I have a question I am phrasing in terms of expansion of a function in terms of Hermite polynomials but it applies to other expansions as well.
First I establish my convention for Hermite polynomials which is
$$ (-1)^n \partial_z^n \varphi(z)= H_n(z) \varphi(z) $$
where $\varphi(z) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{z^2}{2}}$. Under this the first few Hermite polynomials are
$$ \begin{eqnarray} H_0 &=& 1 \\ H_1 &=& z \\ H_2 &=& z^2-1 \\ \dots \end{eqnarray} $$
Now we can expand a function in Hermite polynomials
$$ f(z)=\sum_{n=0}^\infty c_n H_n(z) $$
and we can extract the coefficients by
$$ c_n = \frac{1}{n!} \int_{-\infty}^\infty dz~ \varphi(z) H_n(z) f(z) $$
All this is fine. What I want to know what is the problem with the following. Suppose I have the function value only in the interval $z \in [a,b]$ and we compute
$$ \begin{eqnarray} \tilde c_m &=& \frac{1}{m!} \int_{a}^b dz~ \varphi(z) H_m(z) f(z) \\ &=& \frac{1}{m!} \int_{a}^b dz~ \sum_{n=0}^\infty c_n ~\varphi(z) H_m(z) H_n(z) \\ &{\bf \color{blue}{\stackrel{?}{=}}}& \frac{1}{m!} \sum_{n=0}^\infty c_n \int_{a}^b dz~ ~\varphi(z) H_m(z) H_n(z) \\ &=& \frac{1}{m!} \sum_{n=0}^\infty \gamma_{m,n} c_n \end{eqnarray} $$
where I have defined
$$ \gamma_{m,n} := \int_{a}^b dz~ ~\varphi(z) H_m(z) H_n(z) $$
which can be evaluated explicitly. My question is I can have two functions that are the same in the interval $z \in [a,b]$. $\gamma_{m,n}$ are independent of the function and therefore I cannot see how the above can be true. Said differently it seems I can invert the relation and from $\tilde c_m$ get the true $c_n$ which is ridiculous because I should not be able to read of the full function on the real line from the value inside an interval. So what am I doing wrong? I suspect the problem is with the step that I put a question mark on but what is specifically the problem?
You are correct in that a general function cannot be determined by its values on any finite interval, but if you know that the function is given by a power series, for example, then it is determined by its values on any interval, however small. This isn't really all that surprising. If we know that a function is a quadratic polynomial, then it is determined by its values at three points.