Linked Questions

Consider the following function in $\mathbb{R}^3$ $$ \begin{cases} 1 & |x|<1\\ 0 & \text{else} \end{cases}$$ How can we calculate its Fourier transform (again, in $\mathbb{R}^3$). I tried ...

When I read Lebesgue differentiation theorem, I suddenly have the following conjecture, which I can't prove or find a counterexample. Let $f\in L_{\mathrm{loc}}^1(\mathbb{R}^n)$. If $$ \int_{B_r(...

I am trying to show that for any Lebesgue measurable set of finite positive measure $E$, the characteristic function $\chi_E$ is not in $H^{\frac{1}{2}}(\mathbb{R}^n)$. I found somewhere that it would ...

I am trying to calculate the following integral: $ \int \frac{d k_x d k_y d k_z}{(2 \pi)^3} \left[ \exp( - \frac{(k_x^2 + k_y^2 + k_z^2) \sigma^2}{2}) + \frac{1}{2} H(\sqrt{k_x^2 + k_y^2 + k_z^2} - ...

In the context of a cosmology text, I have found the following function, called a spatial top-hat filter: $$W_{TH,R}(r)=\dfrac{3\theta(R-r)}{4\pi R^3}$$ It is claimed that this leads to: $$W_{TH,R}(r)=...

Consider the integral $$I = \int_{||\mathbf{r}|| < d} f(x_n)\ \mathrm{d}x_1 \cdots \mathrm{d}x_n,$$ where $\mathbf{r} = (x_1, \dots, x_n)^\top$. Since $f$ only depends on $x_n$, the integral can be ...

Let $B$ stand for the centered Euclidean ball with radius $\frac{1}{2}$ in $d$ dimensions. What is the value of the Fourier transform of the indicator function of B evaluated at the origin?

I am trying to analytically convolve two spherically symmetric functions in 3D spherical coordinates, a 3D Gaussian and a "box" (really a radial step function). Numerical convolution yields ...

Let $\{x_k\}$ be a $\frac{1}{R}$-separated set of points on $S^{d-1}$. Then $$ \left\|\sum_ka_ke^{ix_k\cdot\xi}\right\|_{L^2\left(B\left(0,R\right)\right)}\lesssim R^{d/2} \left(\sum_k|a_k|^2\right)^{...

I am struggling to obtain the value of the following definite integral: $$g_1(\underline{\xi}) = \int_{\mathcal{B}_+(R)} e^{i \underline{\xi} \cdot \underline{X} } \mathrm{d}\underline{X} $$ with $\...

I was reading this answer: Fourier transform of the indicator of the unit ball and couldn't really follow the computations in the answer. Specifically the equality $$\int_{\|x\| \leq 1} e^{-ix_n\rho} ...