Searcharxiv⌕ Search

arXiv subjects

Sergo A. Episkoposian

Publications and source records attributed to Sergo A. Episkoposian.

9 recordsLinked to original sources
arXiv2026

Walsh Analysis in the Twenty-First Century: Convergence, Summability, Nonlinear Approximation, Universality, and Digital Applications

Walsh analysis connects ordered Fourier convergence, summability, nonlinear approximation, and universal representation through a common character algebra. This survey examines developments from 2000-2026 against their classical foundations. Binary kernel geometry explains Hardy-space losses and summability thresholds; interval and packet methods control spectral selection; localized blocks produce correction and universal representations with distinct coefficient and exceptional-set requirements. Selected proofs and counterexamples compare these mechanisms and identify where an implication needs an additional hypothesis. The same distinctions organize digital integration, sparse measurements, Boolean spectra, and generalized Walsh bases. Quantitative thresholds, source conventions, and published theorem pointers are collected in comparative tables. The concluding problems separate source-listed questions, proposed refinements, and elementary obstructions. Supplementary derivations preserve normalization checks and finite covariance calculations while keeping the main exposition focused on the analytic mechanisms.

math.CA↗
arXiv2026

Diagonal and branchwise criteria for universal Fourier series

We characterize diagonal profiles admitting universal real trigonometric Fourier series with factors in $\mathcal E_s=\bigcap_{p>s}\ell^p$ and $\ell^{s,q}$, $2\le s<\infty$, $1\le q\le\infty$. For nonincreasing profiles we obtain exact dyadic criteria and logarithmic endpoints. For independently regular odd/even sine/cosine branches we characterize simultaneous subseries, sign, and rearrangement universality for finite almost-everywhere targets. One function in every $L^p$, $1\le p<2$, supplies the coefficients. A separation example admits this realization although every compatible nonincreasing majorant with bounded relative cost belongs to $\ell^1$. Residue-class decompositions extend the criterion; a weakest-model condition permits extended-real rearrangement targets. A Walsh--Paley counterpart yields the same diagonal multiplier thresholds for ternary $\{0,\pm1\}$-universality, together with a Fourier--Walsh realization. Trigonometric realization uses Pogosyan's majorant theorem; the multiplier, factorization, projection-transfer, separation, and Walsh Fourier-origin arguments are proved here.

math.CA↗
arXiv2026

Compactness and Spectral Properties of Multiplier Operators in the Walsh System

We investigate compactness and spectral properties of multiplier operators associated with the Walsh system in the spaces $L^p[0,1]$, $1<p<\infty$. Building upon previously established criteria for boundedness of Walsh multipliers, we prove an exact compactness criterion in the $L^p\to L^p$ regime for all $1<p<\infty$(assuming boundedness of the multiplier), and also in the $L^p\to L^2$ regime for $2<p<\infty$. The key result states that compactness is equivalent to the condition $a_n\to 0$ for the multiplier symbol. We also examine in detail the point spectrum and derive strict spectral inclusions; in the Hilbert space case $p=2$ we obtain a complete description of the spectrum. For $p\neq 2$, we emphasize the limitations of transferring "diagonal" arguments and formulate results in a form that does not admit incorrect generalizations.

math.FA↗
arXiv2015

Existence of double Walsh series universal in weighted $L_μ^1[0,1]^2$ spaces

In this paper we consider a question on existence of double Walsh series universal in weighted $L_μ^1[0,1]^2$ spaces. We construct a weighted function $μ(x,y)$ and a series by double Walsh system of the form $$\sum_{n,k=1}^\infty c_{n,k}W_n(x)W_k(y)\ \ \mbox{with} \ \ \sum_{n,k=1}^\infty \left | c_{n,k} \right|^q <\infty\ \mbox{for all}\ q>2,$$ which is universal in $L_μ^1[0,1]^2$ concerning subseries with respect to convergence, in the sense of both spherical and rectangular partial sums.

math.FA↗
arXiv2011

On the existence of universal series by trigonometric system

In this paper we prove the following: let $ω(t)$ be a continuous function, increasing in $[0,\infty)$ and $ω(+0)=0$. Then there exists a series of the form$\sum_{k=-\infty}^\infty C_ke^{ikx}$ with $\sum_{k=-\infty}^\infty C^2_k ω(|C_k|)<\infty$, $C_{-k}=\bar{C}_k$, with the following property: for each $ε>0$ a weighted function $μ(x), 0<μ(x) \le1,| \{x\in[0,2π]: μ(x)\not =1 \}| <ε$ can be constructed, so that the series is universal in the weighted space $L_μ^1[0,2π]$ with respect to rearrangements.

math.FA↗
arXiv2011

On greedy algorithms with respect to generalized Walsh system

In this paper we proof that there exists a function f(x) belongs to L^1[0,1] such that a greedy algorithm with regard to generalized Walsh system does not converge to f(x) in L^1[0,1] norm, i.e. the generalized Walsh system is not a quasi-greedy basis in its linear span L^1[0,1].

math.FA↗