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A Fourier Cosine Series uses an infinite sum of only cosine waves to represent a target function whose left endpoint is $0$, by considering its even extension. The even extension seems to amount to just adding a reflected version of the original function to the original function.

A Fourier Sine Series uses an infinite sum of only sine waves to represent a target function whose left endpoint is $0$, by considering its odd extension. The odd extension seems to amount to just adding a rotated version of the original function to the original function.

On one hand, these cases do seem to have a certain special simplicity to them. However, there are many other types of transformations, as well as many other sinusoid phases. Is it possible, then, to create a different extension by adding a differently-transformed version of the original function to itself, with the end goal of using something other than $sin(x)$ or $cos(x)$ as the fundamental wave in its own Fourier ____ Series?

As a hypothesized example, would it be possible to add a version of a function to itself, which had undergone a transformation "half way between a reflection and a rotation," to create a Fourier Half-way-between-cosine-and-sine Series that uses $\cos(x - \frac{\pi}{4})$ as the fundamental wave? If not, is there any other way to create such a series with $\cos(x - \frac{\pi}{4})$ (or an arbitrary-phase sinusoid) as the fundamental wave?

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  • $\begingroup$ Did the answer I posted below not answer your question? If not, could you please write a precise mathematical definition of your function of interest similar to formulas (1) and (4) for $f(t)$ and $g(t)$ in the answer I posted below? $\endgroup$ – Steven Clark Apr 18 at 19:16
  • $\begingroup$ @StevenClark I think for most practical purposes it does, since it is always possible to make a substitution of the independent variable to shift the target function horizontally. So based on your answer, there is a continuum of options between $\cos(nt)$ and $\cos(nt - 2\pi)$ from which the mathematician can choose to construct the Fourier ____ Series of a function whose interval goes from $0$ to $\pi$. One such option happens to be $\cos(nt - \frac{\pi}{2})$, a.k.a. $\sin(nt)$. $\endgroup$ – user10478 Apr 19 at 19:00
  • $\begingroup$ However, when constructing a Fourier Sine Series, instead of shifting the target function by $\frac{\pi}{2}$, the usual approach is to consider a different extension of the function (the odd extension instead of the even extension), which amounts to applying a different transformation to the graph from $0$ to $\pi$ to be used as the portion of the graph from $-\pi$ to $0$. Is there a different type of transformation we can use to extend a function originally defined from $0$ to $\pi$, such that $\cos(nt - \frac{\pi}{4})$ becomes the appropriate Fourier ____ Series without further shifting? $\endgroup$ – user10478 Apr 19 at 19:00
  • $\begingroup$ Is there a continuum of such transformations that yield extensions "between" even extensions and odd extension (and another continuum cycling back to even extensions), mirroring the cyclical nature of sinusoids? $\endgroup$ – user10478 Apr 19 at 19:00
  • $\begingroup$ It seems to me your first comment above is related to formulas (1) to (3) and Figure (1) in my answer below, and your second comment above is related to formulas (4) to (6) and Figure (2) in my answer below, but note that formulas (2) and (5) in my answer below use $\cos\left(n \left(t-\frac{\pi }{4}\right)\right)$ and $\sin\left(n \left(t-\frac{\pi }{4}\right)\right)$, not $\cos\left(nt - \frac{\pi}{4}\right)$ and $\sin\left(nt - \frac{\pi}{4}\right)$. I really don't understand what you're looking for in your third comment above. $\endgroup$ – Steven Clark Apr 19 at 21:58
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You can represent a shifted square wave with a shifted Fourier series.


For example the shifted square wave with peak-to-peak amplitude $2$ and period $2 \pi$ defined as follows

$$f(t)=\left\{ \begin{array}{cc} -1 & -\frac{\pi }{4}\leq t<\frac{3 \pi }{4} \\ 1 & \frac{3 \pi }{4}\leq t<\frac{7 \pi }{4} \\ f(t-2 \pi\ \text{sgn}(t)) & \text{True} \\ \end{array} \right.\tag{1}$$

can be represented by the following shifted Fourier series:

$$f(t)=\underset{N\to\infty }{\text{lim}}\left(\sum_{n=1}^N a(n) \cos \left(n \left(t-\frac{\pi }{4}\right)\right)\right)\tag{2}$$

where

$$a(n)=\frac{2 \left(\sin \left(\frac{3 \pi n}{2}\right)-\sin \left(\frac{\pi n}{2}\right)\right)}{\pi n}\tag{3}$$


The following plot illustrates formula (2) for $f(t)$ in orange overlaid on the reference function defined in formula (1) in blue where formula (2) is evaluated using an upper evaluation limit of $N=16$.


Illustration of Formula (2) for f(t)

Figure 1: Illustration of Formula (2) for $f(t)$


As another example the shifted square wave with peak-to-peak amplitude $2$ and period $2 \pi$ defined as follows

$$g(t)= \left\{ \begin{array}{cc} -1 & -\frac{3 \pi }{4}\leq t<\frac{\pi }{4} \\ 1 & \frac{\pi }{4}\leq t<\frac{5 \pi }{4} \\ g(t-2 \pi\ \text{sgn}(t)) & \text{True} \\ \end{array} \right.\tag{4}$$

can be represented by the following shifted Fourier series:

$$g(t)=\underset{N\to\infty }{\text{lim}}\left(\sum_{n=1}^N b(n) \sin\left(n \left(t-\frac{\pi }{4}\right)\right)\right)\tag{5}$$

where

$$b(n)=-\frac{2 \left((-1)^n-1\right)}{\pi n}\tag{6}$$


The following plot illustrates formula (5) for $g(t)$ in orange overlaid on the reference function defined in formula (4) in blue where formula (5) is evaluated using an upper evaluation limit of $N=16$.


Illustration of Formula (5) for g(t)

Figure 2: Illustration of Formula (5) for $g(t)$

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