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Gevorg Mnatsakanyan

Publications and source records attributed to Gevorg Mnatsakanyan.

9 recordsLinked to original sources
arXiv2026

Infinite quantum signal processing for arbitrary Szegő functions

We provide a complete solution to the problem of infinite quantum signal processing for the class of Szegő 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 Szegő function. The proof of stability involves solving a Riemann-Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.

quant-ph↗
arXiv2026

$\ell^1$ mapping properties, smoothness and decay for $SU(2)$-valued nonlinear Fourier transform

We prove an analog of Baxter's theorem for $SU(2)$-valued nonlinear Fourier transform (NLFT). That is, we prove that under certain natural conditions on the NLFT data, the potential is in $\ell^1$ if and only if the linear Fourier coefficients of the NLFT data are in $\ell^1$. Furthermore, we prove some smoothness-decay estimates for the NLFT motivated by similar estimates for the linear Fourier transform.

math.CA↗
arXiv2025

Almost everywhere convergence of a wavelet-type Malmquist-Takenaka series

The Malmquist-Takenaka (MT) system is a complete orthonormal system in $H^2(\mathbf{T})$ generated by an arbitrary sequence of points $a_n$ in the unit disk with $\sum_n (1-|a_n|) = \infty$. The point $a_n$ is responsible for multiplying the $n$th and subsequent terms of the system by a Möbius transform taking $a_n$ to $0$. One can recover the classical trigonometric system, its perturbations or conformal transformations, as particular examples of the MT system. However, many interesting choices of the sequence $a_n$, the MT system is less understood. In this paper, we consider a wavelet-type MT system and prove its almost everywhere convergence in $H^2(\mathbf{T})$.

math.CA↗
arXiv2025

One sided orthogonal polynomials and a pointwise convergence result for $SU(2)$-valued nonlinear Fourier series

We elaborate on a connection between the $SU(2)$-valued nonlinear Fourier series and sequences of left and right orthogonal polynomials for complex measures on the unit circle. We show a convergence result for the associated reproducing kernel. This is a universality type result in the vein of Mate-Nevai-Totik, which turns out to be much simpler in the $SU(2)$ case than in the $SU(1,1)$ case. We then relate a.e. pointwise convergence of the product of left and right polynomials and their squares with both behavior of their zeros as well as behavior of some local parameters for these polynomials. We conclude by proving almost everywhere convergence along lacunary sequences of the functional $(a_n ^* +b_n)(a_n - b_n ^*)$ of the partial $SU(2)$-valued nonlinear Fourier series $(a_n, b_n)$ under the assumption that the nonlinear Fourier series $(a,b)$ itself satisfies both $\|b\|_{L^{\infty} (\mathbb{T})} < 2^{- \frac 1 2}$ and $a^*$ is outer.

math.CA↗
arXiv2024

Quantum signal processing and nonlinear Fourier analysis

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

quant-ph↗
arXiv2022

On almost everywhere convergence of Malmquist-Takenaka series

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 $L^p$ 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.

math.CA↗
arXiv2021

Sharp weighted estimates for strong-sparse operators

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.

math.CA↗