Potential directions post-integration

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Sequences of functions

Define $f_n:X \to \RR$ to be a sequence of functions with common domain $X$. We say the sequence $(f_n)$ converges pointwise to the function $f:X \to \RR$ if

\[\lim_{n\to \infty} f_n(x) = f(x)\]

for all $x$. Also of importance is uniform convergence. Sequences of functions arise everywhere in analysis, typically as a means to approximate the function $f$.

[example=Fourier series] Let $f:[0, 2\pi) \to \RR$ be defined by

\[\begin{align*} f(t) = \begin{cases} +1 & \text{ if } 0 \le t < \pi, \\ -1 & \text{ if } \pi \le t < 2\pi. \end{cases} \end{align*}\]

Now define $f_n:[0, 2\pi) \to \RR$ by

\[f_n(t) = \sum_{\substack{k\text{ odd} \\ |k| \le n}} \frac{-4i}{2\pi k} e^{ikt}.\]

Via some Fourier theory (Dirichlet-Jordan test, in particular), one can prove that $(f_n)$ converges pointwise to $f$. Here’s a code implementation to convince you of that claim:

[/example]

A good goal here would be to understand the basics of:

  • Pointwise and uniform convergence
  • Taylor series
  • Fourier series
  • Weierstrass approximation theorem

Fourier analysis

(Classical) Fourier analysis is the branch of mathematics that deals with sequences of functions like in the example I gave above. In addition, it also studies Fourier series on various types of domains (spheres, tori, etc.), their convergence properties, Fourier transforms, and their linear algebraic properties (via functional analysis).

A good goal here would be to understand the basics of:

  • Fourier series
  • Convergence and the Dirichlet kernel
  • Convolution
  • Fourier transforms

Measure theory

Measure theory is a more robust framework for studying limits, derivatives, and integrals as well as the vector spaces associated with these objects. A classic example that is used to motivate measure theory is to consider a sequence $(f_n)$ of Riemann-integrable functions $f_n:[a, b] \to \RR$ that converge pointwise to the function $f:[a, b] \to \RR$. Then is

\[\begin{align*} \lim_{n\to \infty} \int_a^b f_n(x) \dd{x} = \int_a^b f(x) \dd{x}? \end{align*}\]

In general, the answer is no and often it can be difficult or annoyingly ad-hoc to show the above is true (in fact, $f$ might not even be Riemann-integrable). Measure theory has an analogue of the above problem that is very easy (relative to Riemann integration) to prove it holds. See: Dominated convergence theorem and monotone convergence theorem.

A good goal here would be:

  • Lebesgue measure on $\RR$
  • Lebegue integration on $\RR$
  • Monotone and dominated convergence theorems

Functional analysis

Functional analysis studies various vector spaces of interest to analysts (or analysis-adjacent fields) by extending the methods of linear algebra to infinite dimensions.

[example=($L^p$ spaces)] Given a subset $X \subseteq \RR$, we define $L^p(X)$ be the class of functions $f:X \to \RR$ such that the norm

\[\|f\|_p \coloneqq \left(\int_X |f(t)|^p \right)^{1/p} \dd{t} < \infty.\]

when using Lebesgue integration from measure theory, it turns out that this space is complete as a topological space (and is what is called a Banach space – i.e. a complete normed vector space).

A very special case is when $p = 2$ in which there is an inner product structure given by

\[\langle f, g\rangle \coloneqq \int_X f(t)g(t) \dd{t}\]

which turns $L^2(X)$ into a Hilbert space (complete inner product space).

$L^p$ spaces, in general, are extremely important and vast amounts of analysis and applications of analysis directly use them or similarly-defined spaces (e.g. Sobolev spaces). [/example]

[example=(Spectral theory)] In linear algebra, we know that symmetric matrices are important because they are orthogonally diagonalizable (i.e. there is an orthonormal basis of the vector space using the eigenvectors) and the eigenvalues are real.

Very similar results show up when the vector space is allowed to be infinite-dimensional and this leads to the huge area of functional analysis called operator theory. [/example]

A good goal here would be:

  • $L^p$ spaces
  • Operator-theoretic tools (like Hahn-Banach)
  • Riesz representation theorem and how it ties functional analysis into measure theory

Partial derivatives

Another direction to go in the basics of multivariable calculus. Here, a lot of pathological behavior can develop but the added dimensions make the relationship between linear algebra and calculus much clearer. In particular, the Jacobian matrix is the derivative of maps $\RR^n \to \RR^m$ (with respect to the standard bases of $\RR^n$ and $\RR^m$). Since matrices are coordinate representations of linear transformations, this shows that derivatives are actually linear transformations.

The goal of this direction would be to understand the Inverse Function Theorem and Implicit Function Theorem.

Lie groups and lie algebras

Lie groups are groups equipped with a differentiable structure (i.e. one can do calculus on them). So one can study what it really means to take the derivative of a matrix, take exponentials, take logarithms, square roots, etc. Lie algebras arise as the tangent space of the lie group at its identity element. There are many directions one can go from here:

  • Differential geometry
    • Perhaps matrix groups or basics of Riemannian manifolds
  • Algebraic geometry
    • Affine/projective structures on matrix groups is one possibility
    • Schubert or flag varieties is another possibility
  • Differentiation equations
    • Can be pretty intense. A good goal would be to just understand how Lie groups can be used to recover all of the MATH 57 ODE solving methods.
  • etc.

Dynamical systems

Dynamical systems are another possibility. Due to how vast the field is (and also how differently-flavored it can be), I refer the reader to wikipedia’s overview and examples page and then discuss avenues to pursue if people are interested.

Differential equations

In the DEs realm, we could study existence-uniqueness (not super enlightening) of initial-value problems or dynamical systems (see above). Another possibility is to read a couple sections out of Evans’ Partial Differential Equations book (standard graduate PDEs book).

Stochastic analysis

Stochastic analysis, roughly speaking, is the study of stochastic processes (which can be naively thought of as time-varying random variables). There are a couple directions we could go here:

  • Markov chains
  • Martingales
  • Ito calculus and stochastic differential equations
  • etc.