We consider the recovery of functions of the weighted Wiener classes $W_s^\mu$ from perturbed Fourier--Hermite coefficients in the metric of $L_\infty(\mathbb R)$. For the summation methods of the class $\mathcal S^\theta$ we find error estimates on $W_s^\mu$. The estimates take into account the behaviour of the Hermite functions both in the oscillatory region and near the turning point: the maxima of functions with different indices are attained at different points, and the supremum of a sum over a block of indices is substantially smaller than the sum of the suprema. Owing to this, in particular, we establish a common lower bound for all finite diagonal methods; for $s\ne4/3$ and $p\ne4/3$ it matches the upper bound and gives the exact order of the error in this class, while at the critical points the gap between the bounds does not exceed $(\log(e/\delta))^{1/4}$ up to a constant. If, in addition, $s>4/3$ or $p<4/3$, then this order is also optimal among all recovery operators that use the whole sequence of data.
Let $(F_n)$ be a sequence of random vectors with identity covariance matrix whose components belong to the same fixed Wiener chaos, and assume that $F_n$ converges in law to a standard Gaussian vector. We prove that, for every integer $m\geq0$ and every $p\in[1,\infty]$, the optimal rate of convergence of the densities...
We develop stability and error estimates for multistep convolution quadrature (CQ) under general radial bounds on the Laplace symbol. Instead of the classical single-power condition $|F(z)|\leq M|z|^{-\mu}$, we assume $|F(z)|\leq C\Phi(|z|)$ and consider two classes of radial functions: monotone bounds satisfying a sca...
We study bounded operators whose prescribed powers remain uniformly close to the identity, and resolvent conditions sampled along sequences of powers. The displacement bound $\sup_{n\ge1}\|I-T^n\|\le q<2$ forces a finite peripheral spectrum consisting of odd-order roots of unity and makes an explicitly determined odd p...
We consider the problem of numerical differentiation of functions defined on the half-line: the derivative $f^{(r)}$, $ r=1,2,\ldots$, is recovered from a finite set of perturbed Fourier--Laguerre coefficients of a function $f$, whose error is measured in $\ell_p$, while the accuracy of approximation is measured in the...
We study the high-frequency Fourier coefficients $C_n$ of critical Gaussian multiplicative chaos $M$ on the circle, for logarithmic covariances with an arbitrary admissible smooth, possibly nonstationary remainder. For every fixed finite set of integer offsets, the corresponding vector of neighboring coefficients, mult...
Yin-Qi Cai, Bo-Nan Chen, Xiang Fang et al.· 0 citations
We investigate the eigenvalues of Hermitian Toeplitz matrices generated by the symmetric Fisher-Hartwig symbol $\hat a(z) = (1-z)^{\alpha/2}(1-z^{-1})^{\alpha/2}$ for $\alpha \in (0,2)$. Using Dirichlet-Neumann bracketing of the discrete Laplacian, we establish explicit, non-asymptotic bounds for the individual eigenva...
Jacek Wszoła· Linear Algebra and its Appli...· 0 citations
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