# Toronto Math Forum

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

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 $(-\infty,\infty) was discussed in http://www.math.toronto.edu/courses/apm346h1/20129/L16.html (http://www.math.toronto.edu/courses/apm346h1/20129/L16.html) Again see below: Title: Re: About HA5 Post by: Chiara Moraglia on November 10, 2012, 04:07:41 PM I'm confused over why the sketches for the full Fourier series and the sine Fourier series are different. They have the same formula. I would have expected them both to look like the plot for the sine Fourier series. Thanks. Title: Re: About HA5 Post by: Zarak Mahmud on November 10, 2012, 05:07:56 PM I'm confused over why the sketches for the full Fourier series and the sine Fourier series are different. They have the same formula. I would have expected them both to look like the plot for the sine Fourier series. Thanks. For the function$f(x) = x$? Yes I think they should look the same. Title: Re: About HA5 Post by: Victor Ivrii on November 10, 2012, 05:25:45 PM If we are decomposing into full F.S.$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)