Exponential bases on multi-rectangles in $\mathbb{R}^d$
We construct explicit exponential bases on finite unions of disjoint rectangles of $\mathbb{R}^d$ with rational vertices.
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arXiv subjects
Laura De Carli
Publications and source records attributed to Laura De Carli.
One-parameter groups of operators and discrete Hilbert transforms
We show that the discrete Hilbert transform and the discrete Kak-Hilbert transform are infinitesimal generator of one-parameter groups of operators
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Pitt inequalities and restriction theorems for the Fourier transform
We prove new Pitt inequalities for the Fourier transforms with radial and non-radial weights using weighted restriction inequalities for the Fourier transform on the sphere. We also prove new Riemann-Lebesgue estimates and versions of the uncertainty principle for the Fourier transform.
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Stability of exponential bases on d- dimensional domains
We find explicit stability bounds for exponential Riesz bases on domains of R^d. Our results generalize Kadec theorem and other stability theorems in the literature.
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Parseval frames with n+1 vectors in R^n
We construct a Parseval frame with $n+1$ vectors in $\R^n$ that contains a given vector. We also provide a characterization of unit-norm frames that can be scaled to a Parseval frame.
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Split functions, Fourier transforms and multipliers
We study the effect of a splitting operator S_t on the L^p norm of the Fourier transform of a function f and on the operator norm of a Fourier multiplier m. Most of our results assume p is an even integer, and are often stronger when f or m has compact support.
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Minimal potential results for the Schrodinger equation in a slab
Consider the Schrodinger equation -Δu =(k+V) u in an infinite slab S= \R^{n-1}x (0,1), where V is a bounded potential supported on a set D of finite measure. We prove necessary conditions for the existence of nontrivial admissible solutions. These conditions involve the sup. of |V|, the measure of D, and the distance of k from the "special set" {π^2 m^2, m positive integer}. In many cases, these inequalities are sharp.
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Exponential bases on two dimensional trapezoids
We discuss existence and stability of Riesz bases of exponential type of L^2(T) for special domains T called trapezoids. We construct exponential bases on L^2(T) when T is a finite union of rectangles with the same height. We also generalize our main theorems in dimension d\ge 3.
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Minimal support results for Schrödinger equations
We consider a number of linear and non-linear boundary value problems involving generalized Schrödinger equations. The model case is $-Δu=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. We use the Sobolev embedding theorem, and in some cases the Moser-Trudinger inequality and the Hardy-Sobolev inequality, to derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of $V$, the measure of $D$, and a sharp Sobolev constant. In most cases, these inequalities are best possible.
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