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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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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20 views

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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96 views

$f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R) \implies |f(x)| \to 0$ 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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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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79 views

What is the importance of phase spectrum in Fourier transform

For any given signal using Fourier transform, we can compute it's magnitude and phase spectrum. But I have found that while discussing Fourier transform ,only frequency spectrum or magnitude ...
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31 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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33 views

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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52 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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21 views

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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29 views

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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23 views

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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47 views

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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44 views

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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26 views

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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37 views

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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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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33 views

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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51 views

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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42 views

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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1answer
21 views

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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40 views

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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71 views

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
24 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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19 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
44 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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111 views

Features of phase and magnitude spectrum?

I have read in many books that whether the signal is 1D or multidimensional , The magnitude spectrum tells you how strong are the harmonics in the signal and The phase spectrum tells where this ...
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184 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
5
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1answer
59 views

Decay of Fourier Transform

I encountered the following statement, and I cannot see why it is true(if it is). Suppose $f$ is a nonnegative, bounded, compactly supported and measurable function with the following properties: ...
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Decay of amplitude integral

Consider the function $$ f(\vec{x}) = \int_{\Bbb R^3} {\frac{ e^{-i\,\vec{x}\cdot\vec{k}}}{\sqrt{\vec{k}^2 + m^2}}} d^3 k $$ from Zee's Quantum Field Theory in a Nutshell. He argues like this: ...
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Maximal Magnitude of Fourier Transform

Assume you are given a length-$n$ vector $x\in\mathbb{C}$ with elements $x_0$ through $x_{n-1}$. Define the Fourier transform of $x$ as $$ X(e^{j\theta}) = \sum_{k=0}^{n-1} x_k e^{-jk\theta}. $$ I'm ...
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23 views

Fourier Transform - Conditions

I've been looking at the topic in the Farlow book on PDEs, and it says: "The major drawback of the Fourier Transform is that not all the functions can be transformed; for example even simple ...
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32 views

spherical wave expansion

In the paper, Sheen, David M., Douglas L. McMakin, and Thomas E. Hall. "Three-dimensional millimeter-wave imaging for concealed weapon detection." Microwave Theory and Techniques, IEEE Transactions ...
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1answer
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Fourier Analysis question on convolution [closed]

Let $f(x) = \operatorname{sinc}(x)^2$. Find $(f*f)(x)$? This is what I tried $f(x)=\mathrm{sinc}(x)^2$ $$ \begin{align} f\ast f(x) &=\int f(u)\cdot f(x-u)\,\mathrm{d}u\\ ...
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41 views

Does a Plancherel Style Theorem for the Hardy Space $\mathbb{H}^2$ on the Unit Circle Exist?

I am working on a problem regarding Toeplitz operators, and it involves trying to prove $\mathbb{H}^2$ boundedness of the operator (defined in terms of its Fourier coefficients). Now normally when I ...
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2answers
170 views

How could I continue to show the inequality?

Let $g: [0, \pi]\rightarrow \mathbb{R}$ a $C^{\infty}$ function for which the following stands: $$g(0)=0 \ \ , \ \ g(\pi)=0$$ I have to show that $$\int_0^{\pi}g^2(x)dx \leq ...
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39 views

Differentiate between Fourier analysis and Fourier decomposition

I am a beginner. I am confused between two terms i.e. Fourier analysis and Fourier decomposition.I don't understand when to use Fourier analysis term and when to use Fourier decomposition term. It ...
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1answer
31 views

How can we find the sums ?

We have the function $$g: [0, 2\pi] \rightarrow \mathbb{R} \\ g(x)=\frac{(x-\pi)^2}{4}, x \in [0, 2\pi]$$ I found that the Fourier series of $g$ is the following: $$g \sim ...
3
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122 views

What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?

We know that a Fourier series for signal $x(t)$ is given as $$\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T)$$ So my question is what ...
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1answer
82 views

$\sin$ and $\cos$ are the basis of what space?

When learning Fourier expansions, we learn that $\{\sin(mx), \cos(mx)\}_{m \in \Bbb N}$ is an orthonormal basis for our space and thus we can expand functions in it. My question is what space is this ...
4
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1answer
69 views

Fourier transform not surjective using oppen mapping theorem.

I know that it is possible to prove that the Fourier transform $\displaystyle\mathcal{F}: (L^1(\mathbb R),\|\cdot\|_1) \to (\{f\in C(\mathbb R): \lim_{|x|\to\infty} f(x) = 0\}, \|\cdot\|_\infty)$ is ...