In this second paper on descent methods as proof mechanisms in harmonic analysis (the first treated Bourgain's $\Lambda(p)$ selection theorem), entropic mirror descent gives an improved restricted isometry bound, improving sampling and recovery bounds in the Fourier Ratio program. Let $H$ be a complex $n\times n$ matri...
Let $(\phi_i)_{i=1}^n$ be an orthonormal system on a probability space with $\|\phi_i\|_\infty\leq L$. We prove that, for every $2\leq k<n$, there is a set $I$ of at least $n-k$ indices such that $$ \|f\|_2\leq CM_k\sqrt{\log(e+M_k)}\|f\|_1 $$ for every $f$ in the span of $(\phi_i)_{i\in I}$, where $M_k=L\sqrt{\frac{n}...
In this second paper on descent methods as proof mechanisms in harmonic analysis (the first treated Bourgain's $\Lambda(p)$ selection theorem), entropic mirror descent gives an improved restricted isometry bound, improving sampling and recovery bounds in the Fourier Ratio program. Let $H$ be a complex $n\times n$ matri...
We prove quantitative polynomial Szemer\'edi-type theorems involving polynomial progressions with shift parameter restricted to the set of shifted primes $\mathbb{P}-1$. The types of configurations covered are distinct degree progressions and progressions involving integer multiples of a fixed polynomial. For nonlinear...
Ben Krause, Hamed Mousavi, Terence Tao et al.· 2 citations· ⚡2
We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_{\lambda} \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 \pi i \lambda y^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}...
A. Fragkos, Ben Krause, Michael T. Lacey· 0 citations
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