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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Significance of the complex conjugation symmetry of the DFT for real-valued input

For real-valued input $\mathbf{x} = (x_0, ..., x_{N-1})$ and its discrete Fourier transform (DFT) $\mathbf{X} = \mathcal{F}(\mathbf{x})$ we have that $$X_{N-k} = X_k^*$$ where * denotes complex ...
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60 views

Find the solution of the Dirichlet problem in the half-plane y>0.

Find the solution of the Dirichlet problem in the half-plane $y>0$. $${u_y}_y +{u_x}_x=0, -\infty<x<\infty,y>0$$ $$u(x,0)=f(x),-\infty<x<\infty$$ $u$ and $u_x$ vanish as $$ \lvert ...
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39 views

Folland 8.20 (Fourier Analysis)

I'm stuck a bit on this problem from Folland: The first part I can't figure out at all. The second part, I know: $\|Pf(x)\|_1 = |Pf(x)| = |\int f(x,y)dy| \leq \int |f(x,y)|dy$. If the last term is ...
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Fourier Sequence Converges Uniformly Implies Almost Everywhere Pointwise Convergence

I'm trying to understand this problem: Let $f$ be Riemann integrable on $[0,2\pi]$ Suppose that the Fourier Series of $f$, $S_{n}^{f}(x)$, converges uniformly on the interval. I want to show that ...
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33 views

Schwartz function whose Fourier transform is compactly supported and $\geq 1$ on the unit ball.

I need to construct such a function but the closest I have come to is to take $f(t) = e^{-|t|}, t\in\Bbb{R^d}$. But its Fourier transform is not compactly supported as is $\hat{f}(x) = ...
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16 views

DFT confusion about complex conjugation in forward process

I learning about the DFT and there's one thing that's sort of confusing me, I hope it's not too dumb of a question! I understand that the DFT involves a process of taking an input signal vector and ...
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1answer
18 views

Computing the Fourier Transform of the square pulse

The function in question is $f(x) = H(a - |x|)$, where the Fourier transform is given by $F(k) = (\frac{2}{k}) \sin(ak)$. Initial attempt: $F(k) = ( H(a - |x|), e^{-ikx} )$ = $- (\delta(a - |x|), ...
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93 views

How is the Fourier transform a generalization to the Fourier series?

I have taken a self-tought course on the subject of Fourier series and Fourier transform and I got the message that the latter is a generalization of the first. I know that the idea that the Fourier ...
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19 views

Coefficients of N-dimensional Chebyshev polynomial using discrete cosine transform

I am using Chebyshev polynomials to interpolate a multidimensional function f(x,y,z). I sample f on the Chebyshev roots grid and want to obtain the coefficients of the interpolation polynomial. I now ...
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509 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
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59 views

Fourier transform of distribution

Let $f\in S_{\infty}$ be a Schwartz function and let us define a linear functional,for any $\varphi\in S_{\infty}$, $S_{\infty}\to\mathbb{C}$, $\varphi\mapsto (f,\varphi)$ ...
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684 views

Does $f(x)\in L^1$ imply that $\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x \omega } \, dx=0$?

Suppose that $f(x)$ is $L^1$ and R- integrable function, problem is to resolve if it is possible existence of such a $f(x)$ that: $$\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x ...
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32 views

Fourier transform of translation in $L^2$.

For a function $f : \mathbb{R} \longrightarrow \mathbb{R}$, let $(\tau_y f)(x) = f(x - y)$. If $f \in L^1(\mathbb{R})$, then it is straightforward to show that $\widehat{\tau_y f}(\xi) = e^{-2\pi j ...
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78 views

Fourier sine transform of $\frac{1}{x}$

Fourier sine transform of $\displaystyle \frac{1}{x}$ is ..(fill in the blanks).. My thoughts: By Definition, $\displaystyle F_s(s)=\int_{0}^{\infty}f(x)\sin sxdx $ $\displaystyle ...
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30 views

Finding a Fourier Transform

I need some help with the following question. If f has a Fourier transform F(k), what is the Fourier transform of cos(x)f(2x+1). I have made pretty much no progress on this. This seems ...
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23 views

Function Singularity in a Sobolev Space

For which $t$ and $p$ does function of the form $$ f(x) = 1/|x|^{\alpha} ,\quad 0<\alpha<1$$ belong to the fractional Sobolev space $H_p^t(\mathbb{R})$, $p>1,t\geq 0$? UPD: actually I am ...
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38 views

Integral with Dirac delta function

We are given that: $$u(x,t)=\frac{2}{\pi} \int_0^\infty e^{-k^2t}G_s(k)\sin {kx}\space\text{d}k,$$ where $G_s(k)$ is the Fourier sine transform of $g(x)$. Find the solution $u(x,t)$ when ...
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16 views

Poissions Equation (Laplace)

$$\begin{align} u''_{xx}&+u_{yy}= x, \quad 0<x<1, \quad 0<y<1,\\ \\ u(x,0)&=u(x,1) = 0, \\ u(0,y)&=u(1,y) = 0,\\ \end{align}$$ Having some problems with Poissons Equation. ...
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How to solve the following partial differntial equation using fourier transform?

How to solve this equation? $$2\iota n_0k_0 \frac {\partial E_x}{\partial y}=\frac {\partial^2 E_x}{\partial x^2} + \frac{\partial^2 E_x}{\partial z^2} $$ where, $E_x(x,y,z)$, $n_0$ and $k_0$ are ...
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19 views

If D_m is the mth Dirichlet kernel, $||D_m||_1\to\infty$ as $m\to\infty$

I was working on this problem for my own studying but am stuck on how to solve it. Let $D_m$ be the $m$th Dirichlet kernel. Show that $||D_m||_1\to\infty$ as $m\to\infty$. Anything would help. Thanks
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How can a complex exponential represent a real world quantity?

Equations containing complex exponentials are mysterious. The complex exponential merely embodies a complex number but in a more compact form where doing maths is easier. Right? If this complex ...
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Is $\{(\epsilon + \cos(x))^{2k}\}_{k\in\mathbb{N}}$ a family of good kernels?

Show that for any $0<\delta<\pi$, $$\lim_{k\to \infty} c_k\int_{\delta<|x|<\pi} \left(\epsilon + \cos(x)\right)^{2k} dx = 0 $$ where $\epsilon >0$ is some small number (for ...
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113 views

Evaluating infinite series $\sum_{n=0}^{\infty} \frac{1}{a^{2}+(2n+1)^2}$

I have no idea to approach this problem. Mathematica gave the sum to be $$ \sum_{n=0}^{\infty} \frac{1}{a^{2}+(2n+1)^2} = \frac{\pi}{4a} \tanh(\frac{a \pi}{2}) $$ How can I analyze this?
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9 views

Calculating mean and standard deviation of frequency given time displacement data

For example, using this data set (sample rate = 200 Hz): https://my.mixtape.moe/mtyocd.gz If I take the power DFT of it (either unwindowed or with the Blackman-Harris window) and zoom in, I can see ...
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12 views

For fourier series g(x), prove that the fourier series for the integral G(x) can be found by term-by-term integration of g(x)

I want to prove that if I have a fourier series of the form $g(x) = a_0/2 + {\sum_i}^\infty a_icos(ix) + b_isin(ix) $, the fourier series of G(x) $-x*a_0/2$ can be found by simply integrating g(x) ...
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23 views

Fourier series for absolute value of sin functiom

If we take the absolute value for sin function, then it becomes even. However, isn't period of this function pi? To find fourier series, 1.Even 2. period 2 pi. Can we just treat this function as ...
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Planar sets in R^{2} with bounded Fourier transforms

I have a question which I'd be greatly happy to hear an answer for (because it sounds really cool!). Say I'm given some "nice" region $\Omega$ in $[0,1]^{2}$, how can one determine all the $p>0$ ...
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21 views

Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
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37 views

What is wavelet tranform in simple words?

I have read wiki and other sources and have still problem understanding the wavelet transform. What is the basic idea in simple words? Does the Fourier uncertainty hold for wavelet transform?
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Inverse Fourier Transform

I have to calculate the following Inverse Fourier-Transform, which describe the potential function for a point force on a half-space: ...
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19 views

Fourier transform of $\cos(x) f(2x+3) $

So when trying to compute the Fourier transform I believe I can use the convolution theorem to evaluate this as a whole. $$ \widehat{g*h}(x) = 2\pi \hat{g}(k) \cdot \hat{h}(k) $$ If it let $ g(x) = ...
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26 views

Given Fourier coefficients of a function , find the function

Given these Fourier coefficients: $$ X[k]=\begin{cases} 1 & \text{, k even}\\ 2 & \text{, k odd}\\ \end{cases} $$ I want to find the analytical expression for the function. What i tried was ...
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35 views

Proving $\|[b,T](f)\|_{p}\le C\|b\|_{BMO(\mathbb{R}^{n})}\|f\|_{p}$ using the Fefferman-Stein inequality

Let $1<p<\infty$ and $1<r<\infty$ and let $\mathcal{M}_{r}(g)$ denote $\mathcal{M}(|g|^{r})^{\frac{1}{r}}$, where $\mathcal{M}$ is the Hardy-Littlewood maximal function. Also let $T\in ...
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Fourier transform of Si[$x^2 + y^2$]; Energy integrals involving sin integral functions

Problem Statement I'm trying to prove( or disprove ) the following identity \begin{equation} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\Big[\text{Si}[x_1^2 + y^2]- \text{Si}[x_2^2 + ...
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35 views

Dilation of Fourier transform

Let $f\in \mathcal{S}(\mathbb{R}).$ The Fourier transform of $f$ is defined by $\hat{f}(w) := \int_{-\infty}^\infty f(x) e^{-2\pi i x w} dx$. We use the notation $f(x) \longrightarrow \hat{f}(w)$ ...
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Sequence $\{ e^{2 \pi i n x}\}_{n\in\mathbb N}$ and Paley-Wiener space $PW(0,1)$.

Let us consider the Paley-Wiener space: $$PW(0,1):=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset (0,1) \}.$$ Let us consider $\{ e^{2 \pi i n x}\}_{n\in\mathbb N}$, for ...
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1answer
15 views

Stuck finding inverse Fourier transform.

I have the equation $u_t - u_{xx} = f(x,t),\; x\in\mathbb{R},\;t>0$, with the initial condition $u(x,0) = 0$. I think I see where this is going but I want to make sure I'm not going in the wrong ...
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Find the function $f(x)$ by using its fourier coefficient

It is easy to find the fourier coefficient and fourier expansion of $f(x)$ function. But I want solve the inverse problem How to find the function $f(x)$, if I know its fourier coefficient (or ...
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Class of Functions , that admit Fourier transforms

For which class of functions/distributions is it sensible to take a Fourier transform ?
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46 views

Convolution of Schwartz and test function approximated by partition of unity.

Let $\rho\in\mathscr{D}$, $0\leq\rho\leq 1, \rho(0) = 1$, and $\sum_{n\in\mathbb{Z}^d}$ $\rho(x-n) = 1$. Denote, $\rho_{n,\epsilon}() = \tau_n\rho(\frac{x}{\epsilon})$, where $\tau$ is the translation ...
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103 views

Several questions about the proof of this lemma

I am reading this paper. I have Several questions about the proof of the lemma 3.3. How to Prove 1,2,3,4?
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24 views

Fourier Series and Fourier Transform confusion.

I dont understand the following paragraph after the proof. In particular, how does that theorem above give us that the Fourier transform maps $L^2$ onto $l^2$? all that theorem says is that this set ...
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76 views

Fourier Transform leading to $\delta$: How does the Integration work?

So it is well-known that the complex exponential $$f(t) = e^{i\omega_0t}$$ has Fourier transform $$F(\omega) = 2\pi \delta(\omega-\omega_0) \ .$$ The transformation integral $$F(\omega) = ...
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8 views

Deconvolution by disks

I have a function $$f(x,y) = \begin{cases} 1, \|(x,y)-p\| < r\\0, \|(x,y)-p\|\geq r\end{cases}$$ where $p$ is some unknown point in $[0,1]^2$; i.e. $f$ is the characteristic function of some disk ...
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25 views

Gaussian is a rapidly decreasing function.

Definition of rapidly decreasing function $$\sup_{x\in\mathbb{R}} |x|^k |f^{(l)}(x)| < \infty$$ for every $k,l\ge 0$. Given the Gaussian function $f(x) = e^{-x^2}$, I know that its derivatives ...
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29 views

Dirac function expansion

In my book it is said that Dirac function $\delta(\tau)$ can be expanded as: $$ \delta(\tau)=(\beta \hbar)^{-1}\sum_{n \in even} e^{-i\omega_n\tau} $$ where $\omega_n=\frac{n\pi}{\beta\hbar}$, and ...
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31 views

How to get from $\sum_{n=0}^N (a_n \cos{nx} + b_n \sin{nx})$ to $\sum_{-N}^{N} c_n e^{inx}$?

I'm currently reading Baby Rudin, and I'm in the section of Chapter 8 that covers Fourier series. There is one line that I just can't figure out for the life of me, and I can't find anything online ...
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15 views

What if we define function of moderate decrease as function satisfying $|f|\le\frac{A}{x^2}$

In Stein's Fourier Analysis, he defines the a continuous function $f$ as of moderate decrease if there exists $A>0$ such that $$|f(x)| \le \frac{A}{1+x^2} \forall x\in\mathbb{R}$$ I am wondering ...
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26 views

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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42 views

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 ...