I have a problem which I have not been able to solve. I want to plot a function and and operator which approximates it when you let w
to infinity. I will give all needed information for MWE and my faults.
MathJax
$$ \operatorname{Fejer}(x):= \dfrac{1}{2} \operatorname{sinc}^2\left(\dfrac{x}{2}\right) \quad (x\in \mathbb{R})\\ \operatorname{sinc} x:= \begin{cases} \dfrac{\sin \pi x}{\pi x}, & x\in \mathbb{R} \backslash \{0\}\\ 1, & x=0 \end{cases}\\ \operatorname{Function}(x):=\begin{cases} \dfrac{9}{x^2},& x<3\\ 2,& -3<x<-2\\ -\dfrac{1}{2},& -2<x<-1\\ \dfrac{3}{2}, & -1<x<0\\ 1,& 0<x<1\\ -1,& 1<x<2\\ 0,& 2<x<3\\ -\dfrac{50}{x^4},& 3<x\\ \end{cases} $$
Code
sinc[x_] := Piecewise[{{1, Equal[x, 0]}, {Sin[Pi x]/(Pi x), True}}]
Fejer[x_] := 1/2*sinc[x/2]^2
function[x_] :=
Piecewise[
{{9/(x^2), x < -3},
{2, -3 <= x < -2},
{-1/2, -2 <= x < -1},
{3/2, -1 <= x < 0},
{1, 0 <= x < 1},
{-1, 1 <= x < 2},
{0, 2 <= x < 3},
{-50/(x^4), 3 <= x}}]
constant[x_] := 1
With above code I am defining the functions which I gave mathematically above to make it easier for you. Now, I'm trying to define my operator.
operator[w_, kernel_, func_, x_] :=
Sum[
w *
Integrate[func[u], {u, k/w, (k + 1)/w}, Assumptions -> k ∈ Integers] *
kernel[w*x - k],
{k, -Infinity, Infinity}]
I'm not sure about the above code. I used Assumption
in the integral because I was getting errors like "integral limits may not reals, please add assumption". I also show it in MathJax so you can understand what I'm trying to do.
Operator
$$ (S_wf)(x):=\sum_{k\in \mathbb{Z}} \chi(wx-k) w\int_{k/w}^{(k+1)/w} f(u)du , \quad x\in \mathbb{R}, \, w>0$$
When you take function to br $1$ for every $x\in \mathbb{R}$, operator gives $1$. Anyway when I running code
operator[w, Fejer, constant, x]
or
operator[5, Fejer, constant, x]
it gives nothing. When I tried plot
Plot[Operator[5, Fejer, cons, x], {x, 0, 5}]
it quits the kernel without an error.
When I tried
Plot[Operator[5, Fejer, function, x], {x, 0, 5}]
It gives many errors and some of them are:
NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate::nlim: u = 0.2 k is not a valid limit of integration.
General::stop: Further output of NIntegrate::nlim will be suppressed during this calculation.
Finally, I'm adding a result which I'm trying to reach.
GSKS
is not defined. $\endgroup$