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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Absolute value Sinus Cardinalis integral

The existence of the Fourier Transform integral is conditioned for some functions. An important example is the sinc (sinus cardinalis) function, which although does not satisfy certain conditions, it ...
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25 views

Example of function not Fourier Invertible in $L^1$

It is well known that, if $1 < p \le 2$, then, for every $f \in L^p$, $$ \int_{[-R,R]^n} e^{-2\pi i x \cdot y} \hat{f}(y) dy \rightarrow f(x) $$ As $R \rightarrow \infty$, in the $L^p$ sense. ...
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How to prove that $L_2[0,\infty)$ space is linearly isomorphic to $\mathcal{H}_2$ the space of analytic in $Re(s)>0$ functions?

I want to know how to prove that $L_2[0,\infty)$ space is linearly isomorphic to $\mathcal{H}_2$ the space of analytic in $Re(s)>0$ functions. Please help me. Thanks very much.
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If $\gamma$ is irrational, then $\frac{1}{n}\sum_{k=1}^nf(2\pi k \gamma)\to \int_T f(t)\,dt$

I need to show that $$ \lim_{n\to+\infty}\frac{1}{n}\sum_{k=1}^nf(2\pi k \gamma)=\int_T f(t)\,dt. $$ Here $\gamma$ is any irrational number on the real line and $f(t)$ is any continuous periodic ...
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On a property of Fourier coefficients

I need to prove the following: If $(\Phi_n)_{n\ge0}$ is an orthonormal system of integrable functions defined on some interval $[a,b]$, and $(c_n)_n$ is a sequence of reals such that $\sum c_n ...
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Properties of Certain Example of Nonuniqueness to Heat equation

We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that $$\sup_{x\in ...
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An⇀̸A in L1[−π;π] ( An is partial fourier sum )

Let \begin{equation*} (A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k \cos(kt) + b_k \sin(kt), \\ a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} x(t) \cos(kt) dt, \\ b_k = \frac{1}{\pi} \int_{-\pi}^{\pi} ...
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Is the Fourier transform of a continuous and compactly supported function summable?

Let $\varphi$ defined on the real line be continuous and with compact support. What can we say about the summability of $\hat{\varphi}$? I've gone through some theorems such as Parseval's without ...
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20 views

Approximate identity for periodic integrable functions

I'm studying Fourier analysis now and learned the concept of approximate identity. $$h_n\ge 0,\quad \int_{\mathbb{T}}h_n=1,\quad \lim_{n\to\infty}\int_{\mathbb{T}\setminus[-\delta,\delta]}h_n=0\quad ...
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Existence of Certain Locally Integrable Function Defining a Tempered Distribution

We say that a distribution $T$ is tempered if for every sequence $\left\{\phi_{n}\right\}$ in $C_{c}^{\infty}(\mathbb{R}^{n})$ tending to $0$ in the topology of the Schwartz space ...
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How to prove for a system with rational stable transfer function, the output is square integrable?

I want to know for a system with rational stable transfer function, i.e. H(jw)=1/[(a1+jw)(a2+jw)...(an+jw)] (a1,a2,..,an>0), why a square integrable (L2 integrable) input must generate a square ...
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Is the following property of a Fourier Transform valid?

We know that $$\mathscr{F}\left\{f*g\right\}=\mathscr{F}\left\{f\right\}\mathscr{F}\{g\}$$ so I was wondering whether the inverse is true: ...
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Fourier transform of exponential of a function

I am wondering what $\mathcal{F}[\exp(f)]$ is in terms of $\mathcal{F}[f]$. The farthest I have got is using the series expansion of $\exp$, such that I end up with $\mathcal{F}[\exp(f))] = ...
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How to solve a particular PDE (which reminds of heat equation)

I suddenly ran into this equation: Let $u:[a,b]\times \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying: $$\partial_t u = -u' + \frac{1}{2}u''$$ with bountary conditions $u(0,x)=g(x)$ where ...
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randomly rough surface by ifft : real output from ifft

I'm trying to generate a randomly rough isotropic surface with predefined roughness amplitudes (standard deviation of heights). Suppose I have the absolute values of fourier components of the surface ...
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How to get h(t) using direct inverse Fourier transform formula for H(jw)=1/(a+jw) (only when |w|< W)?

I want to get the expression of signal in time domain by using inverse fourier transform. The signal in frequency domain is a little special: H(jw) is 1/(a+jw) only when |w|< W; and when |w|> W ...
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$\phi_{\epsilon} \ast \mu \rightarrow \mu$?

Let $\phi$ be a non-negative function on $\mathbb{R}$ with $\int_{\mathbb{R}} \phi = 1$. Define $\phi_{\epsilon}(x)=\epsilon^{-1}\phi(\epsilon^{-1}x)$ for $x \in \mathbb{R}, \epsilon > 0$. For $f ...
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Limit of Cosine and Sine Fourier Transforms

If I define the cosine and sine Fourier transform as (skipping constant prefactors $(2\pi)^{0.5}$): $$\mathcal{F}_C\{f(x)\}=\int_0^{\infty}\,f(x)\,\cos(\omega x)\,dx$$ and ...
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Fourier transform of an integrable odd function

I'm trying to prove a proposition about the Fourier transform of an odd function. Let $f\in L^1(\mathbb{R})$ be an odd function. Then there is $M>0$ such that for any $a,A>0$, ...
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$f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R) \implies |f(x)| \to 0$ as $|x| \to \infty$?

Suppose $f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R)\cap L^{\infty}(\mathbb R), (1<p<\infty).$ My Question: Can we expect $\lim_{|x|\to \infty} |f(x)|=0$ ? (In other words, If $f$ and its ...
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How Fourier decomposition is performed?

The Fourier decomposition explains a time series entirely as a weighted sum of sinusoidal functions and with the Fourier series,it is possible to do it. Suppose a sinusoidal periodic signal is ...
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Existence of Fourier Transform for Implicit function

Given an "explicit" function $f:\mathbb{R}^n\to\mathbb{R}^n$, (e.g $F(x_1,\dots x_n)=\cos(x_n)+x_1^2e^{x_2}$) under some assumptions one can allegedly develop a Fourier transform given by ...
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What information is contained in the phase spectrum of a signal?

For any given signal using Fourier transform, we can compute it's magnitude and phase spectrum. In that I want to give focus on phase spectrum. But for phase spectrum, I don't have much data ...
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28 views

How to get $h(t)$ using direct inverse Fourier transform formula for $H(jw)=1/(a+jw)$?

I want to find the inverse Fourier transform of $H(jw)=1/(a+jw)$. We know from the Fourier table that $$ F(e^{-at}) = 1/(a+jw). $$ So that $$ h(t)=e^{-at}. $$ But can we get $h(t)$ directly using ...
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question on Fourier Transformation

I have to find the Fourier Sine transform of $f(x)=1$ when $|x|<a$ and $f(x)=0$ when $|x|\ge a$ and hence show that $$\int_0^\infty {\sin(t)\over t} dt =\pi/2$$ and $$\int_0^\infty ...
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40 views

Fourier coefficients of a symmetric function in $\pi$

I want to show that the Fourier coefficients $\int_{-\pi}^\pi e^{ij \lambda} f'(\lambda) d \lambda$ of the derivative of a continuously differentiable function $f: [ - \pi, \pi] \rightarrow ...
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Show that lamda is greater than or equal to zero for a sturm liouville problem

To show that this problem can be put into S-L form for an eigenvalue problem, Observe that The S-L form is of $$\text{p'(x)}\phi _x\text{+p(x)}\phi _{\text{xx}}\text{+q(x)$\phi $+$\lambda \phi ...
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Using fourier transforms to solve odes

In the very last line of the solution that i have given i got the residue to be (e^(iaz))/2i which when multiplied by 2pii gives me pie^(iaz) now from this i don't understand how they got rid of z ...
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Can a DTFT have a period different of $2\pi$?

I think almost everything is in the title. In an exercise, a DTFT is given : $$X(e^{j\Omega}) = \sin(\Omega) + \cos(\Omega/2)$$ The period of this DTFT is $4\pi$. Is that possible? I mean, the ...
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Fourier transform of a product of two rect functions

I am trying to evaluate the following expression $$\mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\mathrm{rect}_{L_{y}}(y)\}$$ which denotes the 2-dimensional Fourier transform (reciprocal variables $k_x$, ...
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Judging whether a function is not in the range of Fourier transformation

(1) First, I have to show that if f is an odd function that is integrable on the rea line, then there exists a positive number M such that for any a,A (where A is bigger) the following holds. (2) ...
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Fourier transform in three dimensions getting out of hand

I have the following integral I wish to compute, it transforms a quantum position wave function into momentum space: $$\phi(\mathbf p)=\int\frac{\mathrm d^3r}{(2\pi\hbar)^{3/2}}e^{-i\mathbf{p\cdot ...
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Multiple convolutions

Let $\phi(x)=1$ on [0,1] and 0 anywhere else. Is there a was to say what the support of the n-times convolution of phi with itself, that I wann to denote by $B_n$, is? In especially is it possible to ...
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Fourier series sketching

Whenever I am asked to draw fourier series, is it correct to first draw the function on the interval first (in this case 0<= x < pi), then extend the the graph to the desired interval ...
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Fourier transform and $L^1,$ $L^2$ convergence

Let $\phi \in L^2(\mathbb{R})$ and $\hat{\phi}$ be the Fourier transform of $\phi.$ Does this mean that $\sum_{m \in \mathbb{Z}} |\hat{\phi}(x + 2 \pi m)|^2$ converges in the $L^1$ sense on each ...
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Is there an explicit formula for the Fourier transform of $(z-|\xi|^{\alpha})^{-1}$ on $\mathbb{R}$?

Let $0<\alpha< 2$, and $z=\lambda+i\mu$, where $\mu\ne 0$. Consider the following Fourier transform on $\mathbb{R}$ $$g(z,x)=\int_{\mathbb{R}}\frac{e^{ix\cdot\xi}}{(z-|\xi|^{\alpha})}d\xi$$ ...
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50 views

Calculate $\int_{-T}^T {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau$.

Let $\lambda$ and $\nu$ be real numbers. Then, it has \begin{equation} \int_{-\infty}^\infty {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau= ...
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Fourier transformations and the inversion formula

I am working through the above question in preparation for an upcoming exam. I have completed part (a) and quoted the inversion formula for part (b), but I cannot see how to find a form to evaluate ...
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How i can find the fourier transform of $\frac{\sinh(ax)}{\sinh(\pi x)}$ where,$ |a| < \pi$

Using a rectangular contour in the complex plane, bypassing the poles at $z=0$ and $z=i$, i got $$\int_{-\infty}^{+\infty}\frac{\sinh(ax)e^{ikx}}{\sinh(\pi x)}dx - ...
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Books covering the basics of Fourier Transform for image processing

I am studying computer science and I would like to improve myself on the subject of image processing. There is just one obstacle, Fourier transformations. Is there any material which covers basics of ...
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Please explain the $*$-operator in $x^*[n]$

I have to calculate the $IDFT$ for a signal $y_2[n]$: \begin{align*} y_2[n] = DFT^{-1} \Big\{ \Im m \{ \tilde{X}[k] \} \Big\} \end{align*} and I am allowed to use some formulas from a collection ...
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Fourier transform and recursion

Starting with the first derivative of a continuous general function $u(x)$, say $\frac {du}{dx}$ and I take the Fourier Transform of it, I know the solution is $i\cdot k\cdot U$, where $ U$ is the ...
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How to choose $f\in C_{c}^{\infty}(\mathbb R)$ so that $ \hat{g}\in \ell^{1}(\mathbb Z)$, where $g(x)=f(x+2\pi)$?

Suppose $K$ is compact proper subset of $[0, 2\pi]$ with the property $K\subset V \subset [0, 2\pi]$ where $V$ is open . My Question: Is it possible to choose $f\in C_{c}^{\infty}(\mathbb R)$ such ...
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Why are these functions called “kernels”?

In the last years while studying numerical analysis I came across different "kernels", like the Dirichlet Kernel $$D_n(x) = \sum_{k=-n}^n e^{ikx}$$ the Fejer-Kernel $$F_n(x) = \frac{1}{n} ...
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Fourier Transform method to solve a parabolic PDE in $\mathbb{R^n}$

Let $b\in \mathbb{R^n}$ and $c>0$. Assume $g \in C(R^n)$ has compact support and $f = f(x,t)$, $f \in C_1^2(R^n \times [0,\infty))$ has compact support. I'm trying to solve the following IBP via ...
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1answer
19 views

Relationship between Inverse Fourier and Inverse Laplace Transform?

Suppose we are given a fourier transform $$ F(\omega) = \frac{1}{\omega^2+4} $$ Can we use inverse laplace tranform by taking $i\omega = p$ to find the inverse fourier transform? I did this and got ...
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15 views

How do I find the Fourier transform of $\mathcal{F}[\log(a^2+s^2)](s)$

For $a>0$ i have managed to show that this is the Fourier transform of the function. $$ \mathcal{F}[e^{-a|x|}](s) = \frac {2a}{\sqrt{2{\pi}}(a^2+s^2)}. $$ How do I now use this to find the ...
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Mode of convergence for partial Fourier series in $B( L_p[-\pi; \pi ])$, $p \in [1; \infty]$

Which mode of convergence takes place, strong, weak, or in norm? If we have sequence of continuous linear operators in $L_p[-\pi; \pi]$: $(A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) ...
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1answer
39 views

Show that an analytic function in a strip has a complex Fourier expansion

I am a self-studier, and this is a homework problem from a course in Complex Analysis. First, let me give a plug for the course as it is outstanding. Taught by Jerry Shurman at Reed. Great lecture ...
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Can anybody give justification about features of phase and magnitude spectrum in case of Fourier transform?

I have read that from Fourier transform we obtain magnitude and phase spectrum. The magnitude spectrum tells you how strong are the harmonics in a image and the phase spectrum tells where this ...