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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Geometrical interpretation of complex exponential integral

Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ ...
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Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty $) of: $$ f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases} $$ My ...
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upsampling and plotting a signal in matlab

I want to upsample by 5 a signal in frequency domain, and then plot(stem) it. I figured how to upsample, Fk=(1/5)*upsample(ak_new,5) now this creates a vector ...
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Uniform convergence of Fourier series given certain conditions

If $f$ is a continuous, $a$-periodic and piecewise differentiable function on $[0,a]$ with piecewise continuous derivative on $[0,a]$, then $(f_N)$ converges uniformly to $f$ over $\Bbb R$. ...
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Coercitivity of an elliptic operator with constant coefficients

We are given an elliptic operator $P=\sum_{|\alpha|\leq m}a_\alpha\partial^\alpha$ that is elliptic in $\Omega$. $a_\alpha$ are constants. I am supposed to show that $$\|u\|_s\leq ...
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Prove that if $f \in C^r(T)$, then $\hat{f}(n) = o(\frac{1}{|n|^r} )$ as $n \rightarrow ±∞$

I searched through everything that came up when I searched this question, but didn't come with anything. I'm used to typing in latex, so please excuse any formatting errors.
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If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$.

If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$. I am not quite sure how to start this problem. ...
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Which functions are tempered distributions?

Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function $$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$ where ...
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How do I calculate the Fourier Transform of this signal?

The Context: Find $X(ω)$ which is the frequency domain representations of $x(t)$. $$ x(t) = \sum_{k=-\infty}^\infty \delta(t-4k) $$ This my professor's solution: As we can see, the ...
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Use trigonometric polynomial to approximate periodic function.

From page 53 of Fourier Analysis by Stein, we have If $f$ is integrable on the circle, then the Fourier series of $f$ is Cesaro summable to $f$ at every point of continuity of $f$. Moreover, ...
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48 views

Contradiction between Fourier and Laplace transforms?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has both Fourier and Laplace transforms. Also let $f(t)=0$ for all $t<0$. The Fourier transform of $f$ is ...
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76 views

How the second and third equalities can be achieved?

I am reading this paper. In the Proof of Lemma 3.3, How the second(*) equality can be achieved? How can i use Parseval's identity in third(**) equality?
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30 views

What is duality argument for the operator on $L^p-$ spaces?

Suppose that the operator $T: L^{p}(\mathbb R) \to L^{p}(\mathbb R)$ (say for instance, some nice convolution operator) is bounded for $1\leq p \leq 2.$ At various, places we see that (for ...
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1answer
65 views

Fourier transform of isotropic Laplace distribution (2D)

How would I evaluate the Fourier transform of an isotropic 2D Laplace distribution? $F(\omega_x,\omega_y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp(-b \sqrt{x^2+y^2})\exp(-j\omega_x ...
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Fourier series in spherical coordinates?

I'm reading an article and he just state: let $f\left(\theta,\varphi\right)$ be of this form $$f\left(\theta,\varphi\right)={\sum}g_{m}\left(\theta\right)e^{im\varphi},$$ I'm on the unitary ...
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Given that $\sum\frac{1}{n^2} = \frac{\pi^2}{6}$, how can I find $\sum\frac{1}{n^6}$?

Given that $\sum\frac{1}{n^2} = \frac{\pi^2}{6}$, how can I find $\sum\frac{1}{n^6}$? I know that you can prove the first equality using Fourier analysis. For the second one, do I try to use a ...
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34 views

Determing an inverse Fourier transform

The inverse Fourier transform is defined as: $$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(k) e^{i k x} d k$$ I can't get an inverse Fourier Transform to ...
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486 views

Using partial fraction decomposition to find inverse Fourier transform

I've reduced my problem to $H(w) = \dfrac{1}{(1-\frac{1}{4}e^{-jw})(1-\frac{1}{3}e^{-jw})}$. I need its inverse discrete Fourier transform. My thinking is that I could use partial fraction ...
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1answer
16 views

inverse fourier transform of w*e^w

I have the function \begin{align} F^{-1}\{{λe^{-|λ|}}\} \end{align} How can we find the inverse Fourier transform? The correct answer is: \begin{align} \frac{-2ix}{π(1+x^2)^2} \end{align} Can ...
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Solving Viscous Burgers using spectral method

I am trying to solve the Viscous Burgers equation using the spectral method. $$u_t+uu_x = Du_{xx}$$ where $D$ is a constant (chosen to be zero) and with the initial condition $$u(x,0) = ...
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Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of ...
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Fourier Tansform of derivative on Wolfram Alpha

If I'm not mistaken, the Fourier Transform of the $n$th order partial derivative of a function with respect to $x$, using the transform variable $k$ is: $$(i*k)^n * [F(k)]$$ so for the $1$st order ...
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Fourier series of dirac delta

Let $f \in S(\mathbb{R}^n)$ is it true that $$\frac{1}{(2\pi)^n} \lim_{\epsilon \rightarrow 0} \sum_{z\in \mathbb{}{Z^n}} \int_\mathbb{R^n} f\left( \frac{x}{\epsilon} \right) e^{iz (x-a\epsilon)} dx = ...
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Can we relax the hypothesis of Uniqueness theorem for Fourier series?

I know this fact: "Suppose that $f\in L^{1}(\mathbb T)$ and $\hat{f}(n)=0$ for all $n\in \mathbb Z,$ then $f=0 $ all most everywhere on $\mathbb T$." My Question is: Suppose that $f\in ...
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Question about Dirichlet kernel of Fourier transform for $f\in L^p$ with $p\in [1,2]$, help needed in understanding proof.

I am trying to understand the proof that the following two statements are equivalent. For fixed $R>0$ and $f\in L^p(\mathbb{R}^n)$ let $$S_Rf(x)=\int_{|\xi|<R} \hat{f}(\xi)e^{2\pi i x . ...
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How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
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13 views

Sobolev space membership of logarithmic function

Determine the largest $s\in(0,1)$ for which the following integral converges $$\int_0^1\int_0^1\frac{\Big|\log|x-\frac{1}{2}|-\log|y-\frac{1}{2}|\Big|}{|x-y|^{1+2s}}^{2}dxdy$$
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Is $A(D)$ a complemented subspace of $C(T)$?

Let $T$ be the unit circle and $D$ the open unit disk. A function $f$ belongs to $C(T)$ if it is continuous at $T$. A function $g$ belongs to $A(D)$ if it is continuous at $\overline{D}$ and ...
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Fourier transform of integral with isotropic kernel

The textbook I'm reading claims that this integral: $$ A = \int_V \,d\mathbf{r} \int_V\,d\mathbf{r}' f(\mathbf{r}) K (| \mathbf{r} - \mathbf{r}'| ) f(\mathbf{r}')$$ can be written in Fourier ...
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How can I convince myself of the Fourier scaling property via inverse FT?

I have this function $f(at)$, and I want to Fourier-tranform it. I proceed in the following way, for $\quad\alpha<0 \Longrightarrow a=-|a|$: \begin{align} \ \mathcal{F}_{t \rightarrow \xi}[f(at)]= ...
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Fourier transform of product of twho functions that includes characteristic function

I need to find the fourier transform of f(x)=(1-abs(x))(Chi_[-1,1] (x)). In words, I need the fourier transform of one minus the absolute value of x multiplied by the characteristic function on ...
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What effect does sampling time have on a Fourier Series sum?

What effect would the sampling time of this Fourier sum have on it's accuracy? Is this to do with Nyquists theorem? or am I heading in the wrong direction with this question? Cheers
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Sobolev and Fourier

If we have $f=1_{[a,b]} \varphi$ with $\varphi \in \mathcal{D}(\mathbb{R})$, we found that the sufficient and necessary conditions to have $f\in H^1$ is that $\varphi(a)= \varphi(b)=0$. If we take ...
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Difficulty in understanding a part in a proof from Stein and Shakarchi Fourier Analysis book.

Theorem 2.1 : Suppose that $f$ is an integrable function on the circle with $\hat f(n)=0$ for all $n \in \Bbb Z$. Then $f(\theta_0)=0$ whenever $f$ is continuous at the point $\theta_0$. Proof ...
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What am I doing wrong when I try to deduce the Laplace transform formula?

The Laplace transform of a function $f(t)$ is the projection of $f(t)$ vector (indexed with $t$) onto the linearly independent set of vectors $e^{st}$. The projection of a vector $\vec{v}$ onto ...
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Fourier sine and cosine series: reconstruction is shifted with respect to measured data

I am working in strain analysis. Strain in a mechanical testing machine is captured by strain gages. Signals are like the slim line in the graph below showing strain versus time. The data are of the ...
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Proof that fixed points of a null field are zero

In this context, a null field means a field constructed of planar waves $e^{I k_{\mu} x^{\mu}}$ that satisfy the null condition $k_{\mu} k^{\mu} = 0 \implies c^2 k^2 = \omega^2 $ Suppose we have a ...
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How to solve $u_{tt}+\Delta u + x^{2}u=0$?

Let $u:\mathbb R \times \mathbb R \to \mathbb C$ be some function so the everything in the following make sense. Consider the following PDE: $\frac{\partial^{2}}{\partial t^2} u(x,t) + ...
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26 views

A $H^p$ function

Set $\mathbb U=\{x+iy|\;y>0\}$. A function $f:\mathbb U\to\mathbb C$ is called a $H^p$ function if $f(z)$ is holomorphic and $\|f\|_{H^p}:=\sup_{y>0} \left(\int_{-\infty}^{\infty} |f(x+iy)|^p ...
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Relation between Fourier Transform Duality and other properties.

I'm having a hard time with Fourier Transform's Duality Property. The Duality Property states that, if $$\mathcal{F}\left\{x(t)\right\} = X(\nu),$$ then $$\mathcal{F}\left\{X(t)\right\} = 2\pi ...
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3k views

Fourier transform in cylindrical coordinates

I must implement Fourier transform in cylindrical co-ordinates. Matlab offer fft function. How can I use this function ?
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35 views

Asymptotic behavior of Fourier transform and Hölder condition

I'm trying to solve this question. Following the hint, the Fourier inversion formula gives me : $$ \big| f(x+h) - f(x)\big| = \left| \frac{1}{2\pi} \int_{-\infty}^\infty \widehat{f}(\xi) ...
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Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
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Computing the complex fourier series for triangular wave from trigonometric coefficients

I'm trying to figure out how to compute the complex Fourier series for the triangular wave, given the trigonometric coefficients. The book gives as a result for the complex series the following: $$ ...
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Fourier coefficients of a discontinuous function

For the following given function $J$ It is not continuous but nonetheless integrable. My professor mentioned that it is still possible to "find a Fourier series corresponding to $J$" and his proof ...
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41 views

laplace transform of $t^nf(t)$

I have: $$\mathcal{L}(t^nf(t)) = \int_0^\infty t^nf(t)e^{-st}\ dt = \left(-\dfrac{d}{ds}\right)^n \int_0^\infty f(t) e^{-st}$$ I don't understand where the derivative came from
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If a sequence of function converges in $L^2$ sense, then its Fourier series converges in $l^2$ sense?

I have the following material from my lecture notes and I am trying to prove it, but I am not sure how to do the second part. Suppose we have a sequence of continuous functions $\{q_n (x)\}$ ...
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23 views

Periodic Foricing Terms

The question asks to find the solution for the initial value problem: $ y''+\omega^2y=sin(nt),\quad y(0)=0,\quad y'(0)=0 $ where $n$ is a positive integer when a) $\omega^2\neq n^2$ and b) ...
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show that $\sum P(x,y) e^{x^2 + y^2}$ is a modular form over $\Gamma_0(4)$ where $P(x,y) = x^4 - 6 x^2 y^2 + y^4$

In these lecture notes of Zagier, I read that generalized theta functions are still modular forms. Let $q = e^{2\pi i z}$ $$\theta(z) = \sum_{(x,y) \in \mathbb{Z}} \Big[ x^4 - 6 x^2 y^2 + y^4 \Big] ...
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Show that f is constant by using fourier coefficient

Let f be a 2$\pi$ periodic, Riemann integrable function and let $\alpha$ be an irrational number. Suppose that $f(x+2\pi\alpha)=f(x)$ for all x. Show that f is constant almost everywhere. I know that ...