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).

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Deriving existence of classical Fourier transform via the space of temperate distributions

If for some measurable function $f:{\bf R}^n\rightarrow{\bf R}$ the functional $g\mapsto\int fg$ is in ${\scr S}'$ (space of temperate distributions) and there exists some measurable $f'$ such that ...
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Is this derivation of the Dirichlet Integral using a derivative under the integral sign, incorrect?

To find the integral of the Sinc function: Start with, \begin{equation} I(a)=\int_{-\infty}^{\infty}\frac{\sin\ ax }{x}dx %\hspace{20.0} ; (a>0) \end{equation} \begin{equation} \Longrightarrow ...
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Inverting a complex function

I am facing the following problem. I know that the following relationship holds $$A(\omega) = C + \int_{0}^{\infty} \frac{L(\tau)}{1 + i\omega \tau}\mathrm{d}\tau$$ where $C$ is a positive constant ...
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1answer
31 views

Fourier transform of Gaussian?

For the Fourier transform defined as $$\frac {1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-i\alpha x}\,dx$$ I know there is simple formula for the Fourier transformation and inverse transformation ...
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2answers
26 views

Using Fourier transform to compute Fourier series.

I have found an exercise on a signal processing book that asks to compute the Fourier series of a function by using its Fourier Transform, let: $$ x(t) = \sum_{n=-\infty}^{\infty} \Lambda \left( ...
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52 views

Question regarding Fourier Series

Things I understand (scroll down to see question in bold): Let $T$ be the function's period Let $w_0 = \frac{2π}{T}$ A function $x(t)$ can be written as the sum of its even and odd parts, that is ...
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inverse fourier transform of unit impulse function of omega

What is the inverse fourier transform of the unit impulse function of omega. Sorry I've not got the symbol in my phone. It Should looks like §(W).. Sorry for the special symbols.
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Couldn't find answer using the Fourier Transform and Fourier series

I am fond of Fourier series &transform. In Fourier domain we can come to know what frequency components are present and the contribution of each component in forming the given signal.But I ...
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1answer
57 views

Inverse Fourier transform of $\exp(4\pi^2i|\xi|^2t)$

I would like to compute the inverse Fourier transform of $\exp(4\pi^2i|\xi|^2t)$. \begin{equation} f(x,t) = \int_{\mathbb{R}^n} e^{2\pi i x\cdot\xi} e^{4\pi^2i|\xi|^2t} \,\mathrm{d}\xi \end{equation} ...
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1answer
19 views

Inequality for Ratio of Hardy-Littlewood Maximal Function over Balls and Cubes

Let $M$ denote the centered Hardy-Littlewood maximal function using balls, and let $M_{c}$ denote the centered Hardy-Littlewood maximal function using cubes. Exercise 2.13 in [L. Grafakos, ...
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Heat equation with heat source in form of delta function

Consider the problem \begin{equation} \left\{\begin{array}{cc}u_t-u_{xx}=\delta_0,&0<x<1,\ t>0\\ u_x(0,t)=u_x(1,t)=0,&t>0,\\ u(x,0)=0,& 0\leq x\leq 1.\end{array}\right. ...
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Relation between Laplace and Fourier transform

I have a function that has the property $\tilde f(s) = \tilde{f}(abs(s))$. For this function, I need the inverse Fourier transform. I actually know the inverse Laplace transform of $\tilde f$ and I ...
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114 views

What are the limitations of the Fourier Transform and Fourier series?

I am fond of Fourier series & Fourier transform. In Fourier domain, we can come to know what frequency components are present and the contribution of each component in forming the given signal. ...
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43 views

How to show the following inequality with f(x) = 1/(1+x^2)?

How to show the following inequality : Let, $f(x) = \frac{1}{1+x^2}$ . Then show that $$\int_{\mathbb{R}\setminus (-1,1)} \left( \sqrt{f(x+y)}-\sqrt{f(x)}\ \right)^2 \ \frac{dy}{y^2} \leq C f(x) ...
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1answer
29 views

Inverse Fourier transform of Gaussian

I need to calculate the Inverse Fourier Transform of this Gaussian function: $\frac{1}{\sqrt{2\pi}} exp(\frac{-k^2 \sigma^2}{2})$ where $\sigma > 0$, namely I have to calculate the following ...
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1answer
19 views

Evaluating limit of a characteristic function (Fourier Transform) in $R^k$

I am trying to evaluate this limit: $$\lim_{n \to \infty} \left[(\text{det}\ \Gamma_n)^{-\frac{1}{2}}\exp \left\{ {-\frac{1}{2}(x-m_n)\cdot (\Gamma_n)^{-1}(x-m_n)} \right\} \right]$$ where ...
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A uniform bound by an integrable function for a Fourier series' partial sums.

Consider \begin{equation} \sum\limits_{n=1}^\infty\frac{\cos(nx)}{n}=-\log|2\sin x/2|~~~ \big(x\in(0,2\pi)\big), \end{equation} and its $2\pi$-periodic extension $f$ (for a proof of the above ...
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Three-dimensional fourier transform for Biot-Savart law

While working on a proof of Biot-Savart law in three dimensions (2D case, though simpler, should be provable in a similar manner) - if $curl \ (v) = \omega$ then $v=\int K(x-y) \omega(y,t) dy$, where ...
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1answer
19 views

Self similarity function

My self-similarity function is defined by : R(t) = $$ \int_{-\infty}^\infty \mathrm y(x+t)y^\ast(x)\,\mathrm{d}x $$ which is apparently equal to R(t) = $$ \int_{-\infty}^\infty \mathrm ...
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1answer
47 views

$L^{p}$ Boundedness of Fourier Multiplier without Littlewood-Paley

Suppose $\zeta\in\mathcal{S}(\mathbb{R}^{n})$ is a Schwartz function such that $\widehat{\zeta}$ is compactly supported away from the origin (say in the annulus ...
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3answers
51 views

Find the Fourier Transform of $2x/(1+x^2)$

I tried doing this the same way you would find the Fourier transform for $1/(1+x^2)$ but I guess I'm having some trouble dealing with the 2x on top and I could really use some help here.
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1answer
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does frequency scaling property of Fourier transform not work for Fourier series?

So frequency/time scaling property of Fourier transform says that: fourier transform of $|c|f(ct)$ is $F(\omega/c)$. (I am using angular frequency $\omega = 2\pi f$ here) However, this doesn't seem ...
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46 views

Show that $\sum_{n \neq 0} \frac{(-1)^{n+1}}{in} e^{in\theta} = 2 \sum_{n=1}^\infty (-1)^{n+1} \frac{\sin n\theta}{n}$.

Show that $$\sum_{n \neq 0} \frac{(-1)^{n+1}}{in} e^{in\theta} = 2 \sum_{n=1}^\infty (-1)^{n+1} \frac{\sin n\theta}{n}.$$ This is not an exercise. It is an example from Stein, Fourier Analysis ...
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29 views

Fourier transform of indicator function

Given a set of complex numbers $\mathcal A$, is there a convenient solution for the Fourier transform of its indicator function $\chi_{\mathcal A}(z)$? More specifically, if $\mathcal A$ is a set of ...
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23 views

Find the supremum of the function

Hi I'm trying to figure out for which values of $w$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is going to be all ...
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24 views

Show a function is periodic and find the period

Let $x(t)$ be a continuous signal, and $\hat x(u)$ be the fourier transform of $x(t)$. We define $\sigma_T(u)=\frac{1}{T}\sum_{n=-\infty}^{\infty}\hat x(u-\frac{n}{T})$ Show that $\sigma_T$ is ...
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53 views

Fourier Transform in $L^2(\mathbb{R})$

I have found a proof to the following theorem which is a fair bit shorter than the proof in my notes. I would be very grateful if someone could tell me whether this way works or whether I've made an ...
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Eigenfunctions of the Fourier transform on locally compact abelian groups

The eigenfunction theory of the Fourier transform on $\Bbb R$ is well-understood. For example, the Hermite-Gauss functions are eigenfunctions with eigenvalues $i^n$; in fact, this comprises the ...
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53 views

Finding value of exponential sum

I'd like to find the value of the following sum $$S(u) = \sum_{n=0}^\infty \frac{e^{iu2^n}}{2^{n+1}}$$ for $u \in \mathbb R$, but I can't seem to do it. Unfruitful work Writing $$S = ...
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40 views

Elementary question about convolution

If $f,g \in L^1(\mathbb{R})$, my textbook proves that the convolution $f*g$ is also in $L^1(\mathbb{R})$. But it doesn't say why, for any $x \in \mathbb{R}$, we have that $y\mapsto f(x-y)g(y)$ is in ...
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Pisier's $\epsilon$-net condition

I'm reading a book about Sidon sets and I'm stuck on the following proof. In order to facilitate the comprehension of my problems I will give the full proof and the context. Let $G$ be a compact ...
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Divergence of Fourier series

Given $f(x)$ is the characteristic function of the interval $[a,b]\subset [-\pi,\pi]$ ($a\neq b$), so $f(x) = 1$ for $x\in [a,b]$ and $f(x)=0$ otherwise. From this definition, I obtained the Fourier ...
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Graphing Fourier transforms on a frequency versus intensity plot (how to deal with complex numbers)

I am trying to understand how Fourier transforms are used to make HNMR plots. HNMR basically consists of hitting some molecules with some radiation and listening to the radio signal that results. ...
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22 views

Fourier transform from Laplace transform

So what I did was Laplace transform $f(t)$ to $F(s)$ and then plug in $s=jw$. However, when I tried this for $cos(3t)$, $$F(jw)={jw\over9-w^2}$$ was the answer. I don't know if this is correct, and ...
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Plancherel's theorem clarification

Folland states Plancherel's theorem as follows: If $f \in L^1 \cap L^2$ then $\widehat{f} \in L^2$ and $\mathcal{F} | (L^1 \cap L^2)$ extends uniquely to a unitary isomorphism on $L^2$ where ...
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35 views

Riemann Lebesgue Lemma Clarification

If $f$ is continuous real-valued function, does the Riemann Lebesgue Lemma give us that $\int_{m}^k f(x) e^{-inx}\,dx \rightarrow 0\text{ as } n\rightarrow \infty$ for all $m\le k$? Specifically, is ...
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30 views

Fourier inversion Lemma (Lars Hörmander)

I always like to have more than one proof for the same theorem. The other day I was browsing through my copy of Lars Hörmander's book on PDE (volume 1). When proving the fourier inversion formula (on ...
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23 views

Fourier cosine transform of $e^{-ax}$

What is the Fourier cosine transform of $e^{-ax}$ I got $$ \int_{0}^{\infty}\cos(kx)e^{-ax}dx = \frac{e^{-ax}(k\sin(kx) -\cos(kx))} {a^{2}+k^{2}}\Bigr|_{0}^{\infty} $$ But how do you continue ...
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Non periodic Fourier Series Point Convergence

If $f$ is a real-valued non-periodic continuous function that is differentiable at the point $x_0$, is it true that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the ...
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28 views

What is the Fourier transform of $1/f(x)$?

Given $F(t) = \mathcal{F}\{f(x)\}$ is the Fourier transform of $f$, how can one express $\mathcal{F}\left\{\dfrac{1}{f(x)}\right\}$ in terms of $F(t)$? EDIT: To be more concrete, I want to compute ...
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25 views

Compute the solutions of the following equation in Fourier space:

$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$ I know this is a ODE so this is what I came up with so far: ...
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Riesz Projection as a Cauchy type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
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30 views

The maximum value (peak) of multiple self-convolution of rectangular function

In Multiple self-convolution of rectangular function - integral evaluation, formula for self-rectangular function of rectangular function seems to have been derived. How do we prove that this formula ...
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3D Fourier Transform - Angle between $\mathbf{k}$ and $\mathbf{r}$

The definition of the Fourier transform for three dimensions is $$\mathcal{F}[f(\mathbf{r})](\mathbf{k})=\int e^{-i\mathbf{k}\cdot \mathbf{r}}f(\mathbf{r})\,d^3 r$$ If the function $f(\mathbf{r})$ ...
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Help establishing a bound on the Fourier coefficients of a bounded $2\pi$ periodic function that is discontinous at the end points?

This is from a practice midterm, and I'm having trouble with the first part. Suppose $f$ is a $2\pi$-periodic function that is continuous and differentiable on the interval $[-\pi, \pi]$, but has jump ...
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21 views

Diagonal of multidimensional DFT

If $X$ is a $n\times n$ square matrix and $F$ its Discrete Fourier Transform, is there a way to compute the diagonal $(F_{1,1},\ldots,F_{n,n})$ without explicitly computing the full DFT? How about ...
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Fourier transform on fractional Sobolev spaces

We say that a tempered distribution $f$ satisfies $f \in H^s(\mathbb R)$ for some $s \in \mathbb R$ if $(1+|\xi|^2)^{s/2} \hat f \in L^2(\mathbb R)$. Here, $\hat f$ denotes the Fourier transform of ...
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How this result in archived in Fourier series

I was reading some notes about functions of symmetry in Fourier series and came across the following result for a function with symmetry of an odd quarter wave $$\begin{align} ...
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10 views

Substitution in complex-valued Fourier integral

In Knapp (Representation theory of semisimple groups, 86'), on page 34 it is shown by means of Euclidean Fourier transform that the principal series representation of $SL(2, \mathbb C)$ is irreducible ...
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Intuition Behind the Riesz Transform

Define the Riesz transform in singular integral form \begin{equation*} R_jf(x)=\lim_{\epsilon\to 0}\pi^{\frac{-(n+1)}{2}}\Gamma(\frac{n+1}{2})\int_{|y|>\epsilon}\frac{y_jf(x-y)}{|y|^{n+1}}dy. ...