Periodic Extensions

If the function we want to investigate is not periodic or is only defined on an interval, we will want to use the something called the periodic extension of the function. We will limit ourselves to functions defined on [-L, L] with a finite number of jump or removable discontinuities.

Some ideas you may (or may not) have seen in calculus:

The left-hand limit of f at a point x0 is
lim (x->x0 from the left) f(x).
We denote this as f(x0-).

The right-hand limit of f at a point x0 is
lim (x->x0 from the right) f(x).
We denote this as f(x0+).

Note that if f(x) is continuous at x0, then f(x0-)=f(x0+)=f(x0)

With these ideas we can define left and right handed derivatives by taking left and right handed limits of the difference quotient.

To define the periodic extension of f(x), written f^(x) and pronounced "f hat", we first redefine f at points of discontinuity:
If f is discontinuous at x0, define f^(x0) as the average value of f(x0-) and f(x0+):
f^(x)=1/2 [f(x0-)+f(x0+)]
(Note that this matches the convergence of orthogonal series at points of jump discontinuity.)

If f is continuous at x0, we do not redefine it:
we take f^(x0)=f(x0).

To define f^ outside of [-Pi, Pi] simply extend the function to make it periodic:
If x is not in [-Pi, Pi], f^(x+2Pi)=f^(x)

This is actually quite easy. Consider f(x)=x on -Pi to Pi.


  Plot[x,{x,-Pi,Pi}];

Let's find the periodic extension.


  part1=Plot[x+2Pi,{x,-3Pi,-Pi},DisplayFunction->Identity];
  part2=Plot[x,{x,-Pi,Pi},DisplayFunction->Identity];
  part3=Plot[x-2Pi,{x,Pi,3Pi},DisplayFunction->Identity];
  part4=ListPlot[{{-3Pi,0},{-Pi,0},{Pi,0},{3Pi,0}}
  ,PlotStyle->{RGBColor[1,0,0],PointSize[.02]},DisplayFunction->Identity];
  graphoff=Show[part1,part2,part3,part4,
  DisplayFunction->$DisplayFunction,
  AspectRatio->Automatic,Ticks->{{-3Pi,-Pi,Pi,3Pi},Automatic}];

It is pretty simple once you see it.

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