What are the applications of sequence of functions?

Question

This might be a "silly" question, but before starting my studies, I need a motivation.

In Analysis books, there are the subjects such as "sequence of functions, uniform convergence etc." which deals with basically the sequence of functions, but up to now [I'm a 2. year physics & mathematics student], I haven't seen any real application of the concept of sequence of functions. I mean, for example, I have seen lots of application of sequences in defining continuity, compactness etc. (our instructor does the whole analysis based on sequences), but this is not the case for the sequence of function.

Therefore, my question is that what are the applications of the concept of "sequence of function" in both mathematics and physics ?

Answer 1

The best example of "real application" is for numerical approximation of solutions. A lot of problems can't be solved with a nice formula and that's why numerical mathematics exists. For example, you can use the Galerkin method to get a sequence of functions which converges to the solution of a differential equation.

Answer 2

The most common way that you can "view" a sequence of any kind is an approximation. One way that could motivate you studying sequences of functions is that we use sequences of functions to approximate a weird and not-usual function by better ones, e.g. the Taylor expansion tells you good you can approximate your given function with polynomials.

One of the most important topics of such areas is Fourier Theory. It really is a subject that interwines all kinds of mathematics from representation theory to ergodic theory. You can find many videos on youtube showcasing the basic idea. Also wikipedia is a good place to start .

Answer 3

Given a subset $X\subset \mathbb{R}^n$, you seem to understand the usefulness of sequences in $X$ in order to study properties of $X$ (openness, compactness, completeness,$\dots$) and properties of maps $F\colon X \rightarrow \mathbb{R}$ on $X$ (like continuity). This usefulness persists in a more general context, say in case $X$ is merely a metric space.

Many collections of functions can be given such a metric structure, for example:

• $X_1=\{x\colon [0,1]\rightarrow \mathbb{R}^n\vert ~ x \text{ is continuous}\}$, with $d(x,y)=\sup_{t\in[0,1]}\vert x(t) - y(t)\vert$.
• $X_2=\{x\colon [0,1]\rightarrow \mathbb{R}^n\vert ~ x \text{ is continuously differentiable}\}$, with $d(x,y)= \sup_{t\in[0,1]}\vert x(t) - y(t)\vert + \sup_{t\in[0,1]}\vert x'(t) - y'(t)\vert$.
• See the list here under functional analysis.

Then a convergent sequence $(x_n)_n\subset X_1$ is simply a sequence of continuous functions that converges uniformly.

Application 1: The Picard-Lindelöf-Theorem gives a condition under which an ODE admits a (unique) solution. Its proof goes as follows: One shows that a continuous function $x\in X_1$ satisfies a certain ODE if and only if it satisfies the fixed point equation $Tx=x$, where $T$ is a certain map $T\colon X_1\rightarrow X_1$. Then one shows that $X_1$ is complete (using sequences of functions) and applies the Banach fixed point theorem. The proof also implies that, starting with an arbitrary $x_0 \in X_1$, the sequence $x_n=T \circ \dots \circ T(x_0)$ ($n$-times) converges uniformly to the solution $x$.

Application 2: Consider the subspace $X_2'=\{x\in X_2\vert ~ x(0)=p,x(1)=q\}\subset X_2$ (with the same metric) for some points $p,q\in\mathbb{R}^n$ and define $F\colon X_2'\rightarrow \mathbb{R}$ as $$ F(x) = \int_0^1 \vert x'(t)\vert dt $$ Interpret $x$ as a curve from $p$ to $q$, then $F(x)$ is exactly the length of this curve. Now one can show that $F$ is continuous (using sequences of functions), i.e. wiggeling a curve a little bit also only changes the length a little bit.

One can also show that $F$ has a unique minimum, which happens to be a straight line (not surprising, this is the curve of shortest length between two points). This has more interesting generalizations: If you replace $\mathbb{R}^n$ in the definition of $X_2$ by a more interesting space, say a sphere, the extreme points of $F$ correspond to geodesics.

Upshot: You can study interesting problems in analysis and geometry and (...) if you understand the spaces of functions $X$ well enough. In order to do so, you often use sequences.

Answer 4

To add another application of sequences of functions to the many other good applications already mentioned, convergence of sequences of functions is at the core of mathematical statistics' two most foundational theorems.

In modern probability theory, random variables are defined as (measurable) functions on a sample space. In statistics, we like to take a sample $X_1,\ldots,X_n$ and construct a statistic $Y_n$ which "summarizes" our sample. Convergence of functions allows us to study how our statistic $Y_n$ behaves as we take larger and larger samples.

For example, if you have random variables $X_1,X_2,\ldots$ that are independent, identically distributed, and have a defined expectation (mean) $\mu = E(X_1) = E(X_2) = \cdots$, then the law of large numbers states that the averages

$$ \bar{X}_n := \frac{\sum_{i=1}^n X_i}{n} $$

converge to the the constant function with value $\mu$.¹ This is a very formal mathematical statement that proves the intuitive notion that if you average a large number of trials of a random experiment that eventually converge to the actual average.

The central limit theorem, states that if $\sigma$ is the standard deviation of $X_1,X_2,\ldots$ and you compute

$$ \xi_n := \frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} $$

then $\xi_n$ (a sequence of functions) converges in distribution to a standard normal random variable. The importance of this result is hard to overstate as it (along with the law of large numbers) forms the backbone of statistics.

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¹ In fact, $\bar{X}_n$ converges to the constant function $\mu$ in two seperate senses, corresponding the weak and strong laws of large numbers. In the strong law, the functions $\bar{X}_n$ converge to $\mu$ "almost everywhere" which loosely means $\bar{X}_n(x)$ converges to $\mu$ for all but a very "small" set of values $x$. This form of convergence should be discussed in most books on Lebesgue measure and integration theory.

Answer 5

Both space-filling curves and continuous nowhere differentiable functions are often described as limits of sequences of continuous functions. Sometimes the best way to construct an interesting counterexample is as a limit of a sequence of functions.

Answer 6

As you are interested in physics, it is worth mentioning that in quantum mechanics, the wavefunction of a particle can be expanded as an infinite sum of eigenfunctions of the Hamiltonian operator via the Spectral Theorem (or an integral if the particle is unbound.) So, for example, if our Hamiltonian operator is $\hat{H}$, our eigenvalues are $E_1,E_2,\ldots$, and our eigenfunctions are $\psi_1,\psi_2,\ldots$ (that is, $\hat{H}(\psi_n) = E_n\psi_n$ for every $n \in \Bbb N$) then any wavefunction can be expanded as an infinite sum of these eigenfunctions. That is, for any wavefunction $\phi$, there exists $(c_n)$ such tht

$$ \phi = \sum_{k=1}^\infty c_k\psi_k, \tag{1} $$

which means that the sequence of functions

$$ \phi_N = \sum_{k=1}^N c_k \psi_k \text{ converges to } \phi. $$

By studying convergent sequences of functions, we learn important questions about when you can manipulate infinite series of functions term-by-term. Provided you can apply the Hamiltonian term-by-term, then

$$ \hat{H}\phi = \sum_{k=1}^\infty c_k \hat{H}\psi_k = \sum_{k=1}^\infty c_k E_k\psi_k. $$

With a little more elbow grease, you can use this idea to reduce the full time-dependent Schrödinger equation so solving for all the eigenfunctions $(\psi_n)$ and expanding your initial conditions in the form of (1), upon which the time-dependent solution is

$$ \phi(t) = \sum_{k=1}^\infty c_ke^{-iEt/\hbar} \psi_k. $$

All of this only works because you can apply operators term by term to (1), which relies on theorems you will prove when studying series of functions.