All Questions
57 questions
7
votes
2
answers
437
views
Radial limit does not exist almost everywhere
Problem 4 in Chapter 4 of Stein's book "Real Analysis" says
$\sum_{n\geqslant 0}z^{2^n}$
doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...
- 43.7k
10
votes
1
answer
760
views
Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?
Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $j'_{n+1/2,1}$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$.
...
- 135
1
vote
0
answers
154
views
variation norm of a Fourier transform
Motivated by certain uniform estimate in oscillatory integrals, I am now trying to calculate the Fourier transform of the function ${\large e^{i|t|^{\epsilon}}/t}$ on $\mathbb{R}$, where $\epsilon\in (...
- 331
5
votes
0
answers
913
views
Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?
Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
- 103
1
vote
1
answer
289
views
Compactly supported smooth function with Laplace transform bounded on a cone
My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
- 91.8k
2
votes
1
answer
433
views
bounding the absolute value of a trigonometric polynomial
Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$
\begin{equation*}
f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})}
\...
CommunityBot
- 1
5
votes
1
answer
550
views
Weakest assumption for pointwise convergence of Fourier series
This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS!
Suppose $f(x)=\sum_{...
- 31.5k