Intereting Posts

Value of $y=\sqrt{4 + \sqrt{4+\sqrt{4+\sqrt{4+\ldots}}}}$
Understanding Borel sets
Examples of Separable Spaces that are not Second-Countable
Is $\sqrt 7$ the sum of roots of unity?
Order statistics of $n$ i.i.d. exponential random variables
Quaternions and Rotations
Algebraic numbers that cannot be expressed using integers and elementary functions
Help finding a closed form
Property of critical point when the Hessian is degenerate
An intuitive approach to the Jordan Normal form.
Is the space of $G$-maps $G/H \to X$ naturally homeomorphic to $X^H$?
Good Book On Combinatorics
Why isn't there a good product formula for antiderivatives?
A question on the proof of 14 distinct sets can be formed by complementation and closure
Projection map being a closed map

Suppose $f(x)$ is a $d$-dimensional real function and $\int_{R^{d}}|f(x)|^2dx=1$. Show that

$$ (\int_{R^{d}}|x|^2|f(x)|^2dx)(\int_{R^{d}}|\xi|^2|\hat f(\xi)|^2d\xi)\geq\frac{d^2}{16\pi^2}$$

I derived that $$1=\int_{R^{d}}x(\frac{d}{dx})|f(x)|^2dx$$

but I lost my way. I need your help.

- About the intersection of any family of connected sets
- Proving the last part of Nested interval property implying Axiom of completeness
- $f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$
- Problem similar to folland chapter 2 problem 51.
- Examples of statements that are true for real analytic functions but false for smooth functions
- Puzzled by $\displaystyle \lim_{x \to - \infty} \sqrt{x^2+x}-x$

- Any even elliptic function can be written in terms of the Weierstrass $\wp$ function
- Why did we define the concept of continuity originally, and why it is defined the way it is?
- Fourier transform as diagonalization of convolution
- Computing fundamental forms of implicit surface
- If a Laplacian eigenfunction is zero in an open set, is it identically zero?
- Find the coefficients of the Fourier series that minimise the error.
- Logic behind continuity definition.
- Picking a $\delta$ for a convenient $\varepsilon$?
- When is the difference of two convex functions convex?
- How to do contour integral on a REAL function?

Consider the equation

$$

\sum_{i=1}^n\frac12x_i\frac{\mathrm{d}}{\mathrm{d}x_i}|f|^2=\mathrm{Re}\left(\nabla f\cdot\overline{xf}\right)\tag{1}

$$

Integrating $(1)$ over $\mathbb{R}^n$ and then integrating by parts on the left side:

$$

\begin{align}

\frac n2\|f\|_2^2

&=\mathrm{Re}\left(\int_{\mathbb{R}^n}\nabla f\cdot\overline{xf}\,\mathrm{d}x\right)\\

&\le\left|\int_{\mathbb{R}^n}\nabla f\cdot\overline{xf}\,\mathrm{d}x\right|\\[6pt]

&\le\|\nabla f\|_2\|xf\|_2\\[9pt]

&=2\pi\|\xi\hat{f}\|_2\|xf\|_2\tag{2}

\end{align}

$$

Thus,

$$

\|\xi\hat{f}\|_2\|xf\|_2\ge\frac{n}{4\pi}\|\hat{f}\|_2\|f\|_2\tag{3}

$$

The last inequality says that the $L^2$ support radius for $f$ and $\hat{f}$ cannot have a product less than $\frac{n}{4\pi}$. This inequality is sharp as can be seen using the function $f(x) = e^{-\pi x\cdot x}$, whose Fourier Transform is itself, and whose $L^2$ support radius is $\sqrt{\frac{n}{4\pi}}$.

- What is the origin of the formula: $\rho_x (x)=\left|\frac{{d}x}{{d}\alpha}\right|^{-1}\rho_\alpha(\alpha)$ that relates random variables?
- If $f$ is continuous and injective on an interval, then it is strictly monotonic- what's wrong with this proof?
- Path connectedness is a topological invariant?
- Find the Green's Function and solution of a heat equation on the half line
- Closed form of a partial sum of the power series of $\exp(x)$
- Finding supremum of all $\delta > 0$ for the $(\epsilon , \delta)$-definition of $\lim_{x \to 2} x^3 + 3x^2 -x + 1$
- Does $\pi_1(X, x_0)$ act on $\tilde{X}$?
- Are there (known) bounds to the following arithmetic / number-theoretic expression?
- Showing the exponential and logarithmic functions are unique in satisfying their properties
- Universal Definition for Pullback
- Missing solution in quadratic equation
- Can we permute the coefficients of a polynomial so that it has NO real roots?
- Verify for $f(x,y)$, homogeneous of degree $n$: $xf_x+yf_y=nf$
- Continuous unbounded but integrable functions
- Integral inequality (Cauchy-Schwarz)