Question Fourier half range expansion of function on 0 < x < p

[yungman]

I am looking into Fourier series expansion of function that is non periodic on 0< x< p. This is under the tittle of "Half range expansion".

All the books only talked about even or odd extension of f(x) to -p <x< p using either Fouries cosine series expansion or sine series expansion. But never using both for extension.

If f(x) has both sine and cosine series expansion, it is possible to expand with both sine and cosine. the difference is with both expansion, f(x) = 0 for -p<x<0, but still the expansion is valid for 0<x<p. To me this is more closer to the original f(x)... Instead of odd of even extension where the part in -p<x<0 is totally different shape from the f(x). To me I rather just null out the part -p<x<0.

Can anyone tell me a better reason?

 

[LCKurtz]

Your choice of how to define the function on (-p,0) can affect the rate of convergence of the resulting series. For example, suppose you want to get x² on (0,p). If you define f(x) = 0 on (-p,0) or use the odd extension, the extended periodic function you get will have discontinuities at multiples of p. But the even extension gives a continuous periodic function. Its FS will converge faster then either of the others. Try it. You will get coefficients of order 1/n² for the even extension and 1/n for the others.

[Edit] I should add for completeness there are some unstated additional assumptions like the Dirichlet conditions. Otherwise a continuous periodic function can have a FS that doesn't converge to it.

 

[yungman]

Thanks for the reply. But I think there is a place for having both sine and cosine expansion as I presented. Like you said, it is also driven what outcome you want eg. even or odd, faster convergence and in my case zero for -p<x<0.

 

[LCKurtz]

Thanks for the reply. But I think there is a place for having both sine and cosine expansion as I presented. Like you said, it is also driven what outcome you want eg. even or odd, faster convergence and in my case zero for -p<x<0.

If you specify something other than even or odd extension, you are not doing a half range expansion. You are just doing an ordinary full range FS which will have both sine and cosine terms.

 

[yungman]

If you specify something other than even or odd extension, you are not doing a half range expansion. You are just doing an ordinary full range FS which will have both sine and cosine terms.

I see, so the definition of Half range expansion is either sine or cosine expansion only.

Thanks for clearing this up.

 

[sivasankarm]

Find the half-range cosine and sine expansions of the given function:

f(x) = 0 if 0<x<1/2, f(x) = 1 if 1/2<x<1


sir please send the solution as early as possible.
thanking you sir.....

 

[LCKurtz]

Find the half-range cosine and sine expansions of the given function:

f(x) = 0 if 0<x<1/2, f(x) = 1 if 1/2<x<1


sir please send the solution as early as possible.
thanking you sir.....

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