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Given an even $f \in L^2([-\pi,\pi]),$ we can write $f$ as a Fourier Cosine Series as such

$$f(x)= \sum_{n=0}^{\infty} \alpha_n \cos(nx),$$ where $$\alpha_0= \int_{- \pi}^{\pi} \frac{f(x)}{ 2 \pi} \ dx, \\ \alpha_n=\int_{- \pi}^{\pi} \frac{f(x) \cos(nx)}{ \pi} \ dx, n \in \mathbb{N}.$$ The reason why the normalization constants $\frac{1}{2 \pi}, \frac{1}{\pi}$ are in $\alpha_0, \alpha_n,$ respectively is that

$$\int_{-\pi}^{\pi} \frac{\cos(mx) \cos(nx)}{\pi} \ dx= \delta_{m,n}, m,n \in \mathbb{N}$$ and $$\int_{-\pi}^{\pi} \frac{\cos(nx)}{2\pi} \ dx= \delta_{0,n}, n \in \mathbb{N}.$$

Now suppose $f \in L^2( \mathbb{R})$ is even. We define the Cosine Transform as

$$\mathcal{F}_c(f(x))= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(x) \cos(xy) \ dx.$$ How can we obtain the $\frac{1}{\sqrt{2 \pi}}$ in a manner similar to that of the series ? I know it turns out to make the Fourier Inversion Formula nice, but assuming I hadn't known about the Fourier Transform Properties, how can we obtain this normalization constant ? How do we know it could not be any other value like $\frac{1}{\sqrt{4 \pi}},\frac{1}{\sqrt{6 \pi}},\frac{1}{\sqrt{8 \pi}},$ .. etc ?

I also informally made some other observations.

The structure of the transform seems to indicate it is the continuous analogue of the $\alpha_n$ used in the Cosine Series, and by that logic, it looks like the double integral $$f(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(z) \cos(zy) \cos(xy) \ dz \ dy,$$ is the continuous analogue of the entire series. Are these observations "correct" ?

I noticed while playing around with the cosine integral that $$\delta(m-n)= \frac{1}{ 2 \pi} \int_{-\infty}^{\infty} \cos(mx)\cos(nx) \ dx$$ if we now let $m,n \in \mathbb{R}.$ So in a sense, can the Dirac Delta Distribution can be viewed as a continuous analogue of the Kronecker Delta Function, and as before, why does the $\frac{1}{ 2 \pi}$ make things work ?

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  • $\begingroup$ It seems to me that there are some set-up issues: for one, the function $f$ is not equal in an $L^2$ sense to "its cosine expansion" unless it is not only $2\pi$ periodic but also even. Yes, the usual "Fourier series" can be expressed either in terms of exponentials or in terms of cosines and_sines... If there are further tacit assumptions on $f$, etc., could you make them explicit, please? This (possibly) subordinate issue pervades the rest. $\endgroup$ – paul garrett Mar 29 '17 at 23:59
  • $\begingroup$ I think I have made everything explicit, as I have edited it. $\endgroup$ – Vivek Kaushik Mar 30 '17 at 1:54
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Indeed, that normalizing constant is often presented as though it were a magical insight. In fact, the $2\pi$ can be inserted in various fashions into "Fourier transform" or "cosine transform" on even functions. Once we discover that a normalizing constant must occur, and what it is, the only question is where to put it.

To discover the necessary constant when we have fixed other choices (for example, $\cos(xy)$ rather than $\cos(2\pi xy)$), apply a non-normalized transform and inverse transform to something whose transform is easy to understand, such as a Gaussian $f(x)=e^{-x^2}$, and discover the constant needed to make things match up.

A slightly new issue arises, namely, that if we (reasonably-enough, as a heuristic) interchange the order of integration, we seem to find that $\int_{-\infty}^\infty \cos(mx)\,\cos(nx)\;dx$ is some multiple of Dirac's $\delta$. This is entirely correct, if interpreted properly, e.g., not as a numerical integral, but "integral/cosine transform" in an extended sense (e.g., on tempered distributions, extended by continuity).

Again, the $2\pi$ is an artifact that must appear somewhere in these formulas, as we discover. No, it is completely not obvious that your (extended-sense) integral for $\delta$ needs the $2\pi$ to be correct. That is really only discovered by looking at Fourier inversion for nicer functions (e.g., Schwartz functions).

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  • $\begingroup$ What are some sources you recommend I look at to understand the things you have mentioned in greater detail ? $\endgroup$ – Vivek Kaushik Apr 12 '17 at 18:13
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    $\begingroup$ Strichartz' book on Fourier transforms and distributions is a very nice introduction, with good explanations, and at the same time some of the more technical proofs are delayed (which is a good strategy, in my opinion). Folland's TIFR notes on PDE (available on-line) introduce Fourier transforms and tempered distributions in a very practical (but rigorous) fashion. Don't look in a "general" functional analysis book, because such things would typically be too mixed in with other issues, if they appear at all. And unfortunately many "real analysis" books do not mention such things. $\endgroup$ – paul garrett Apr 12 '17 at 20:49

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