1
vote
2answers
113 views

What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
0
votes
0answers
36 views

Are there orthogonal functions with continuous parameters for multivariate functions harmonic analysis?

So I was working with transforming a two-variable function $f(\theta,\phi)$ into an expansion of spherical harmonics $Y_l^m (\theta,\phi)$ such that: \begin{equation} f(\theta,\phi) = \sum_{l=0}^{L} ...
1
vote
0answers
42 views

The Fourier Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
4
votes
2answers
130 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
2
votes
1answer
228 views

Decaying Fourier transform and smoothness

Suppose that $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$ |\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}. $$ I want to show ...
5
votes
1answer
240 views

Convolution square root of $\delta $

I want to somehow classify the distributional solutions of the equation $$ f \ast f = \delta $$ where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we ...
2
votes
0answers
97 views

Different proofs of support theorem for Radon Transform

Let $\mathcal{H}$ be a set of all hyperplanes in $\mathbb{R}^n$. Radon transform of function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ is defined as function $R[f] \colon \mathcal{H} \to \mathbb{R}$ ...