Toronto Math Forum
APM3462012 => APM346 Math => Home Assignment 5 => Topic started by: Victor Ivrii on November 01, 2012, 05:10:42 AM

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 (http://www.math.toronto.edu/courses/apm346h1/20129/L16.html)
Again see below:

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.

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

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