Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

learn more… | top users | synonyms

1
vote
0answers
19 views

Approximating Fourier transform for range of output frequencies

(This may be an elementary question, I am new to Fourier analysis.) I am working on a visualization tool. I have a real function $f(x)$, given by N samples on some interval, and vanishing outside ...
1
vote
0answers
45 views

Is there a Plancherel-type identity for generalized Fourier Transforms?

Let $S$ be in $\mathcal{T}$, the set of tempered distributions, and $\mathcal{F}S$ be its Fourier Transform. Is there some relationship for such distributions, analogous to the Plancherel Theorem for $...
2
votes
2answers
177 views

About the Fourier-Legendre series of $f(x)=e^{-x}$

So for the function $f(x) = \exp(-x)$ I have the formula for the coefficients of $$f(x) = \sum_{n=0}^{\infty}a_n P_n(x)$$ which is(by using Rodrigues formula) $$a_n = \frac{2n+1}{2} \int_{-1}^{1}\...
1
vote
0answers
39 views

$L^1$ functions approximated by non-decreasing continuous sequences

Actually the origin problem is: Suppose $f \in L^1([0,1])$, prove that there are two non-decreasing sequences of continuous functions ${g_k},{h_k}$ which are $a.e.$ bounded, and $$f(x)=\lim_{k \to \...
0
votes
1answer
18 views

Fourier Transform pdes

I have an exam next week and I was hoping someone might be able to help me out with this question. Show that the Fourier transform of the function $f(t+a)$ is $e^{iwa}\hat{f}(w)$ . There is a list of ...
1
vote
1answer
16 views

Confusion with fourier coeffients

Consider $f(t) = \frac{\pi - t}{2}$, $t \in [0, 2\pi]$ The complex fourier coefficients are $c_n = \frac{1}{2\pi}\int_0^{2\pi}\frac{\pi - t}{2}e^{-int}dt$ Which turns out to be $-\frac{i}{2n}$ if im ...
0
votes
1answer
35 views

When the fourier series equal to the original function?

Let $f\in L^2([-1/2,1/2])$. Define $a_n=\int_{[-1/2,1/2]} f(x) e^{-2\pi i n x} dx$ for each $n\in\mathbb{Z}$. Define $S_N(x)=\sum_{n=-N}^N a_n e^{2\pi i n x}$ for each $N\in \mathbb{Z}^+$ and $x\in \...
0
votes
0answers
19 views

Recommend resources for understanding Phase spectrum

I am learning Fourier transform. if we apply Fourier transform on a signal, we get magnitude spectrum and phase spectrum. I want to learn phase spectrum part in detail. So can anyone recommend any ...
0
votes
2answers
63 views

Coefficients of a cosine series

Let $u$ have the cosine series representation $$u = \sum_{k_1=0}^{\infty} \sum_{k_2=0}^{\infty} a_\underline{k} \cos\left(\frac{2\pi k_1 x }{L_1}\right) \cos\left(\frac{2\pi k_2 y }{L_2}\right) $$ ...
1
vote
3answers
59 views

Find the Fourier transform of $\sin x^2$.

I've tried it by applying integratrion by parts, but I'm not getting the answer correct. Its answer is $$\frac{1}{\sqrt{2}}\,\sin\left(\frac{k^2}{4} +\frac{\pi}{4}\right).$$ Please help in this.
2
votes
0answers
28 views

How does the Fourier transform of a “zero avoiding” function look?

Let $n$ be a very large positive integer. Let $f \in\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function, satisfying $0\leq f\leq1$, and supported on $[-n,-\frac{1}{n}]\cup[\frac{1}{n},n]$ such ...
0
votes
0answers
17 views

Fourier Transform of a radial function in $L^1(\mathbb{R}^2)$ [duplicate]

Let $f \in L^1(\mathbb{R}^2)$ be radial, i.e. there exists $g: [0,\infty) \rightarrow \mathbb{R}$ such that $f(x) = g(|x|)$. Prove that $f$ is also radial. (Note that this result is true for $\mathbb{...
0
votes
0answers
16 views

Fourier Transform of operator

Let $A = (a_{jk})$ be a real $n \times n$ matrix. Let $u \in C^2(\mathbb{R}^n)$ and define $L_Au \in C(\mathbb{R}^n)$ by: $$L_Au = -\sum_{j,k=1}^n a_{jk}\partial^2_{x_jx_j}u$$ Assume $u(\infty) = 0$ ...
3
votes
0answers
30 views

Understanding Theorem $7$.$23$ of Rudin's Functional Analysis

The Theorem states the following: (a) If $ u\in D'(\mathbb{R^n})$ has its support in $rB$, if $u$ has order $N$ and if $f(z)=u(e_{-z})$ where $z \in \mathbb{C^n}$, then $f$ is entire, the resriction ...
1
vote
1answer
29 views

Discrete Fourier transform implementation giving results that are order of magnitude off

I tried implementing a Discrete Fourier transform in Matlab, but I found my results an order of magnitude off. I used next definition of DFT: $$ F(u) = \frac{1}{2N} \sum^{N-1}_{x=-N} f(x) e^{- \pi i ...
1
vote
1answer
51 views

Evaluate $\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$

Find a closed form expression for $$\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$$ I know that $\displaystyle\sum_{r=1}^{\infty} \dfrac{\sin(r \pi x)}{r} = \dfrac{\pi}{2} - \...
0
votes
0answers
39 views

Given a finite number of Fourier coefficients, can we construct a corresponding intergrable function?

Let $\xi_1,\dotsc,\xi_n$ and $\eta_1,\dotsc,\eta_n$ be real numbers. Is there a complex valued function $f\in L^1(\mathbb{R})$ such that: $\int f(x)e^{2\pi i\xi_k x}dx=1$ for every $1\leq k\leq n$. ...
0
votes
0answers
17 views

Fractional Sobolev space on an interval

Consider a fractional Sobolev space $H_p^t(I)$ defined on an interval $I\subset \mathbb{R}$. When $I=\mathbb{R}$ the space can be defined via Fourier transform. Is it possible to do it when $I=(-1,1)...
2
votes
0answers
12 views

Sobolev spaces on non-compact manifolds — independence on charts

Are there some standard references where basic facts about fractional-order (or at least integer-order) Sobolev spaces on non-compact manifolds are treated? More precisely I would like to be able to ...
2
votes
1answer
82 views

How to show Plancherel's Theorem for Fourier Transform implies $L^2$ Transform Convergence.

The Plancherel Theorem for the Fourier transform $\hat{f}(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-ist}dt$ on $\mathbb{R}$ states that $$ \int_{-\infty}^{\infty}|\hat{f}(s)|^...
1
vote
1answer
23 views

Fourier decomposition of solutions of the wave equation with respect to the spatial variable

Say I have a wave equation of the form $$\nabla^{2}f(t,\mathbf{x})=\frac{1}{v^{2}}\frac{\partial^{2}f(t,\mathbf{x})}{\partial t^{2}}$$ which is clearly a partial differential equation (PDE) in $\...
0
votes
0answers
12 views

Fourier transform of a continuous periodic spectrum of frequencies

Suppose I have a function of the form $$ f(t) = \exp(i\phi(t)) $$ where $$ \phi(t) = \int_0^t\omega(t) \ dt + \phi_0 $$ is the phase of the function and $\omega$ is the angular frequency, which is ...
0
votes
1answer
35 views

Théorie de Fourier in Sontag`s book

I was reading Sontag`s In America and she mentions: "La théorie de Fourier sur les douze passions radicales.." What is this theorem about?
0
votes
0answers
40 views

Evaluate $\frac{1}{2\pi}\int_{-\pi}^\pi g(x) dx$ where $g(x) = \int_{0}^x f(t) dt$

Let $f$ be a $2\pi$-periodic function such that $\int_{-\pi}^\pi f(t) dt = 0$. Define $g(x) = \int_{0}^x f(t) dt$. Evaluate $$\frac{1}{2\pi}\int_{-\pi}^\pi g(x) dx$$ I hope the integral is equal ...
1
vote
1answer
46 views

Prove that $\mathscr{F}[f] \in L^2(\mathbb{R})$

Let $f \in L^2(\mathbb{R})$ (square integrable functions), I'm trying to prove that his Fourier transform also does: $\mathscr{F}[f] \in L^2(\mathbb{R})$. I have tried to bound it \begin{align} \...
1
vote
0answers
24 views

Show that $\int_{-\infty}^{\infty}f(\xi+i\eta,z_2,\ldots,z_n)e^{i[t_1(\xi+i\eta)+t_2z_2+\cdots+t_nz_n]}d\xi$ is independent of $\eta$

Show that $$\int_{-\infty}^\infty f(\xi+i\eta,z_2,\ldots,z_n) e^{i[t_1(\xi+i\eta) + t_2z_2+\cdots+t_nz_n]} \, d\xi$$ is independent of $\eta$, for arbitrary real $t_1,\cdots,t_n$ and complex $z_1,\...
0
votes
0answers
12 views

Scale of Oscillations

I'm reading an article which claims the following result : (paragraph 2.2) for a function $f = \sin (N g(x)) h(x) $ where $g$ and $h$ are $C^{\infty}$ scalar functions non oscilattory and $N$ a large ...
1
vote
1answer
37 views

Why is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here. I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the ...
1
vote
1answer
20 views

Probability of measuring the label of representation in quantum Fourier transformaton

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the following function. $$ f : G \to \mathbb{C} $$ Then ...
1
vote
1answer
37 views

Why is the sum of irreducible representations nonzero only when the irreducible representation is trivial?

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In section 3, the authors discuss the probability of measuring the irreducible representation $\...
0
votes
0answers
8 views

How describe functions with finite bandwidth?

What is the sufficient and necessary conditions for a $f:\mathbb R\to\mathbb R$ has finite bandwidth (Fourier spectrum is non-zero on a bounded interval)? I'm guess this equivalent $f$ is continuous ...
0
votes
0answers
24 views

What is support and spectrum of this nonnegative trigonometric function (or Finite Fourier Sum )?

This is a follow up of another question. The zeros of the following cosine sum shows the prime distribution, and the gap between the zeros can help to study the gap between prime numbers. $$ P(p_i,x)=\...
0
votes
0answers
22 views

An example that does not satisfy the conditions of the Fourier inversion theorem?

Here is the Fourier inversion theorem page in Wikipedia. It states that for every function $f(x)$ that satisfy some conditions ($f(x)$ can be a function such as a Schwartz function, an integrable ...
2
votes
0answers
31 views

Non-uniform convolution with discrete wavelet transform

I understand that if you have a circular N-dimensional convolution matrix, it can be diagonalized with the Fourier transform of the convolution operator. This makes it easy to calculate the density of ...
1
vote
1answer
45 views

Discrepancy in Discrete Fourier Transform Algorithm Formula?

I'm having a bit of trouble with a small part of the following formula (taken from this page): $$F_k=\sum_{n=0}^{N-1}f(x_n)e^{-(2\pi i)k\frac{n}{N}}\tag{1}$$ This formula is supposed to ...
0
votes
0answers
14 views

Support of the Fourier transform of $\int_{[0, 1]}f(\xi)e^{-2\pi i(\xi x_1 + \xi^2 x_2)}\, d\xi$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a Schwartz function. Let $$F(x_1, x_2) := \int_{[0, 1]}f(\xi)e^{-2\pi i(\xi x_1 + \xi^2 x_2)}\, d\xi.$$ In other words, $F$ is the Fourier transform of $f$...
0
votes
0answers
24 views

Controlling $\dot W_{k,1}$ norm of a schwarz function

If $\phi \in \mathscr{S}(\mathbb{R})$ then does it follow that there is $c$ s.t. $||\nabla^k \phi||_{L_1} \leq c^k$? From the definition I have only been able to get that this is true for all $k$ in ...
0
votes
1answer
20 views

Question Concerning Fourier Series

I was following the derivation of the basic Fourier series using orthogonal function. For the set of orthogonal functions $\{\phi_n\}$, say the function $f$ can be defined as: $$f(x) = c_0 \phi_0(x) +...
1
vote
0answers
29 views

Computing Hilbert transform and envelope of a function

The following is a function with $\alpha$ being a real constant $$f(t) = \frac{\sin(\alpha t)}{\alpha t}.$$ Determine the analytic signal $f_a (t),$ Hilbert transform $\hat{f}(t),$ and the envelope ...
1
vote
1answer
22 views

Proving an identity involving Fourier coefficient

If $f \sim \sum A_n e^{inx}$ and $g \sim \sum a_n e^{inx}$ and $f,g$ are continuous $2\pi$ periodic functions, show that $$\int_{-\pi}^\pi f(t) \overline{g(t)} dt = \sum_{-\infty}^\infty A_n \...
2
votes
1answer
25 views

Does $\sum_{i=1}^n \alpha_ie^{2\pi i\xi_ix}\equiv0$ imply $\forall i,\alpha_i=0$?

Let $\alpha_1,\dotsc,\alpha_n$ be complex numbers. Let $\xi_1,\dotsc,\xi_n$ be distinct real numbers. Define a function $f:\mathbb{R}\rightarrow\mathbb{R}$ by $f(x)=\sum_{i=1}^n \alpha_ie^{2\pi i\...
0
votes
0answers
15 views

Variant: Bounding Fourier coefficients in terms of supremum norm

This is a variant on this answered question. Let $\alpha_1,\dotsc,\alpha_n$ and $\beta_1,\dotsc,\beta_n$ be real numbers satisfying: $\alpha_i,\beta_i \geq 0$ for every $i$, $\sum_{i=1}^n\alpha_i=1$ ...
2
votes
0answers
83 views

What is the name of that theorem?

Here is the statement : Let $f:\mathbb{R}\to \mathbb{C}$ a continuous map which is $\mathcal{C}^1$ by pieces and such that $f\in \mathcal{L}^1(\mathbb{R})$. Moreover, $\hat f \equiv 0$ in $\...
0
votes
0answers
15 views

Significance of the complex conjugation symmetry of the DFT for real-valued input

For real-valued input $\mathbf{x} = (x_0, ..., x_{N-1})$ and its discrete Fourier transform (DFT) $\mathbf{X} = \mathcal{F}(\mathbf{x})$ we have that $$X_{N-k} = X_k^*$$ where * denotes complex ...
1
vote
1answer
45 views

Linearspan of Gaussians dense in Schwartz space

as the title already says I am trying to show that the linear span "A" of the gaussians $e^{\frac{-|x|^2}{2}}$ and their translations/ dilations are dense in the Schwartzspace. This is the space of ...
0
votes
1answer
40 views

Folland 8.20 (Fourier Analysis)

I'm stuck a bit on this problem from Folland: The first part I can't figure out at all. The second part, I know: $\|Pf(x)\|_1 = |Pf(x)| = |\int f(x,y)dy| \leq \int |f(x,y)|dy$. If the last term is ...
1
vote
0answers
20 views

Fourier Sequence Converges Uniformly Implies Almost Everywhere Pointwise Convergence

I'm trying to understand this problem: Let $f$ be Riemann integrable on $[0,2\pi]$ Suppose that the Fourier Series of $f$, $S_{n}^{f}(x)$, converges uniformly on the interval. I want to show that $...
1
vote
1answer
34 views

Bounding Fourier coefficients in terms of supremum norm

Let $\gamma_1,\dotsc,\gamma_n$ be nonnegative real numbers. Let $\eta_1,\dotsc,\eta_n$ be real numbers. Consider the function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by: $$f(x)=\sum_{i=1}^n \...
3
votes
1answer
53 views

If Fourier sequence $\{H_n\}$ of a function $f$ converges almost everywhere to a function $g$, then $f=g$ almost everywhere?

if $\{H_n\}$ fourier sequence of an $L^2$ function $f$ converges almost everywhere to a function $g$ then $f=g$ almost everywhere Where $H_n = n$th fourier series of $f$. True or False?
1
vote
1answer
31 views

$x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$

I am trying to establish the following $x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$ The right sight looks the the Fourier expansion of ...