APM346-2012 > Home Assignment 5

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Victor Ivrii:
Some of subproblems are just tricks:

Example: Decompose into full F.s. $\cos (mx)$. Answer: $\cos (mx)$ as it is one of the basic functions (however it would not be so if we decompose into $\sin$-F.s., or F.s with respect to $\sin((n+\frac{1}{2}x)$, $n=0,1,\ldots$.


What is the sum F.s. on $(-\infty,\infty) was discussed in http://www.math.toronto.edu/courses/apm346h1/20129/L16.html

Again see below:

Chiara Moraglia:
I'm confused over why the sketches for the full Fourier series and the sine Fourier series are different. They have the same formula. I would have expected them both to look like the plot for the sine Fourier series. Thanks.

Zarak Mahmud:

--- Quote from: Chiara Moraglia on November 10, 2012, 04:07:41 PM ---I'm confused over why the sketches for the full Fourier series and the sine Fourier series are different. They have the same formula. I would have expected them both to look like the plot for the sine Fourier series. Thanks.

--- End quote ---

For the function $f(x) = x$? Yes I think they should look the same.

Victor Ivrii:
If we are decomposing into full F.S. $f(x)=x$ on interval $[-l, l]$ then it is the same as to decompose into sin-F.S. $f(x)=x$ on interval $[0, l]$.

But if we decompose into full F.S.  $f(x)=x$ on interval $[0, l]$ then we have different picture: function is $l$-periodic and exactly this is on the picture (note that bolder line shows graph on the original interval)

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