Some of the following may be nonsense but...
The way I interpret the article is that \{exp(inx)\}_{n\in \mathbb{Z}} is a complete orthonormal basis for the space of real valued square integrable smooth function on (-\pi,\pi) of domain (-\pi,\pi) with inner product defined by
\langle f,g \rangle =\int_{-\pi}^{\pi}f(x)g(x)dx.
This allows us to write, for any function f of that space,
f(x) = \sum_{n\in \mathbb{Z}}\langle f,\exp(inx) \rangle \exp(inx)
Using the language of vector spaces, <f,exp(inx)> is the projection of the vector f in the direction of the unit vector \exp(inx), i.e. the component of f in the direction of \exp(inx)
More generally,
\left\{ \exp \left( in\frac{2\pi}{P}x \right) \right\} _{n\in \mathbb{N}}
is a complete orthonomal basis for the space of real valued periodic functions of period P.
For functions that are NOT periodic, but that have the property that the integral
\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}f(x)\exp (-in\omega)dx = F(\omega)
converges, we can write them in a kind of "continuous" form of a Fourier series, i.e. as its Fourier transform:
f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}F(\omega)\exp (inx)d\omegaNow let's make the analogy with the Laplace transform. Suppose a function f:(0,+\infty)\subset D \rightarrow \mathbb{R} can we written as the Laplace transform of some function a(t):
f(s) = \int_0^{+\infty} a(t) \exp(-st)dt
Note that \exp(-st)=(e^{-s})^t. Make the substitution e^{-s}=x and the equation becomes a restriction of f to the positive real line:
f|_{\mathbb{R}^+} = f(x) = \int_0^{+\infty} a(t) x^t dt
This is a kind of "continuous" form of a power series, right?
So to answer your question...
jbusc said:
I am curious, though, if something similar exists for laplace transform
...the analogue is the set \{x^n\}_{n\in \mathbb{N}}, which is a complete orthonormal basis for, say, the space of real valued function developable in a Taylor series of convergence radius R and of domain (-R,R), with the inner product defined as...
...as what exactly? Also, maybe \{x^n\}_{n\in \mathbb{N}} is not orthoNORMAL, but just orthogonal. What inner product would yield
\langle f,x^n \rangle = \frac{f^{(n)}(0)}{n!}
and
\langle x^m,x^n \rangle = 0
except for m=n
??