Different ways to represent functions other than Laurent and Fourier series?

Question

In the book "A Course of modern analysis", examples of expanding functions in terms of inverse factorials was given, I am not sure in today's math what subject would that come under but besides the followings : power series ( Taylor Series, Laurent Series ), expansions in terms of theta functions, expanding a function in terms of another function (powers of, inverse factorial etc.), Fourier series, infinite Products (Complex analysis) and partial fractions (Weisenstein series), what other ways of representing functions have been studied? is there a comprehensive list of representation of functions and the motivation behind each method?

For example , power series are relatively easy to work with and establish the domain of convergence e.g. for $ \sin , e^x \text {etc.}$ but infinite product representation makes it trivial to see all the zeroes of $\sin, \cos \text etc. $

Also if anyone can point out the subject that they are studied under would be great.

Thank you

Answer 1

You can add the following to your list of representations

1. Continued fractions. See Jones & Thron or Lorentzen's books
2. Integral representations (Mellin–Barnes, etc). See the ECM entry
3. Exponentials. See Knoebel's paper Exponentials reiterated
4. Nested radicals. See Schuske & Thron's Infinite radicals in the complex plane
5. Rational or polynomial approximation (e.g Padé approximants)

I guess the general topic is studied under Complex Analysis, Asymptotic Analysis, Harmonic Analysis. AFAIK, there is no single book which covers all representations.

Answer 2

There are literally dozens of ways to represent "arbitrary" (given or unknown) functions $f$ in terms of "special" functions. Each of these ways responds to the particular geometrical situation at hand, to ways of encoding the available information about $f$, to a-priori-conditions that $f$ must fulfill, etc. The special functions used in each particular case are "taken from a catalogue", they are well understood and usually have an algebraically describable behavior with respect to the natural operations (differentiation, shifts, rotations, etc.) present in the given environment.

Answer 3

This may not be exactly what you are looking for, but in physics a common example of this shows up in quantum mechanics.

The time-independent schroedinger equation in one dimension is:

$ - k \frac{\partial^2 \psi}{\partial x^2} + V(x) \psi = E \psi $

which is essentially an eigenvalue problem for the differential operator

$ - k \frac{\partial^2}{\partial x^2} + V(x) $

where $k$ is a parameter and $V(x)$ is a given function to describe a given physical situation (in physics terms it is the potential of the system).

What happens is that (for suitable $V(x)$) if one imposes a boundary condition on $\psi$, such as $\psi$ must be zero at infinity, then one finds that the eigenvalues of the above operator form a discrete set $\lbrace E_n \rbrace$, and the corresponding eigenfunctions $\lbrace \psi_n(x) \rbrace $ form a complete set for $L^2(\mathbb{R})$. (complete here means that any function can be decomposed as a series $\sum a_n \psi_n$ much like a fourier series).

In fact, if $V(x)$ is taken to be the infinite square well potential ( http://en.wikipedia.org/wiki/Infinite_square_well ) one gets that

$\psi_n(x) = \sin(n \pi x / L)$

and that the $\psi_n$ form a complete set is just the statement that functions can be decomposed into a fourier series. At any rate, one way to think of this whole thing is that by choosing different $V(x)$'s you can generate some fairly weird looking complete sets. For example take a look at the eigenstates portion of:

http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

Here $\psi_n = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right) $ forms a complete set.

Also I believe the study of this kind of problem falls under Functional Analysis.

Answer 4

To add to Sandeep's list, one can (formally) expand a given function $f(x)$ in terms of a basis function $g(x)$ through the so-called (Lagrange-)Bürmann series.

If in addition to the two previously mentioned functions, you are given an expansion point $a$, and the assumption that $g^{\prime}(x)$ is nonzero over some interval containing $a$, such that $g(x)-g(a)$ is monotonic as $x$ increases over said interval, you can obtain the Bürmann series of $f(x)$ with respect to basis function $g(x)$ and expansion point $x=a$ is

$$f(x)=f(a)+\sum_{k=1}^{\infty} \frac{\beta_k(a)}{k!}(g(x)-g(a))^k$$

where the $\beta_k(x)$ satisfy the recursion

$$\beta_0(x)=f(x),\qquad \beta_k(x)=\frac1{g^{\prime}(x)}\frac{\mathrm d}{\mathrm dx}\beta_{k-1}(x)$$

The series may or may not converge, of course; you're on your own to check the convergence of this expansion.

In addition, one can also consider a continued fraction analog of the Bürmann series:

$$f(x)=f(a)+\cfrac{g(x)-g(a)}{\varphi_1(x)+\cfrac{g(x)-g(a)}{\varphi_2(x)+\cdots}}$$

where the $\varphi_k(x)$ satisfy the recursion

$$\varphi_0(x)=f(x),\qquad p_{-2}(x)=p_{-1}(x)=0,\qquad p_k(x)=p_{k-2}(x)+\varphi_k(x),\qquad \varphi_k(x)=k\frac{g^{\prime}(x)}{\frac{\mathrm d}{\mathrm dx}p_{k-1}(x)}$$

Items 3 and 4 (along with Taylor series and continued fractions) in Sandeep's list can be unified under the heading "continued function representations".