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-1
votes
0answers
15 views
Equidistribution of sequence $an^b$, 0<b<1, a>0 [migrated]
This is Exercise number 8, Chapter 4 in Stein's book on Fourier Analysis. I'm supposed to solve it by giving an asymptotic bound to the exponential sum created from Weyl's Criterion, but so far have ...
0
votes
0answers
24 views
decay rate or bound of short-time Fourier transform of nonlinear waves
Suppose we have a nonlinear wave $f(x)=e^{2\pi i N(x+\epsilon \sin(x))}$ with positive $\epsilon$ small enough. Let $w(x)$ be a smooth window function supported in a unit ball. Define the short-time ...
1
vote
0answers
96 views
Obtaining a pointwise bound on the convolution of two singular measures
I am confused about a passage in the paper by T. Tao A sharp bilinear restriction estimate for paraboloids.
We are in Section 7, near equation (34) (pag.16 of the arxiv).
Notations and ...
4
votes
0answers
243 views
Riemann hypothesis and Kakeya needle problem
The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...
7
votes
1answer
112 views
Kakeya and Nikodym maximal functions
I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more ...
3
votes
1answer
97 views
Matched pair of locally compact groups
In measure theoretic language there is a notion of matched pair of locally compact (l.c.) groups due to Baaj-Skandalis-Vaes. A pair $(G_{1}, G_{2})$ is called a matched pair of l.c. groups if there ...
1
vote
2answers
108 views
Tangent vectors on the algebra of trigonometric polynomials
Let $G$ be a compact real Lie group and ${\sf Trig}(G)$ the algebra of trigonometric polynomials on $G$ (defined in the Hewitt-Ross, Abstract harmonic analysis, (27.7)), i.e. the algebra of functions ...
0
votes
1answer
55 views
local moments of measures whose Fourier transform vanish in an interval
Assume h is a measure whose Fourier transform vanishes in an interval $[-\Omega,\Omega]$. I'm interested in obtaining inequalities of the form
\begin{equation*}
\int_{-\delta}^{+\delta}|h|(dt)\le ...
3
votes
2answers
109 views
On lower bounds of exponential frames in l1 norm
Let $\{t_k\}_{k=-\infty}^\infty$ be a sequence of real numbers.
I'm interested in finding the largest number A such that
\begin{equation*}
\int_{-\Omega}^\Omega|\sum_{k=-\infty}^{+\infty}c_ke^{2\pi i ...
6
votes
2answers
182 views
the convolution of integrable functions is continuous?
The question is simple but I still can't prove it or contradict it. Here it goes:
Suppose $f$ and $g$ are defined on the circle
(or, equivalently, $2\pi$ periodic functions) and Lebesgue ...
3
votes
4answers
242 views
How does one show the existence of discrete and complementary series for SL(2,R)?
In his book on $\mathrm{SL}(2,\mathbb{R})$, Lang shows that any nontrivial irreducible unitary representation of this group is infinitesimally isomorphic to an irreducible admissible subrepresentation ...
2
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0answers
79 views
A microlocal representation for quantum operator dynamics
In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...
4
votes
2answers
218 views
Operators from $L^{\infty}$ to $L^{\infty}$
If $T$ defined as $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded from $L^{\infty}$ to $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le ...
2
votes
2answers
121 views
Non-global oscillation of banded Fourier transform
Can we say something like monotonicity, growth rate and oscillation of the Fourier transform of a banded function $f$ with support $[0, N]$
$$\mathcal{F}f(\xi) = \int_{0}^N f(x)e^{-ix\xi}dx.$$
Of ...
3
votes
1answer
58 views
To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function
I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the Schrödinger group
\begin{equation}
S_t=e^{i t \Delta}.
\end{equation}
In this context ...
