Gevorg Mnatsakanyan’s research while affiliated with University of Bonn and other places

We provide a complete solution to the problem of infinite quantum signal processing for the class of Szeg\H o functions, which are functions that satisfy a logarithmic integrability condition and include almost any function that allows for a quantum signal processing representation. We do so by introducing a new algorithm called the Riemann-Hilbert-Weiss algorithm, which can compute any individual phase factor independent of all other phase factors. Our algorithm is also the first provably stable numerical algorithm for computing phase factors of any arbitrary Szeg\H o function. The proof of stability involves solving a Riemann-Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.

Elucidating a connection with nonlinear Fourier analysis (NLFA), we extend a well known algorithm in quantum signal processing (QSP) to represent measurable signals by square summable sequences. Each coefficient of the sequence is Lipschitz continuous as a function of the signal.

Recently, Alexei Poltoratski proved (arXive:2103.13349) pointwise convergence of the non-linear Fourier transform giving a partial answer to the long-standing question of Muscalu, Tao and Thiele (arXive:0205139). We quantify his techniques and, in particular, prove an estimate for the de Branges function associated to the NLFT through its zeros and the maximal function of the spectral measure. We push these estimates towards the conjectured weak-$L^2$ estimate of the Carleson operator of the NLFT. As a corollary to the main theorem, we obtain a zero free strip of the de Brange function for potentials with small $L^1$ norm.

We prove the sharp weighted-L2 bounds for the strong-sparse operators introduced in [3]. The main contribution of the paper is the construction of a weight that is a lacunary mixture of dual power weights. This weight helps to prove the sharpness of the trivial upper bound of the operator norm.

The Malmquist-Takenaka system is a perturbation of the classical trigonometric system, where powers of z are replaced by products of other Möbius transforms of the disc. The system is also inherently connected to the so-called nonlinear phase unwinding decomposition which has been in the center of some recent activity. We prove Lp bounds for the maximal partial sum operator of the Malmquist-Takenaka series under additional assumptions on the zeros of the Möbius transforms. We locate the problem in the time-frequency setting and, in particular, we connect it to the polynomial Carleson theorem.

We prove the sharp weighted-$L^2$ bounds for the strong-sparse operators introduced in \cite{KaragulyanM}. The main contribution of the paper is the construction of a weight that is a lacunary mixture of dual power weights. This weights helps to prove the sharpness of the trivial upper bound of the operator norm.

The Malmquist-Takenaka system is a perturbation of the classical trigonometric system, where powers of z are replaced by products of other M\"obius transforms of the disc. The system is also inherently connected to the so-called nonlinear phase unwinding decomposition which has been in the center of some recent activity. We prove $L^p$ bounds for the maximal partial sum operator of the Malmquist-Takenaka series under additional assumptions on the zeros of the M\"obius transforms. We locate the problem in the time-frequency setting and, in particular, we connect it to the polynomial Carleson theorem.

We define sparse operators of strong type on abstract measure spaces with ball-bases. Weak and strong type inequalities for such operators are proved.

We define sparse operators of strong type on abstract measure spaces with ball-bases. Weak and strong type inequalities for such operators are proved.