[[Noteworthy Uses]]:: *[[Noteworthy Uses]]* [[Proved By]]:: *[[Proved By|Crucial Dependencies]]* ---- > [!theorem] Theorem. ([[smoother functions' Fourier coefficients decay faster]]) > Let $k \geq 1$. If $f$ is $C ^{k}$ [[function on the (unit) circle|on the circle]], then $\ex B>0$ s.t. $|\hat{f}(n)| \leq \frac{B}{|n|^{k}}, \ \ \fa n \neq 0.$ ^edaea1 > [!intuition] > We often think about 'smoother' signals as possessing 'less high frequencies'. For example, > - [[Convolution]] of images $f$ with [[Gaussian]] kernels $g$ is used to model the image blurring ('smoothing') process, as each pixel is replaced with a weighted average of its neighbors. A [[Gaussian]] in the time domain corresponds to a gaussian in the frequency domain — a [[low-pass filter]] in light of [[the convolution theorem for Fourier series|the convolution theorem]] — and hence we intuit that smoother signals such as $f * g$ have less high frequencies than 'less smooth' signals such as $f$. > - More generally, [[Convolution creates continuous functions]]; moreover, if $f \in C^{k}$ and $g \in C^{r}$ then $f * g \in C^{\max(k,r)}$. So in the interesting cases, convolution with a low-pass filter (e.g., [[rectangle]], [[Gaussian]], etc) removing high frequencies is done via convolution with a low-pass filter, which incidentally smoothens your function too. > - One often images a prototypical 'noisy signal' as one which 'jitters about' some apparent [[ground-truth]] signal. The 'jittery-ness' of this signal intuitively gives the impression that lots of high frequencies are present... lots of 'sines would be needed' to approximate it > - When approximating certain discontinuous functions, such as the [[Fourier series of a square wave]], [[Gibbs phenomenon]] shows that no matter how many terms you add together, the approximation will struggle heavily at the discontinuities. > - In fact, the approximation will struggle *everywhere* — we see that isolated discontinuities — a *local* phenomenon — affect the *global* approximation. > - This is in part when we need to introduce the notion of [[L2 convergence of Fourier Series|mean-square convergece of Fourier series]] for functions not sufficiently smooth. > - The [[partial sum]]s of the [[Dirichlet Kernel]], in a loose sense, appear to approach $\delta$. In a sense, this is a [[Fourier series|Fourier partial sum]] in which all coefficients are $1$, illustrating that to approximate a not-very-smooth function with a 'discontinuity' as 'dramatic' as $\delta$ at $0$, we need all the high frequencies we can get. > > None of these examples provide anything more than a correlation between smoother functions and less high frequencies, but they do hint at a relationship which we are able to precisely state here. ^4765de > [!proof]- Proof. ([[smoother functions' Fourier coefficients decay faster]]) > Recall that [[derivative of Fourier coefficient of C1 function]]. From this we ascertain that for the $C^k$ $f$ assumed we have $\hat{f}^{(k)}(n)=(in)^{k} \hat{f}(n)$. Thus $\hat{f}(n)=\frac{\hat{f} ^{(k)}(n)}{(in) ^{k}}, \ \ \fa n \neq 0$. $(*)$ \ Since $f$ is $C^k$ [[function on the (unit) circle|on the circle]], $f ^{(k)}$ is [[continuous]] [[function on the (unit) circle|on the circle]], hence [[continuous]] on $[-\pi,\pi]$, hence [[bounded function|bounded]] ([[continuous]] on [[compact]]), hence there exists $B>0$ s.t. $|f ^{(k)}(\theta)| \leq B$ for all $\theta$. Thus by [[bounded function has bounded Fourier coefficients]] we get that $|\hat{f} ^{(k)}(n)| \leq B$ for all $n \in \zz$. Now take the [[modulus]] on both sides of $(*)$ to obtain $|\hat{f}(n)|= |\frac{\hat{f} ^{(k)}(n)}{n ^{k}}| \leq \frac{B}{|n |^k}$ as desired. ^f581a4 ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```