Plotting function and its approximation function

Question

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.

Answer 1

Clear["Global`*"]

$Version

(* "12.2.0 for Mac OS X x86 (64-bit) (December 12, 2020)" *)

The definition of Fejer can be simplified to

Fejer[x_] := 1/2*Sinc[x Pi/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}}]

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}]

constant[x_] := 1

Operator[w, Fejer, constant, x] // FullSimplify

(* 1/4 (2 + (1/(π^2 w^2 x^2))
   Cos[π w x] (-4 + 
      w^2 x^2 (4 PolyGamma[1, w x] + PolyGamma[1, 1/2 - (w x)/2] - 
         PolyGamma[1, 1 + (w x)/2])) + Sec[(π w x)/2]^2) *)

Evaluate the function to be plotted

Plot[Operator[5, Fejer, constant, x] // FullSimplify // Evaluate,
 {x, 0, 5}, WorkingPrecision -> 20, PlotRange -> {0, 1.1}]

This example is quite slow

Plot[Operator[5, Fejer, function, x] // N[#, 30] & // Evaluate, 
 {x, 0, 5}, WorkingPrecision -> 25]

EDIT: Higher precision is needed for negative values of x

Plot[Operator[5, Fejer, function, x] // N[#, 40] & // 
  Evaluate, {x, -5, 5}, WorkingPrecision -> 30]