Sequences of Functions in Application

Sequences of Functions

We have developed our knowledge of sequence of number in the chapter 2. Now we would like to talk about knowledge of sequences of functions and their application in the real life. First lets look at the definition of sequences of functions. Suppose $S \subset \mathbb{R}$ and for all $n \in \mathbb{N}$ there is a function $f_n: S\rightarrow \mathbb{R}$ . The collection { $f_n: n \in \mathbb{N}$ } is a sequence of function defined on $S$ .

Point-wise Convergence and Uniform Convergence.

In the textbook, we learnt that the definition of Point-wise Convergence is: Let $f_n$ be a defined function on $A \subseteq \mathbb{R}$ . The sequence ( $f_n$ ) of functions converges pointwise on $A$ to a function $f$ if, for all $x\in A$ , the sequence of real number $f_n(x)$ converges to f(x).According to the previous definition of converges, it means that let $x \in A$ , $\epsilon >0$ be arbitrary, there is a $N \in \mathbb {N}$ such that whenever $n\geq N$ , $|f_n(x)- f(x)|< \epsilon$ . since we pick the x arbitrary and dis the $|f_n(x)-f(x)|$ , so the $|f_n(x)-f(x)|$ is the vertical distance above x. On the other hand, the uniform convergence states that for arbitrary $\epsilon > 0$ , there is $N \in \mathbb{N}$ such that for any $x\in A$ , $|f_n(x)-f(x)|< \epsilon$ .

Application

Some application of sequences of functions include the signal and image processing. In reality, ‘signals’ are real world parameters such as heat, sounds, temperature, pressure and lights etc. A signal is a real (or complex) valued function of one or more real variable(s).It follows that some ‘transformer’ convert the signal into electrical energy. For example, a microphone converts the changing air pressure into time-varying voltage after receiving the signal and processing the information in it.

As mentioned earlier, we will use the term ‘signal’ to mean a real or complex valued function of real variable(s). Let us denote the signal by x(t). The variable t is called independent variable and the value x of t as dependent variable. We say a signal is continuous time signal if the independent variable t takes values in an interval. For example t (−∞,∞), or t [0,∞] or t [T0, T1] The independent variable t is referred to as time,even though it may not be actually time. For example in variation if pressure with height t refers above mean sea level. When t takes a vales in a countable set the signal is called a discrete time signal. For example t {0,T, 2T, 3T, 4T, …} or t {…−1, 0, 1, …} or t {1/2, 3/2, 5/2, 7/2, …} etc

Before talking about how sequences of functions got involved in this process, we need to talk about delta function. It is defined as $\vec {\delta} (x) =\begin{cases} 1 & t = 0\\ 0 & t \neq 0 \end{cases}$ . When you zoom into the plot, the delta function works by trying to ‘recover’ an exact signal when we sample only discrete values of t. And the ‘recovery’ tool is integration. If you look at the delta function again, you can tell the integration would give us a zero value.

Consider a single delta function as a wave of the signal that we receive.The process of recovering the signal motivate us to use another approach such as sequence of function. In Signal Processing, Fourier Transform would produce sequences based on the amplitude and frequency of the waves of the signal received. And by taking integrations of the sequences that involve delta function, we would have zero as the result, essentially not giving any information helpful of the signal.This came to the motivation to use sequences of function , as we mentioned, the integration of the delta function $\int_{\infty}^{\infty} \delta (t)dt = 0$ .This tool used for signal processing is not good for telling us about what information about the signal being processed. So we would need setting that $\int_{\infty}^{\infty} \delta (t)dt \neq 0$ . Thus define a new function as rect(t)= $\begin{cases} 1 & x \in [-\frac{1}{2},\frac{1}{2}]\\ 0 & elsewhere \end{cases}$ . Now consider the sequence of functions $\delta_n(t)= n* rect(nt)$ . This sequence of function is point-wise convergent to 0 except when $t =0$ . If we plot out the graph of function, $\int_{\infty}^{\infty} \delta_n (t)dt$ is area of the box and this would lead to $\int_{\infty}^{\infty} \delta_n (t)dt =1$ .This would gives us some insight of getting the information of the signal that we received compared with the Dirac Delta Function. This sequence of function behave in a similar way as the delta function but produce a non-zero value after the integration.