Questions tagged [fourier-transform]
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516 questions
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Bounds on functions pullbacked via exponential map
Let us assume that $M$ is a compact Riemannian manifold (without boundary). For any point $x\in M$, we can pullback $C^\infty(M)$ functions to $T_x M$ via the exponential map, by setting
$$ (\exp_x^* ...
- 9,853
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How is the deconvolution of a fat gaussian from a polynomial derived?
We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let:
$\begin{eqnarray}
p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\
G(x,y) &=& c_k\...
2
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Positive Fourier coefficients for a function $f:\{+1,-1\}^n \to \mathbb R$
This is from my research in computer science where the Fourier transform over $GF(2)^n$ is a tool to study functions on the Boolean hypercube.
For example, the majority function on 3 variables is ...
2
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814
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Quantifying the “flatness” of functions which are the Fourier transforms of positive functions
Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...
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Find a collection of values of polynomial
Given a polynomial $f(x)\in \mathbb C[x]$ where $\deg f(x)=n-1.$ Assume that we need to find a collection of values of this polynomial corresponding to the following set of $x$-values: $\{ e^{ik} \}$ ...
- 21
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Fourier transform of a holomorphic function
Question: Is there a simple method for calculating the Fourier transform of a holomorphic complex function ${f{{\left({z}\right)}}}:\Omega\to{\mathbb{{{C}}}}$?
In order for my question to be well-...
- 547
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Fourier transform either changes sign infinitely often far out or is continuous at $x=0$
I am reading a book "Fourier Series and Integrals" by Dym & McKean.
There is an exercise (Page 106):
Exercise: Check that if $f$ is a real, even, summable function and
if $f(0+)$ and $f(0-)$...
- 371
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2
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225
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Show that the kernel $|x -y|^{-1}$ on $\mathbb{R}^3 \times \mathbb{R}^3$ is Hilbert Schmidt with respect to a weighted $L^2$ space
Let $\langle x \rangle := (1 + |x|^2)^{1/2}$, $x \in \mathbb{R}^3$. For $s > 1$, consider the weighted convolution operator
\begin{equation*}
T_s \varphi = \langle x \rangle^{-s} \int_{\mathbb{R}^3}...
- 481
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1
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364
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What is the 3-dimensional Fourier transform of $1/k^4$?
In electrostatics, we often encounter the following 3-dimensional integral:
\begin{equation}
V=\int d^{3}\vec{k}\,\dfrac{e^{i\vec{k}.\vec{r}}}{|\vec{k}|^{2}}
\end{equation}
which yields the Coulomb ...
- 19
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Fourier Transform of an even function
Let $S^n$ be an $n$-dimentional unit sphere.
Consider $f: S^n \longrightarrow R_+$, where $f$ is an even continuous function.
Denote
$$
F(f):=\int_0^{\infty}\int_{S^n}f(y)g\left(\frac{|xy|}{t}\...
- 343
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1
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Closed sets in the space of Fourier transforms $\mathcal{F}L^{1}$
Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is,
$$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$
with the ...
- 1,051
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Characteristic exponent after Girsanov transformation
Let $B$ be a standard Brownian motion. Its characteristic exponent (or Fourier transform) is easily calculated to be
$$ \mathbb E [e^{ixB_t}] = e^{-\frac 12 x^2 t}. $$
Now I want to apply a Girsanov ...
- 245
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1
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217
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Uniqueness of Fourier–Stieltjes transform for finite complex valued measures
Let $\mu$ be a finite complex valued measure on $\mathbb{R}$ and let $\hat{\mu}$ be it's Fourier–Stieltjes transform
$$
\hat{\mu}(\omega)= \int_{\mathbb{R}} e^{it\omega} d \mu(t)
$$
Question: Does $\...
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When are Fourier cosine coefficients convex?
In the question When are Fourier coefficients monotonic it was determined that, if a function $f$ is (the restriction to $[0,2\pi]$) of a completely monotone function, then its Fourier coefficients, ...
- 595
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Fourier transform for $H^2(\mathbb{R}^N)$, $N\geq 5$
How i can prove that if $u\in H^2(\mathbb{R}^N)$ then $u\in \mathcal{F}(L^{p^*}(\mathbb{R}^N))$, where $1/p+1/{p^*}=1,$ $2\leq p<2N/(N-4)$?
- 69
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Fourier transform of a generalized function on the plane
Is there an explicit formula for the Fourier transform of the generalized function of 2 variables
$$\frac{1}{x+y^2+i0}?$$
Remark. Equivalent question: consider the Schroedinger equation one the ...
- 21.8k
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Relating the R-transform in free probability to noncommutative group representations
In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform.
In free (noncommutative) ...
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190
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Generalisation of Paley–Wiener type results for unbounded sets
Do you know an unbounded open set $A\subset \mathbb{R}^d$, $d\geq 2$ with the following property: if some integrable function $f$ on $\mathbb{R}^d$ has its Fourier transform vanishing on $A^c$ and all ...
- 1,352
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Function with non Riemann-integrable Fourier transform
Does there exist a compactly supported continuous function $f$ on $\mathbb R$, such that
$$
\lim_{n\to\infty}\int_{-n}^n\widehat{f}(x)\ dx
$$
does not exist?
Here $\widehat f$ is the Fourier transform ...
- 460
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Why we have $f=0$
Define the Fourier transform for a suitable function $f\in L^1(\Bbb R)$ by $\widehat{f}(\xi)=\int_{\Bbb R}f(x)e^{-ix\xi} dx$.
Assume the condition $$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^...
- 521
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1
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Wavelet momentum identity
I am reading an article on wavelet connection coefficients (G. Beylkin, "On the representation of operators in bases of compactly supported wavelets", 1992 (MSN)) and I came across Equation (3.31):
\...
1
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1
answer
108
views
Maximum of the periodogram of a truncated sequence
Suppose that $Z,Z_1,Z_2,\ldots$ are iid random variables such that $\operatorname EZ=0$, $\operatorname EZ^2=1$ and $\operatorname E|Z|^s<\infty$ with some $s>2$. Let $\tilde Z_t=Z_tI_{\{|Z_t|\...
- 423
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2
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333
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What gives a "Parseval like" equation mixing cosine and sine Fourier transforms ?
Noting $\mathcal{F}^c$ the cosine transform and $\mathcal{F}^s$ the sine transform defined on real functions by:
$$\mathcal{F}^c [f (x)]=\int_0^{\infty} f(t) \cos(xt) dt $$
$$\mathcal{F}^s [f (x)]=\...
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Fourier transform of delta function restricted to sphere [duplicate]
I want to compute $\mathcal{F}^{-1}\{\delta(|\cdot|-1)\}(x)$, which exactly means the following computation:
$$f(x) = (2\pi)^{-n/2} \int_{|\xi|=1}e^{ix\cdot\xi}\mathrm{d}\xi, \mbox{ where }~ \xi \in \...
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1
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625
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What are the spaces for which the Fourier transform is an automorphism? [closed]
this is well-known that the Fourier transform is an automorphism of $L^2(\mathbb R)$ and also of $\mathcal S(\mathbb R)$ (Schwartz space). Is there any other spaces of functions of one real variable ...
- 615
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484
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When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?
Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows:
$$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$
It is ...
- 1,051
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1
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142
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Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,058
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1
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Examining the Hilbert transform of functions over the positive real line
$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(...
- 424
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Hamiltonian particle system and its frequency domain
I am interested in the following question.
So let suppose we have finite number of point particles on plane $\mathbb{R}^2$.
We can assume that every $j$ point is represented by Dirac delta function $\...
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1
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Spectral family associated with the Laplacian operator in $L^{2}(\mathbb{R}^{n})$ [closed]
Let $\lambda>0$ be given. Define
$$G_{\lambda}(\xi) = \chi_{_{\lbrace |\xi|^{2} \leq \lambda \rbrace }}.
$$
and
$$
E_{0}(\lambda)f = \mathcal{F}^{-1}[G_{\lambda}(|\xi|^{2})\mathcal{F}(f)], \ \ f \...
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Explanation of a step in a work by C. E. Kenig and A.D. Ionescu
I am studying the work
Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. ...
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2
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Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?
Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
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Fourier transform of the fractional Poisson kernel
Recall that the extension of function from $u:\mathbb{R}^n\to \mathbb{R}$ can be defined using the Poisson Kernel as follows:
$$u^{\mathrm{e}}(\mathbf{x}):=\gamma_{n} \int_{\mathbb{R}^{n}} \frac{x_{n+...
- 537
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The semigroup of Laplace-Beltrami operator on 3-flat torus
I am studying a recent paper in which the author worked on the rectangular, flat 3 tori. It can be realized, the author explained, as $\mathbb{R}^3 \over (L_1 \mathbb Z \times L_2 \mathbb Z \times L_3 ...
- 159
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Probability of two Boolean functions being equal expressed in terms of the maximum Fourier coefficient
This paper by Maslov et al. uses that the probability of two $n$-bit Boolean functions $l(x)$ and $g(x)$ being equal is bound in terms of $\hat{g}_\text{max}$, the largest Fourier coefficient of $g(x)$...
- 13
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1
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194
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2-dimensional Fourier transform
Let $k=(k_1,k_2) \in \mathbb{Z}^2$. Let $\lambda=(\lambda_1,\lambda_2)\in [0,2\pi]^2$ and $F(\lambda)$ be a bounded real function of $\lambda\in [0,2\pi]^2$.
I am interested in the following equation:...
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Positivity of exponentially bounded characteristic functions
I've noticed that for the classical examples of exponentially bounded, symmetrical distributions (Gaussian, Laplace, Double Exponential, Uniform), their characteristic functions are positive for all ...
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439
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Well-known conditions for the Fourier inversion formula
Let $f\in L^1(\mathbb{R})$.
One may easily check that
$$(*)~~~f', f''\in L^1(\mathbb{R})\Rightarrow \int_\mathbb{R}|\hat{f}| ~\text{is finite} \Rightarrow \int_\mathbb{R}\hat{f}(s)e^{2\pi is x}ds ~\...
- 4,058
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Convolution, Fourier transforms, and area preservation [closed]
Consider the convolution of two functions, f * g. And let us assume, for practicality, some example case where an integral of f or g can be interpreted as the "area under the curve" (or the ...
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337
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Posterior expected value for squared Fourier coefficients of random Boolean function
Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by
$$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
1
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1
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385
views
Interchanging Integration Order involving Fourier Transform
$$f(\omega,u):=\frac1{\omega+iu}$$
where $i$ is the imaginary unit number. We see that the integral of a Fourier transform
$$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
- 2,239
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Which Fourier transform is the correct one?
Given $H(x)$ is the Heaviside Theta, the tables give the following Fourier transforms for it:
$$ H(x+a)\to -PV\frac{i e^{i a w} }{w}+\pi \delta (w)$$
while from Sokhotski–Plemelj theorem it follows ...
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2
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148
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Solution to inhomogenous PDE
Given the equation $(1-\Delta)u=f$ for $f \in S(\mathbb{R}^n)$ (rapidly decreasing functions) we get by taking the Fourier transform that
$u = \left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\mathcal{F}^{-...
- 85
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1
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Restricted Isometry Property for Discrete Fourier Transform Matrix
I was wondering if the Restricted Isometry Property holds for Discrete Fourier Transform. In particular, I am interested in whether a subsampled DFT matrix has such property. Let$W \in \mathbb{C}^{d\...
- 1,127
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86
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Subquadratic multiplication of probability mass functions (with log-convolution?)
We are currently looking for a fast, i.e. subquadratic, algorithm for the following equation:
$z_m = \sum_{i,j :\, (i \cdot j) = m} x_i \cdot y_j$.
That is, we are given two finite input vectors $x$ ...
- 13
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1
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672
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Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation
Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$.
Here, $F$ denotes the ...
- 225
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Fourier approximation error in L^2 for piecewise continuous functions
Let $u:[0,2\pi)\to \mathbb{R}$ be the step function
$$u(x) = \begin{cases}
1 & \text{if } x \in [0,\pi), \\
0 & \text{if } x \in [\pi,2\pi)
\end{cases}$$
By a direct computation, one ...
- 837
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1
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229
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Result of Beurling concerning absolute convergence of Fourier series of |f|
Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put,
$$A(\mathbb T):= \{f\in L^{1}(\...
- 1,051
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1
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367
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How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)? [closed]
We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and
...
- 1,051
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1
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Fourier Transform of compactly supported $L^1$ functions
Background Given a (translation bounded) positive definite measure $\gamma$ lets say on $\mathbb R^d$, its Fourier transform as a tempered distribution is a positive measure $\widehat{\gamma}$.
I am ...
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