a function of class $Lip\alpha$.

Theorem 3.

If f is a periodic function of period $2\pi$ of the class $Lip(\alpha ,1)$ then the degree of approximation by product $(E,1)(N,p,q)$ summability its Fourier series is given by $O\left\{{\displaystyle \frac{1}{{(n+1)}^{\alpha }}}\right\}$ where $0<\alpha <1$.

5. Required Lemmas

We require the following Lemma to prove the theorem.

Lemma 1.

$|K_n\left(t\right)|=O\left({\displaystyle \frac{1}{t}}\right)$ for ${\displaystyle \frac{1}{n+1}}\le t\le \pi$.

Proof. 

Proof: Since for ${\displaystyle \frac{1}{n+1}}\le t\le \pi$, $sin(t/2)\ge {\displaystyle \frac{t}{\pi }}={\displaystyle \frac{t}{\pi }}$ and $|sin(nt\left)\right|\le 1$.

$$\begin{array}{cc}\hfill |K_n\left(t\right)|& ={\displaystyle \frac{1}{\pi 2^{n+1}}}\left|\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left\{{\displaystyle \frac{1}{r_k}}\sum _{v=0}^k{p}_{k-v}{q}_v{\displaystyle \frac{sin(v+1/2)t}{sin(t/2)}}\right\}\right|\hfill \\ \hfill & \ll {\displaystyle \frac{1}{t.2^{n+1}}}\left|\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left\{{\displaystyle \frac{1}{r_k}}\sum _{v=0}^k{p}_{k-v}{q}_v\right\}\right|\hfill \\ \hfill & ={\displaystyle \frac{1}{t.2^{n+1}}}\left|\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\right|(\mathrm{assuming}\sum _{v=0}^k{p}_{k-v}{q}_v=r_k)\hfill \\ \hfill & ={\displaystyle \frac{1}{t.2^{n+1}}}.2^{n+1}\hfill \\ \hfill & =O\left({\displaystyle \frac{1}{t}}\right)\hfill \end{array}$$

□

Lemma 2.

For $0\le t<{\displaystyle \frac{1}{n+1}}$, $K_n\left(t\right)=O\left(n\right)$.

Proof. 

$$\begin{array}{cc}\hfill K_n\left(t\right)& ={\displaystyle \frac{1}{\pi .2^{n+1}}}\left[\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left\{{\displaystyle \frac{1}{r_k}}\sum _{\nu =0}^k{p}_{k-\nu }{q}_{\nu }{\displaystyle \frac{sin(\nu +1/2)t}{sin(t/2)}}\right\}\right]\hfill \\ \hfill & \ll {\displaystyle \frac{1}{\pi .2^{n+1}}}\left|\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left\{{\displaystyle \frac{1}{r_k}}\sum _{v=0}^k{p}_{k-v}{q}_v(v+1/2){\displaystyle \frac{t}{t/2}}\right\}\right|(\sin \mathrm{ce}sinx\approx x\mathrm{for}\mathrm{small}x)\hfill \\ \hfill & \ll {\displaystyle \frac{1}{\pi .2^{n+1}}}\left|\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left\{{\displaystyle \frac{1}{r_k}}\sum _{v=0}^k{p}_{k-v}{q}_v(2v+1)\right\}\right|\hfill \\ \hfill & \ll {\displaystyle \frac{1}{\pi .2^{n+1}}}\left|\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)(2k+1)\right|(\mathrm{assuming}\sum _{v=0}^k{p}_{k-v}{q}_v(2v+1)\approx (2k+1)r_k)\hfill \\ \hfill & ={\displaystyle \frac{1}{\pi .2^{n+1}}}\left(2\sum _{k=0}^{n}k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)+\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\pi .2^{n+1}}}\left(2n{2}^{n-1}+2^n\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\pi .2^{n+1}}}\left(n{2}^n+2^n\right)\hfill \\ \hfill & ={\displaystyle \frac{2^n(n+1)}{\pi .2^{n+1}}}\hfill \\ \hfill & =O\left(n\right)\hfill \end{array}$$

□

6. Proof of Principal Theorem

From Riemann-Lebesgue theorem for the $n^{th}$ partial sum of Fourier series of $f\left(x\right)$ and the following Titchmarch [13], we get

$$s_n(f,x)-f\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{\pi }\varphi \left(t\right){\displaystyle \frac{sin(n+1/2)t}{sin(t/2)}}dt$$

Using (1), the $(N,p,q)$ transform of $|t_n-f|$ is

$$|t_n-f|=t_n(f,x)-f\left(x\right)={\displaystyle \frac{1}{2\pi r_n}}{\int }_0^{\pi }\varphi \left(t\right)\sum _{v=0}^n{p}_{n-v}{q}_v{\displaystyle \frac{sin(v+1/2)t}{sin(t/2)}}dt$$

Denoting the product summability by $(E,1)(N,p,q),$ we have

$$\begin{array}{cc}\hfill |{\tau }_n-f|& ={\displaystyle \frac{1}{\pi .2^{n+1}}}\left|{\int }_0^{\pi }\varphi \left(t\right)\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left\{{\displaystyle \frac{1}{r_k}}\sum _{\nu =0}^k{p}_{k-\nu }{q}_{\nu }{\displaystyle \frac{sin(\nu +1/2)t}{sin(t/2)}}\right\}dt\right|\hfill \\ \hfill & ={\int }_0^{\pi }\varphi \left(t\right)K_n\left(t\right)dt\hfill \\ \hfill & =\left({\int }_0^{{\displaystyle \frac{1}{n+1}}}+{\int }_{{\displaystyle \frac{1}{n+1}}}^{\pi }\right)\varphi \left(t\right)K_n\left(t\right)dt\hfill \\ \hfill & =I_1+I_2\left(\mathrm{say}\right)\hfill \end{array}$$

Using Lemma 2 and the property of $Lip\alpha$ class, $\varphi \left(t\right)=O\left(t^{\alpha }\right)$:

$$\begin{array}{cc}\hfill |I_1|& ={\displaystyle \frac{1}{\pi .2^{n+1}}}\left|{\int }_0^{{\displaystyle \frac{1}{n+1}}}\varphi \left(t\right)\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left\{{\displaystyle \frac{1}{r_k}}\sum _{\nu =0}^k{p}_{k-\nu }{q}_{\nu }{\displaystyle \frac{sin(\nu +1/2)t}{sin(t/2)}}\right\}dt\right|\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & \le O\left(n\right)\left|{\int }_0^{{\displaystyle \frac{1}{n+1}}}\varphi \left(t\right)dt\right|\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & =O\left(n\right){\int }_0^{{\displaystyle \frac{1}{n+1}}}{t}^{\alpha }dt\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & =O\left(n\right){\left[{\displaystyle \frac{t^{\alpha +1}}{\alpha +1}}\right]}_0^{{\displaystyle \frac{1}{n+1}}}\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & =O\left(n\right){\displaystyle \frac{1}{(\alpha +1){(n+1)}^{\alpha +1}}}\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & =O\left({\displaystyle \frac{1}{{(n+1)}^{\alpha }}}\right)\hfill \end{array}$$

(22)

Using Lemma 1 and the property of the $Lip\alpha$ class:

$$\begin{array}{cc}\hfill |I_2|& =\left|{\int }_{{\displaystyle \frac{1}{n+1}}}^{\pi }\varphi \left(t\right)K_n\left(t\right)dt\right|\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & \le {\int }_{{\displaystyle \frac{1}{n+1}}}^{\pi }\left|\varphi \left(t\right)\right||K_n\left(t\right)|dt\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & \le {\int }_{{\displaystyle \frac{1}{n+1}}}^{\pi }O\left(t^{\alpha }\right)O\left({\displaystyle \frac{1}{t}}\right)dt\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & \le {\int }_{{\displaystyle \frac{1}{n+1}}}^{\pi }{t}^{\alpha -1}dt\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & ={\left[{\displaystyle \frac{t^{\alpha }}{\alpha }}\right]}_{{\displaystyle \frac{1}{n+1}}}^{\pi }\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & =O\left({\pi }^{\alpha }\right)-O\left({\displaystyle \frac{1}{{(n+1)}^{\alpha }}}\right)\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & =O\left({\displaystyle \frac{1}{{(n+1)}^{\alpha }}}\right)\hfill \end{array}$$

(29)

Combining Equation (17) and Equation (23), we get the required result: $|{\tau }_n-f|=O\left({\displaystyle \frac{1}{{(n+1)}^{\alpha }}}\right)$. Therefore, $\left|\right|{\tau }_n-f{\left|\right|}_{\infty }=O\left\{{\displaystyle \frac{1}{{(n+1)}^{\alpha }}}\right\}$.