# Rigorous development of a continuous distributional basis for $L^2$

In control and signal processing, it's well-known that the impulse response (forcing by $\delta(t)$) of a linear time-invariant system completely determines its behavior. In discrete time the argument proceeds by constructing a basis of the discrete delta functions

$$\delta(n-m) = \begin{cases} 1, \ \ n = m\\ 0, \ \ n \neq m\\ \end{cases},$$

and observing that an arbitrary signal $u(n)$ has the trivial reconstruction $$u(n) = \sum_{m=-\infty}^\infty u(m)\delta(n-m).$$ This in turn allows the system transfer function $H$ (which maps $\ell^2$ into itself) to act on the basis rather than the coefficients. Since the output $y(n)$ is related to the input $u(n)$ via $y(n) = Hu(n)$, by defining impulse response $h(n) = H\delta(n)$ we obtain $$y(n) = \sum_{m=-\infty}^\infty u(m)h(n-m).$$ Since $u(n)$ is arbitrary, all system specific information (e.g. BIBO stability) can be deduced from $h(n)$.

Speaking informally, the continuous time analog should be obtained by reducing $\delta(n)$ to $\delta(t)$, and the sum to an integral. But in this case we need deal with the notion of a continuous basis of $L^2$--and a basis of distributions at that! What is the state of formal work in this area and is there an easy entry/reference for someone with basic knowledge of analysis / topology / linear algebra and some knowledge of functional analysis?

• What I've seen were just approximations of vectors $L^{2}$ by linear combinations of a sufficiently large, but finite basis. More generally, you probably want dense, countable subsets of the $L^2$ in question. For example, a signal periodic on an interval $[0, T]$ can be decomposed into its Fourier series. Then every Fourier sum is a linear combination of the finitely may Fourier basis functions. – avs Dec 19 '16 at 21:13
• I'm looking for a basis of the form $\{\delta(t-\tau) \ |\ \tau \in R\}$. This is possibly dense but certainly not countable. This is the entire point, as the theory of countable bases is classical. Fourier series works by projecting $R \rightarrow R/TZ$, e.g. the circle group for functions of period $T$, and constructing an eigenseries of the characters $e^{int}$. An interesting (and well trod) topic, but perhaps different than what I am concerned with. – Mortified Through Math Dec 19 '16 at 21:22
• P.S. I am also aware there are plenty of great countable bases of $L^2$. The specific distributional basis I am interested in has specific physical interpretations, which privilege it in the applications with which I am concerned. The question is specifically about either this basis or a class of bases which include this one. – Mortified Through Math Dec 19 '16 at 21:25
• sorry if I didn't grasp the goals you are pursuing by looking for a continuous basis. Could you elaborate on that more? Namely, what goals do countable bases fail to achieve that you hope to achieve by a continuous analogue of a basis? – avs Dec 19 '16 at 21:37
• Are we talking about convolution, then? I.e., the continuous analog of $$u(n) = \sum_{m=-\infty}^\infty u(m)\delta(n-m);$$ namely, $$u(t) = \sum_{\tau = -\infty}^\infty u(\tau)\delta(t-\tau)?$$ – avs Dec 19 '16 at 23:29