1

Schutz's book (p.10) mentions in passing a theorem in functional analysis that says that any square-integrable function $g$ may be approximated by an analytic function $f$ in such a way that the integral of $(g-f)^{2}$ over the domain of concern may be made as small as one wishes. And that this justifies why in physics we don't hesitate much to assume that a function is analytic if it helps establish a result.

My question: what is this theorem exactly (does it have a name)? And does the inverse hold: will an analytic function always be square integrable, say over an open set in $R^{n}$ ? Any physical examples will also be appreciated.

For example: if I have a wave function that is not square integrable over some region in $R$ (or over all of $R$) can I represent it as a Taylor series? Or as Fourier or power series in general?

user135626
  • 1,339
  • 1
    Squaer integrable where? Did you consider the identity fucntion? – Kavi Rama Murthy Jul 07 '21 at 23:16
  • @Kavi apologies. I updated the post and chose some open set in $R^{n}$. – user135626 Jul 07 '21 at 23:37
  • I’d say that’s a bit of an oversimplification on the author’s part, but so be it. Replacing ‘analytic’ with ‘smooth’ it’s a density argument. A nice discussion here. – A rural reader Jul 08 '21 at 00:21
  • I don't have access to the book. Does the author mean analytic in the complex or real sense? If the latter, you can use something like density fo continuous functions and the (Stone-)Weierstrass approximation theorem to see that (real!) polynomials approximate a square integrable function. – Jose27 Jul 08 '21 at 01:27
  • @jose27 the author in this remark was discussing real analyticity, I believe. So, is an analytic function in this sense bound to be always square integrable? For example, if the integral in polar coordinates $\int_{a}^{\infty} [f(r)]^{2}rdr$, with $a>0$, gives infinity, can we still expect to be able to express $f(r)$ by a series (Talyor or any power series)? Or would such function be impossible to write as a series? – user135626 Jul 08 '21 at 03:14
  • I don't understand; are you asking if a non square integrable function can be analytic? For instance $|x|^b$ for $-1\leq b<\infty$ all satisfy your condition and are analytic. – Jose27 Jul 08 '21 at 20:09

0 Answers0