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Results tagged with fourier-analysis
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user 5371
The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.
22
votes
Accepted
When I can safely assume that a function is a Laplace transform of other function?
The answer depends on the class of functions $\phi(t):(0,\infty)\to\mathbb R$ where you want to define the Laplace transform. A standard assumption is that
$$e^{-ct}\phi(t)\in L^2(0,\infty)\tag{1}\lab …
- 22.3k
7
votes
Accepted
Decay of the Fourier transform
This is not true without additional integrability conditions on $f(\cdot+iy)$.
$\hat{f}(w)=o(e^{-a|w|})$ implies that $e^{b|w|}\hat{f}(w)\in L_2(\mathbb{R})$ for all $b < a$. The latter inclusion h …
- 22.3k
5
votes
Fourier transform of fractional differential operator and Plancherel formula equivalent for ...
I'd like to add a little bit to the answers by Dick Palais and Denis Serre.
The standard way to define a fractional derivative of order $\alpha>0$ on the real axis is via the Riemann-Liouville integr …
- 22.3k
3
votes
Why do Littlewood-Paley projections behave like iid random variables
There is a quantitative way to express the somewhat vague notion of "almost independence of the Littlewood-Paley projections".
Let $\mathcal F_n$, $n\in\mathbb Z$, be the minimal $\sigma$-algebra gene …
- 22.3k
10
votes
Why is Fourier analysis so handy for proving the isoperimetric inequality?
I'd like to add a few words on what happens in higher dimensions. First, a convexity assumption becomes essential (as in the second proof of Hurwitz which works only for convex domains). The isoperime …
- 22.3k