I have a piece-wise function in $\mathbb{R}^2$,, which is bounded on $[a,b]$ . This function is bounded on [-5,5] and looks like
the function looks like this:
(the left vertical line is a actually a "curly bracket") defining the set of intervals that build up the piecewise function (which is shown here only in part for graphical limitations).
When I convert this to a Fourier series in Mathematica, I use the sum:
$$ y_k=\frac{a_0}{2}+\sum_{k=1}^\infty \alpha_k \cos nt +\beta_k\sin nt$$
where the components:
\begin{equation} \begin{array}{cc} \alpha_0=\frac{1}{2(b-a)}\int_{a}^b f(t)\text{d}t \\ \alpha_k=\frac{1}{b-a}\int_{a}^b f(t)\cos kt \text{d}t \\ \beta_k=\frac{1}{b-a}\int_{a}^b f(t)\sin kt \text{d}t \end{array} \end{equation}
are solved where $f(t)$ is given in the piecewise function given above.
This gives a nice analytic function, bounded on [-5,5]:
This new function looks like this:
$$0.591549 + (0.0345161 + 0.0266358 I) e^{-I t} + (0.0345161 - 0.0266358 I) e^{-I t} + (0.236242 + 0.227141 I) e^{2I t}+ (0.236242 - 0.227141 I) e^{2I t} + (0.217808 - 0.202757 I) e^{-3I t} + (0.217808 + 0.202757 I) e^{3I t} + (1.09053 - 0.00582322 I) e^{-4I t} + (1.09053 + 0.00582322 I) e^{4I t}+ (1.68527 - 0.355833 I) e^{-5I t} + (1.68527 + 0.355833 I) e^{5I t} + (0.362947 - 0.998453 I) e^{-6I t} + (0.362947 + 0.998453 I) e^{6I t} + (1.01148 - 0.0130695 I) e^{-7I t} + (1.01148 + 0.0130695 I) e^{7I t} - (0.398096 + 0.214324 I) e^{-8I t} - (0.398096 - 0.214324 I) e^{-8I t} + (0.138399 - 0.464199 I) e^{-9I t} + (0.138399 + 0.464199 I) e^{9I t} + (0.0348431 - 0.684121 I) e^{-10I t} + (0.0348431 + 0.684121 I) e^{10I t}$$
I want to define what this Fourier series projection actually is.
I assume this is a projection given by:
$$y_k: \mathbb{R}^2\longrightarrow C^\infty[a,b]$$
since we obtain a continuous infinitely differentiable function from a piecewise function in $\mathbb{R}^2$ by the Fourier series projection.
So would this assumption be correct, that we go from $R^2$ to a space of bounded sequence $C^\infty[a,b]$?
However, this function "repeats" itself outside the boundaries of the original piecewise-functions $f(t)$ , which are [-5,5]. So something tells me we are actually looking at an unbounded sequence, and it does not even converge on $[-\infty,\infty]$ since the plot is showing an infinitely repeated sequence of the Fourier series:
Function on the interval [-10, 10]:
Function on the interval [-50, 50]:
About the scope: I am looking for the right norm and metric, in order to do an analysis of this function on $[a,b]$.
But not being sure, I ask here:
What is the correct definition of the space of this repeating sequence, is it $C^\infty[a,b]$ or $\ell^\infty$? Or is it instead an $L^1$ space?
Clearly the function on $[a,b]$ forms a subspace which can be replicated infinitely many times. So I am thinking also about Hilbert space, but this is repeated infinitely many times on the same dimension, so we do not have infinitely many dimensions.
What I can tell from the plots, is that the norm of the function is finite. However, the function is unbounded if I increase the interval of plotting, so it appears to me to be an unbounded sequence, which is infinitely differentiable. But at the same time, this unboundedness is merely n-repetitions of that Fourier series.
So what is the correct answer here, and why?
Thanks