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.
Questions tagged [fourier-analysis]
1478 questions
71
votes
4 answers
What are fixed points of the Fourier Transform
The obvious ones are 0 and $e^{-x^2}$ (with annoying factors), and someone I know suggested hyperbolic secant. What other fixed points (or even eigenfunctions) of the Fourier transform are there?
pavpanchekha
- 1,461
44
votes
10 answers
Fourier transform for dummies
So ... what is the Fourier transform? What does it do? Why is it useful (both in math and in engineering, physics, etc)?
(Answers at any level of sophistication are welcome.)
Kevin H. Lin
- 20,738
33
votes
4 answers
Range of the Fourier transform on $L^1$
It is well known that the Fourier transform $\mathcal{F}$ maps
$L^1(\mathbb{R}^d)$ into, but not onto, $\overline{C_0^0}(\mathbb{R}^d)$, where the closure is taken in the $L^\infty$ norm. This is a consequence of the open mapping theorem, for…
user17240
- 842
29
votes
6 answers
Does there exist a continuous function of compact support with Fourier transform outside L^1?
Let f be a complex-valued function of one real variable, continuous and compactly supported. Can it have a Fourier transform that is not Lebesgue integrable?
Patrik Wahlberg
- 303
13
votes
3 answers
Fourier Coefficients and Hölder Continuity
Suppose we are given the Fourier coefficients of an $L^2$ function on the circle. Are there necessary and sufficient conditions on the coefficients that allow us to determine that $f$ is Hölder continuous of order $\alpha$?
Note that the necessary…
Matt Jacobs
- 245
13
votes
4 answers
Why sin and cos in the Fourier Series?
Is there any special reason that we use the sines and cosines functions in the Fourier Series, while we know that if we chose any maximal orthonormal system in L2, we would get the same result?
Is it something historical or what?
Thanks in advance.
Axiom
- 520
11
votes
4 answers
Fourier transform of $\exp(-\|x\|_p)$: more general question
David Corfield asked the following questions yesterday: Is the
$n$-dimensional Fourier transform of $\exp(-\|x\|)$ always non-negative,
where $\|\cdot\|$ is the Euclidean norm on $\Bbb R^n$? What is its support?
I want to ask a more general…
Tom Leinster
- 27,167
9
votes
1 answer
Why is the Fast Fourier Transform efficient?
Is there a conceptual way to understand where the Fast Fourier Transform is avoiding redundant computation and thus achieving $O(n\log n)$ instead of $O(n^2)$.
Consider a standard example of the FFT to multiply two polynomials faster. Its not…
user16557
- 1,513
- 2
- 14
- 14
8
votes
1 answer
Fourier transform
Does anyone know what the Fourier transform (in the sense of distributions) of
$$
f(x) = (x^2 - 1)^{1/2}x, \quad |x|\ge 1,
$$
and $f(x) = 0$ otherwise, is?
flavio
- 450
8
votes
2 answers
Fourier transforms of characteristic functions
I am wondering how badly summable the Fourier transform of the characteristic function of a measurable subset of $S^1$ can be.
Question: Let $\alpha \colon \mathbb N \to [1,\infty)$ be a monotone increasing function with $\lim_{n \to \infty}…
Andreas Thom
- 25,252
8
votes
1 answer
Maximal $L_1$ norm of Fourier Transform of a Subset
Let $A_n$ be the following $n\times n$ matrix: $(A_n)_{i,j}= \frac{1}{\sqrt{n}}\omega_n^{i\cdot j}$ for all $0 \le i,j
Ofir Gorodetsky
- 13,395
- 1
- 60
- 75
7
votes
1 answer
Fejer's theorem and convergence of Fourier series in measure
Fejer's theorem says that for any continuous function $f \colon S^1 \to \mathbb C$ with Fourier coefficients $a=(a_n)_{n \in \mathbb Z}$ the sequence
$$\sigma_n(a) := \frac1n \sum_{k=1}^n \sum_{l=-k}^k a_l \exp(2\pi i \cdot l\phi)$$
convergences…
Andreas Thom
- 25,252
7
votes
2 answers
Is the Fourier transform of $\exp(-\|x\|)$ non-negative?
Is the $n$-dimensional Fourier transform of $\exp(-\|x\|)$ always non-negative, where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$? What is its support?
David Corfield
- 4,901
7
votes
1 answer
What does the Fourier transform of an L-infinity function look like locally?
Question:
What does an element of $\mathcal F \big( L^\infty(\mathbb R)\big)$ look like locally?
As formulated, the question might be a bit difficult to answer since the Fourier transform of a function f ∈ L∞(ℝ) is a distribution, and it is not easy…
André Henriques
- 42,480
7
votes
1 answer
Average decay of Fourier coefficients of continuous measures along the sequence $\lfloor n^{3/2}\big\rfloor$
Let $\mu$ be a continuous measure on $[0,1]$ (i.e. each individual point has $0$ measure). As usual, denote by $\hat\mu(n)=\int_0^1e^{2\pi inx}d\mu(x)$ the Fourier transform of $\mu$, and let $\lfloor x\rfloor$ denote the floor of $x\in\mathbb R$.…
Joel Moreira
- 1,701