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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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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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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21 views

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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18 views

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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47 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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15 views

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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18 views

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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33 views

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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42 views

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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33 views

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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29 views

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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23 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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47 views

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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1answer
20 views

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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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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51 views

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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21 views

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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33 views

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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1answer
38 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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25 views

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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1answer
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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21 views

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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34 views

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 ...
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23 views

How to prove the absolute value of this Fourier coefficient is bounded?

The question is: Let $f:[0,1)\to C $ be a step function. Prove that there exists a constant C such that for all integers $k\neq 0$, $$ \mid \,\hat{f}(k)\mid\le C\,\big/\mid k \,\mid.$$( $\hat{f}(k)$ ...
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52 views

What does the overline symbol mean?

So I have got a question in an old exam paper for Fourier Analysis. Let $f:I\to C$ be an integrable function. Prove that$\int_I \overline{f(x)}= \overline{\int_I f(x)}$. The problem is that I ...
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35 views

How to prove the Fourier coefficient of a convergent sequence converges as well

So the question is, $\,f_1, f_2,...,f:T\to C$ are integrable functions with $ \,f_n\to f$ in $\parallel \cdot \parallel_1 $ as $n\to \infty$. Let $k\in Z$. Prove that, the Fourier coefficient, $ ...
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1answer
18 views

Easy Fourier Transform

I am asked to find the fourier transform of $f(x)= a-|x|$ when $x<|a|$ and $f(x)=0$ otherwise. I have done the calculation and end up with $\dfrac{1-ik-e^{-ika}}{k^2}$, the answer shown 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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35 views

Is there a general way to prove this Fourier transform property?

We know that one of the important Fourier transform properties is that, the Fourier transform of a narrow function has a broad spectrum, and vice versa, We can easily see this in this example, the ...
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59 views

Show that $\sup_{x \in \mathbb{R}}|\sigma_nf(x) - f(x)| \leq C\frac{\log n}{n}$, for $f$ $2\pi$-periodic and Lipschitz.

I'm learning about Fourier analysis and need help with the following 2 problems: $(1)$ Show that $\forall t \in (0, \pi], K_n(t) \leq \min\{n +1, \frac{\pi^2}{(n + 1)t^2}\}$. $(2)$ For the ...
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93 views

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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1answer
27 views

Integration by parts (convolution)

I am trying to evaluate the integral $$F(\nu) = \frac{jT}{2 \pi} \int^{\infty}_{- \infty} \delta ' (\nu ') \ \text{sinc} (T (\nu-\nu')) e^{-j 2 \pi (\nu-\nu') (T/2)} \ d \nu'$$ Where $\delta$ ...
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26 views

Fourier transform of sampling function

Calculate the Fourier transform of $f_{ZOH}$ (the zero-order hold reconstruction of a sampled signal). Where $f_{ZOH} (t)= f(kT), \ \ kT \leq t < (k+1)T,$ and the sampled signal is $$f_s = f(t) ...
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45 views

What is $\int^{\infty}_{-\infty} e^{(ik+l)x} d x$?

Let $k$ and $l$ be two real numbers. We know that $\int^{\infty}_{-\infty} e^{ikx} d x = 2 \pi \delta(k)$. Here $i$ is the imaginary unit and $\delta(\cdot)$ is the Dirac delta function. What is ...
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31 views

Some confusion about the application of fourier transforms to derivatives

Assuming we denote Fourier transforms as follows: $\mathcal{F}[f(t)](\omega)=\tilde{f}(\omega)$, then we have the following identity: $\mathcal{F}[\frac{df}{dt}](\omega)=i\omega \tilde{f}(\omega)$ ...
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1answer
30 views

Why are differential equations with sinusoidal source terms easier to solve than others?

I am a software engineer trying to wrap my tiny human brain around Fourier Transforms for a project I'm currently working on. Although I will ultimately use an open source Math library to do all the ...
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1answer
22 views

Fourier coefficients of a triangle function

I'm trying to find the Fourier coefficients ($c_n$), of the following function : for $x$ in $[-\pi/2;\pi/2[$ $f(x)=x$ for $x$ in $[\pi/2;3\pi/2[$ $f(x)=\pi-x$ I think its not that hard, but I keep ...
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2answers
43 views

Value Proposition of Fourier Analysis?

I am a software engineer trying to wrap his head around Fast Fourier Transform (FFT). Specifically, I need to implement it as part of some software I am writing. Now I can handle the implementation of ...
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1answer
34 views

If $f \in L^1[-\pi, \pi]$ is odd and $f(x + \pi) = f(x)$ for $x \in \mathbb{R}$, then $\beta_{2k - 1} = 0, \forall{k} \in \mathbb{N}$

I'm learning about Fourier analysis and need help with the following problem: Suppose $f \in L^1[-\pi, \pi]$ and $\alpha_n, \beta_n$ are the Fourier coefficients of $f$. Show that if $f$ is odd ...
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1answer
16 views

Laplace equation with boundary conditions in polar coordinates

Show that the problem with this boundary conditions $u_{rr}+1/ru_{r}+1/r^2u_{\theta\theta}=0$, $\quad 0 < r < 1, \quad 0 < \theta < \pi$ $u(r,0)=0$ $u(r,\pi) =T_0$ $u(1,\theta) =T_0 $ ...
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56 views

Cesàro sum of the series $\sin x + \sin 2x + \sin 3x + \ldots = \frac{1}{2}\cot\frac{x}{2}$ for $x \neq 2k\pi, k \in \mathbb{Z}$

I'm learning about Fourier series (specifically Cesàro summation) and need help with the following problem: Show that the Cesàro sum of the series $\sin x + \sin 2x + \sin 3x + \ldots$ is equal to ...
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51 views

Why is $A(\mathbb{T})\subset C(\mathbb{T})$?

where $A(\mathbb{T})$ is the space of Absolutely Converging Fourier Series and $C(\mathbb{T})$ is the space of Continuous Functions, both over $\mathbb{T} = [0,1)$. If $ f\in A(\mathbb{T})$ is $f\in ...
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10 views

squared bicoherence matlab parameters

I would like to use a modified version of the squared bicoherence formula in Matlab. HOSA does not contain the squared bicoherence formula. The bispectrum is given by the following equation $$\ ...
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1answer
47 views

Given $n$ points in the plane, prove that less than $2n^{\frac{3}{2}}$ pairs of points are a distance 1 apart.

Given $n$ points in the plane, prove that less than $2n^{\frac{3}{2}}$ pairs of points are a distance 1 apart. It seems like Piegeon Hole Principle but I don't know how to proceed.
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102 views

Does $ \int_a^b |f(x) - f_1(x)| = 0$ imply $ \int_a^b |f(x) - f_1(x)|^2 = 0$?

Context:I'm trying to solve this problem: Suppose $f, f_1, g, g_1$ all Riemann integrable complex valued functions on $[a, b]$ such that $f \sim f_1$ and $g \sim g_1$. Prove $\langle f, g \rangle ...
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20 views

Laplace equation with Boundary value conditions by parts

I don´t know how to procced in this problem by parts $u_{xx}+u_{yy}=0$, $\quad 0 < x < \pi, \quad 0 < y < \pi$ $u(x,0)=0$ $u(0,y) = \begin{cases} y, & \text{for } 0 < y < ...
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23 views

A Hilbert transform that takes several functions

While playing with some PDE I came across a singular integral that looks something like ...
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1answer
38 views

Mean-Square Fourier Convergence

Let $ \left \{X_n\right \} ^{\infty}_{n=1}$ be any orthogonal (in the $L^2$ sense) set of functions. Let $$S_N(f) = \sum^{N}_{n=1} \frac{(f, X_n)}{ \left \|X_n\right \|^2} X_n$$ be the “Fourier ...
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61 views

Taking inverse Fourier transform of $\frac{\sin^2(\pi s)}{(\pi s)^2}$ [duplicate]

How do I show that $$\int_{-\infty}^\infty \frac{\sin^2(\pi s)}{(\pi s)^2} e^{2\pi isx} \, ds = \begin{cases} 1+x & \text{if }-1 \le x \le 0 \\ 1-x & \text{if }0 \le x \le 1 \\ 0 & ...
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30 views

Fourier transformed multiplication operator leaves $L^2([-C,C])$ invariant?

Let $C > 0$ be some constant and $L^2([-C,C])$ the square integrable functions on $[-C,C]$. Let $\delta > 0$ and let $M_{|\cdot |^\delta}$ denote the multiplication operator on $L^2(\mathbb R)$ ...