Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a linear evolution operator. In Dynamical Sampling, both the signal, $f$, and the driving operator, $A$, may be unknown. For example, let $f \in l^2(I)$ where $I=\{1, \ldots, d\}$. Suppose for $\Omega \subset I$ we know $\{{ A^j f(i)} : j= 0, \ldots l_i, i\in \Omega\}$ for some $A: l^2(I) \to l^2(I)$.
An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $ L^{2+\frac{2}{d}}([0,1]^{d}) $ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1}) $ for every $\varepsilon>0 $. It has been fully solved only for $ d=1 $ and there are many partial results in higher dimensions regarding the range of $ (p,q) $ for which $L^{p}([0,1]^{d}) $ is mapped to $ L^{q}(\mathbb{R}^{d+1}) $.
The main topics of this thesis concern two types of approximate Schauder frames for the Banach sequence space $\ell_1$. The first main topic pertains to finite-unit norm tight frames (FUNTFs) for the finite-dimensional real sequence space $\ell_1^n$. We prove that for any $N \geq n$, FUNTFs of length $N$ exist for real $\ell_1^n$. To show the existence of FUNTFs, specific examples are constructed for various lengths. These constructions involve repetitions of frame elements.
Intermittency is a property observed in the study of turbulence. Two of the most popular ways to measure it are based on the concept of flatness, one with structure functions in the physical space and the other one with high-pass filters in the frequency space. Experimental and numerical simulations suggest that the two approaches do not always give the same results. In this talk, we prove they are not analytically equivalent.
We introduce the averages $K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) f(x+n)$, where $d(n)$ denotes the divisor function and $D(N) = \sum _{n=1} ^N d(n) $. We shall see that these averages satisfy a uniform, scale free, $\ell^p$-improving estimate for $p \in (1,2)$, that is
$$ \Bigl( \frac{1}{N} \sum |K_Nf|^{p'} \Bigl)^{1/p'} \leq C \Bigl(\frac{1}{N} \sum |f|^p \Bigl)^{1/p} $$
as long as $f$ is supported on the interval $[0,N]$.
The Bergman projection is a fundamental operator in complex analysis. It is well-known that in the case of the unit ball, the Bergman projection is bounded on weighted L^p if and only if the weight belongs to the Bekolle-Bonami, or B_p, class. These weights are defined using a Muckenhoupt-type condition. Rahm, Tchoundja, and Wick were able to compute the dependence of the operator norm of the projection in terms of the B_p characteristic of the weight using modern tools of dyadic harmonic analysis. Moreover, their upper bound is essentially sharp.
The work deals with the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in the presence of the transport term. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used. We discuss how the introduction of the transport term impacts the regularity of solutions.
For (X,d,w) be a space of homogeneous type in the sense of Coifman and Weiss, suppose that u and v are two locally finite positive Borel measures on (X,d,w). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderon--Zygmund operator T from L^{2}(u) to L^{2}(v) in terms of the A_{2} condition and two testing conditions. The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.
