All Questions
12 questions
15
votes
0
answers
749
views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
- 178
13
votes
1
answer
638
views
A question on the sine function
The Fejer-Jackson-Gronwall inequality involving the sine function is as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
Here I ask the ...
- 15.6k
9
votes
1
answer
410
views
The discrete Hardy-Littlewood-Sobolev inequality
Let $p>1$, $q>1$, $0<\lambda<1$ be such that
$\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that
$(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$.
It is known ([1,2,3]...
- 143
7
votes
2
answers
455
views
On a monotonicity property of Fourier coefficients of truncated power functions
Is it true that
$$a_{k,n}:=\int_0^{2\pi}x^k\cos(nx)\,dx$$
is nonincreasing in natural $n$ for each $k\in\{0,1,\dots\}$?
This question is related to this previous one.
Twice integrating by parts, one ...
- 128k
6
votes
0
answers
129
views
Weak-type inequality for the partial Fourier sum operator
I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark:
For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
- 349
5
votes
2
answers
202
views
Monotonicity of a parametric integral
For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<...
- 128k
5
votes
1
answer
880
views
Generalization of Bernstein’s inequality
I'm using Muscalu and Schlag's textbook to study harmonic analysis and I encountered the following claim:
Given some function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\mathbb{R}^{d})$ ...
- 91
4
votes
0
answers
180
views
Approximation by gaussian mollification in Sobolev spaces
I have been trying to find a good estimate on the constant in the inequality (in dimension $d=3$ to simplify)
$$\label{0}\tag{0}
\|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
- 230
3
votes
1
answer
269
views
Estimate of Hölder Norms (Littlewood–Paley theory)
I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem:
Recall that ...
- 91
3
votes
1
answer
232
views
$L^p$ estimate of a multiplier operator
I'm studying harmonic analysis by myself and I encountered the following claim about multipliers: consider a sequence of complex numbers $\{m_{n}\}_{n \in \mathbb{Z}}$ that satisfies:
$$\sum_{n \in \...
- 349
2
votes
1
answer
158
views
Analogous form of Hardy-Littlewood maximal inequality (weak/strong type) on affine subspaces
I'm using some online notes (Professor Schlag, Yale University) to study harmonic analysis by myself. He introduced the following claim as an exercise:
For any function $f \in L^{1}(\mathbb{R}^{d})$ ...
- 197
1
vote
0
answers
248
views
Uniform bound for an alternating series of functions
I have mainly two questions, the first one being motivated by the second one.
1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ ...
- 266