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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Why is the DTFT (Discrete Time Fourier Transform) unique to each input?

As the title implies. I know the DFT of a signal is unique due to the matrix, but can anyone give a solid explanation as to why the DTFT is unique for each signal input? Thanks for your time!
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44 views

Period of a multivariable function

consider a function $$f(x_1, x_2, \ldots, x_n) $$ is it possible to compute the period of the function as a vector $$\langle l_1, l_2, \ldots, l_n\rangle$$ where each $l$ denotes the period of the ...
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1answer
188 views

Shifting using fouriertransform

I'm doing a small mathematical exercise where I take a function, perform a Fourier transform on it and then multiply the result by $e^{i\alpha w}$. And then take the inverse Fourier transform of the ...
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1answer
433 views

Fourier transform of inverse of Laplace operator

On $L^2(\mathbb{R}^n)$ consider the operator $(-\Delta+z)^{-1}$, for $\Delta$ being the Laplacian and $z\in\mathbb{C}$, on $\mathcal{D}(\Delta^{-1})$. How can one show that for $\phi\in ...
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112 views

Are these Fourier transforms equal?

I believe that since $|x|^2=x^2$ then we have the Fourier transforms $$\int_{-\infty}^{\infty} \mathrm dx \frac{\exp{iux}}{a^2+|x|^2} =\int_{-\infty}^{\infty}\mathrm dx \frac{\exp{iux}}{a^2+x^2}$$ ...
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77 views

Weyl Operator Group Actions

For $g \in L^2(\mathcal{R})$ and real numbers $p$ and $q$, denote $g^{(p,q)}(t) = e^{ipt}g(t-q)$. Calculate $||g^{(p,q)}||, M_t(g^{(p,q)}), M_{\omega}(g^{(p,q)}), \sigma_t(g^{(p,q)}),$ and ...
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102 views

Finding a function from a fourier series

Taken from Apostol Analysis, it says, find a continuous function that generates the fourier series: $$ \sum_{n} \frac{-1^n}{n^3} \sin(nx) $$ I really have no idea how to solve this, instinctively I ...
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95 views

Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
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63 views

Heisenberg Bound

Question Verify $x(t) = e^{i\omega t}e^{-(t-\tau)^2}$ exactly satisfies the Heisenberg bound of $\sigma_t(x)\sigma_{\omega}(x)$. Attempt: I know $\sigma_t(x) = \int_{\mathcal{R}} ...
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162 views

Understanding Dirac delta integrals?

I'm confused as to how exactly to integrate using the Dirac delta function. I have the following example: $$\int \delta (x-4)(x^3-4x^2-3x+4)dx$$ and am told this evaluates to 8. Can anyone please ...
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82 views

DFT example in textbook

There is an example of the use of the DFT formula in my textbook which I don't quite follow. The text goes as follows: Let us define the $N$-periodic and anti-Hermitian series $g_n$ where $g_n = f_n ...
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121 views

Do the Fourier series of a function-valued Hermitian matrix converge?

Let $\mathbf{A}(t):\mathbb{R}\rightarrow\mathbb{C}^{n\times n}$ be a rank deficient periodic function-valued positive semi-definite Hermitian matrix. The entries $a_{ij}(t)$ of $\mathbf{A}(t)$ are ...
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90 views

Calculating this integral?

I'm trying to calculate $$\int\limits_{-\pi}^0e^{-x}\cos(nx)\,\mathrm{d}x$$ as part of a Fourier series calculation. My problem is the calculations seem to loop endlessly - I'm integrating by parts ...
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66 views

Fourier transform extended to $L^2$

Let $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, and let $f_k$ be functions in the Schwartz class such that $\|f-f_k\|_1+\|f-f_k\|_2\rightarrow 0$ as $k\rightarrow\infty$. Define ...
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1answer
117 views

Fourier transform of a cosine function

I was reviewing a homework problem, and I'm trying to figure this out. The Fourier transform of ${1\over 2} cos(3\pi t)$, according to the solution I was given is ${1\over 2}\{\delta(f+{2\over ...
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1answer
86 views

prove the existence of $c\in \mathbb C$ such as $\mathcal F(g)=c\delta + {1\over2i\pi}V_p({\mathcal F(f)\over x}) $

prove that it exist $c\in \mathbb C$ such as $\mathcal F(g)=c\delta + {1\over2i\pi}V_p({\mathcal F(f)\over x}) $ with $\mathcal F$ fourier transform , $f\in \mathbb D(\mathbb R)$ , $\delta$ ...
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211 views

About a Fourier transform of a non- integrable function.

I'm trying to obtain the Fourier transform of the following function: $$F(x)=\frac{x}{1+x^2}$$ I have tried using Residue Theorem, but i think it can't be applied because the difference between the ...
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197 views

Explicit example of a divergent Fourier series?

I've been reading Widder's Advanced Calculus text, which says that there are some continuous functions that have divergent Fourier series, which are summable to the function (C, 1). I'd greatly ...
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78 views

Prove that $f \in C^k$ if $|\hat{f}(n)|\leq C/|n|^{k+a}$

I'm looking at some problems related to Fourier series. This one stumped me a little. Suppose that $f$ is $2\pi$-periodic and piecewise smooth. Show that if there exist $k \in \mathbb{N}, a > ...
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161 views

Fourier transform of convolution for $L^2$ functions

If $f,g\in L^1(\mathbb{R})$, it is not hard to show by definition that $$(\hat{f\ast g)}(t)=\hat{f}(t)\hat{g}(t).$$ But what about if $f,g\in L^2(\mathbb{R})$? The Fourier transform on ...
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47 views

$f(0)$ is integral over Fourier transform for Schwartz class

Let $S$ denote the Schwartz class. Assume without proof that for every $f,g\in S$, we also have $\hat{f},\hat{g}\in S$, and $\int_\mathbb{R}f(y)\hat{g}(y)dy=\int_\mathbb{R}\hat{f}(t)g(t)dt$. Show that ...
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354 views

Use Fourier transform to calculate double integral of harmonic function

Let $$P_y(x)=\dfrac{1}{2\pi}\int_{-\infty}^\infty e^{-y|t|}e^{ixt}dt=\dfrac{1}{\pi}\dfrac{y}{x^2+y^2}.$$ Then $P_y(x)$ is harmonic in the upper half-plane $y>0$ and for $f\in L^1(\mathbb{R})$, ...
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140 views

Fourier series convergence for sum of Schwartz class functions

Let $f$ be a Schwartz class function. Let $F(x)=\sum_{n\in\mathbb{Z}}f(x-2\pi n)$. Then $F$ is periodic of period $2\pi$. How can we show that the Fourier series of $F$ converges to $F$ pointwise ...
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72 views

Fourier inversion formula with truncation

Let $f\in L^2(\mathbb{R})$, and denote $$s_N(x)=\dfrac{1}{2\pi}\int_{-N}^N\hat{f}(t)e^{ixt}dt.$$ Show that $$\lim_{N\rightarrow\infty}\int_\mathbb{R}|s_N(x)-f(x)|^2dx=0$$ So, $s_N(x)$ is the ...
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1answer
375 views

Details for proof of Poisson summation formula

In the proof of the Poisson summation formula, there is a detail which is not clear to me how to resolve. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Schwartz-class function. Let ...
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2answers
67 views

Is any periodic $C^2$ function automatically analytic?

this might be a stupid question, but is any $C^2$ function $f:\mathbb{R}\to\mathbb{C}$ of period $f(x+L)=f(x)$ automatically analytic (and in particular, infinitely often differentiable)? I learned ...
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64 views

Calculating Fourier series of infinite sum in two ways

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be in the Schwartz class. Show that $$\sum_{n\in\mathbb{Z}}f(2\pi n)=\dfrac{1}{2\pi}\sum_{k\in\mathbb{Z}}\hat{f}(k)$$ by calculating the Fourier series of ...
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1answer
117 views

Wirtinger's inequality in higher dimension

Wirtinger's inequality for one-dimensional functions states that if $f(x)$, $f'(x) = \frac{df(x)}{dx}$ $\in$ $\mathcal{L}^2(a,b)$ and either $f(a) = 0$ or $f(b) = 0$ then \begin{equation} \int_{a}^{b} ...
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1answer
106 views

About convolution and Fourier transform

I have some doubts with this question: I we have $f,g\in\cal{S}$ (where $\cal{S}$ is the Schartz space) with $f\ast g=0$, Can we deduce that $f=0$ or $g=0$? What I did is apply Fourier transform, ...
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1answer
292 views

Fourier transform and Laplace transform to solve differential equation

generally we know that both Fourier transform and Laplace transform both is used to solve differential equation,first of all let us recall both form,first Fourier transform: some times instead ...
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1answer
57 views

What is the Fourier transform of an M like function

Given the function $$ f(x)= \begin{cases} \vert x \vert& \text{, for }\;\vert x\vert\le M \\ 0 & \text{, otherwise} \end{cases} $$ for some constant $M$. What would be the form for the ...
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1answer
51 views

Complex Fourier Harmonic Oscillator

I have found the complex Fourier series for my desired force. I now need to find the steady-state forced vibration of my oscillator as a Fourier Series. (The particular solution to the inhomogeneous ...
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1answer
43 views

Continuity/Differentiability of Fourier Series

Possibly stupid question: I'm wondering if there is some trick for evaluating the continuity/differentiability of a Fourier series. In particular, I'm looking at the function $f(x)=\sum_{n=0}^\infty ...
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50 views

Derive Fourier transforms from Fourier expansion. How are they related?

I am just trying to relate Fourier Series expansion to Fourier Transforms. If someone could show how one value on the middle of the table is derived (from expansion) as opposed to deriving their ...
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1answer
128 views

Diagonalization of circulant matrices

Why does the following hold?: $A$ circulant matrix iff it has a representation of the form $F^{-1}DF$ where $D$ is a diagonal matrix and $F$ is a discrete Fourier transformation. I get that $F^{-1}DF$ ...
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Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
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1answer
58 views

Fourier Series of $f(x) = x^n$ - Fast Method?

Is there a fast way to compute the real Fourier series of $$f(x) = x^n \ ?$$ How about the complex fourier series? If there isn't a fast way for arbitrary $n$, how about $n = 5$ or something at ...
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1answer
104 views

Paley Wiener Theorem on sinc function

Use the Paley-Wiener theorem to argue that, although ${\rm sinc}\left(t\right)$ is bandlimited, ${\rm sinc}\left(t^{3}\right)$ is not. Explain how the above result allows reconstruction of some ...
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1answer
51 views

How to pin down the complex integral?

I am working with the following problem: Find the Fourier transformation of the function $$f(x)=\frac{\sin x}{x}.$$ I did not learn any trick in complex analysis (especially various integration ...
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1answer
292 views

About the Fourier transform of the sign function

I'm trying to calculate the Fourier transform of the function $f(x):=sign(x)$. I have read some texts where this is solved approximating the function $f$ by other functions, $f_a$, defined as follows ...
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1answer
103 views

Modulus of Continuity and Fourier Series

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ a $2\pi$-periodic function and define $\Omega(f,h)=|f(x+h)-f(x)|_{L^1}$. Show that exists a constant $C>0$ with $|\hat f(n)|\leq C.\Omega(f,h)$, where ...
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1answer
67 views

Complex Integration of DTFT

Question A discrete-time signal $u \in \mathcal{l}^2(\mathcal{Z})$ has DTFT \begin{equation} \hat{u}(\omega) = \frac{5+3\cos(\omega)}{17+8\cos(\omega)} \end{equation} Use complex integration to find ...
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1answer
68 views

Show that for $0<t<1$, $\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$

Show that for $0<t<1$, $$\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$$ So I derived the following Fourier series: ...
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1answer
322 views

Approximation with complex polynomials on $S^1$ - can it be done?

Can one uniformly approximate a function 'similar' to identity on $S^1$ with complex polynomials? I mean a function like: $f(z)=z \cdot (1+h \cdot \sin(m\cdot Arg(z)))$, for $|h| < 1,\ m \in ...
3
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1answer
835 views

Heisenberg uncertainty principle in $d$ dimensions.

Suppose $f(x)$ is a $d$-dimensional real function and $\int_{R^{d}}|f(x)|^2dx=1$. Show that $$ (\int_{R^{d}}|x|^2|f(x)|^2dx)(\int_{R^{d}}|\xi|^2|\hat f(\xi)|^2d\xi)\geq\frac{d^2}{16\pi^2}$$ I ...
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178 views

Writing function as infinite Fourier sum with sine kernel

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Also, $f$ is continuous and goes to $0$ at $\pm \infty$. Let $K(y)=\dfrac{1}{\pi y}\sin(\pi y)$. Show that ...
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1answer
516 views

Characteristic function of Normal random variable squared

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution While reviewing above, Why do you sub $X^2$ for the $Y$ in $e^{tY}$ and not the density of the normal ...
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1answer
164 views

a plucked string problem from Stein&Rami, Q9 page27

Context: The question is : This is my working: for 0$\leqslant$ x $\leqslant$p, $A_{m}$ = $2\over\pi$$\int^\pi_0 {xh\over p} sin(mx)\,dx$ = $2h\over\pi p$$\int^\pi_0 xsin(mx)\,dx$ = $2h\over\pi ...
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1answer
42 views

Given the Fourier transform pair $h(t) \leftrightarrow H(\omega)$, what is the counterpart of $H(-\omega)$?

Given that $H(\omega)$ is the Fourier transform of $h(t)$, what is $H(-\omega)$ the Fourier transform of? Any help will be much appreciated. Thank you.