arXiv Math @arxiv_math@qoto.org

The aim of the current paper is to determine the necessary and sufficient conditions for the weights $\mathbf{q}=\{q_k\}$, ensuring that the sequence of operators $\left\{ T_{n}^{\left( \mathbf{q}\right) }f\right\} $ associated with Walsh system, is convergent almost everywhere for all integrable function $f$. The article also examines the convergence of a sequence of tensor product operators denoted as $\left\{ T_{n}^{( \mathbf{q})}\otimes T_{n}^{( \mathbf{p})}\right\}$ involving functions of two variables. We point out that recent research by Gát and Karagulyan (2016) demonstrated that this sequence of tensor product operators cannot converge almost everywhere for every integrable function. In this paper, the necessary and sufficient conditions for the weight are provided which ensure that the sequence of the mentioned operators converges in measure on $L_{1}$.

How hard is it to estimate a discrete-time signal $(x_{1}, ..., x_{n}) \in \mathbb{C}^n$ satisfying an unknown linear recurrence relation of order $s$ and observed in i.i.d. complex Gaussian noise? The class of all such signals is parametric but extremely rich: it contains all exponential polynomials over $\mathbb{C}$ with total degree $s$, including harmonic oscillations with $s$ arbitrary frequencies. Geometrically, this class corresponds to the projection onto $\mathbb{C}^{n}$ of the union of all shift-invariant subspaces of $\mathbb{C}^\mathbb{Z}$ of dimension $s$. We show that the statistical complexity of this class, as measured by the squared minimax radius of the $(1-δ)$-confidence $\ell_2$-ball, is nearly the same as for the class of $s$-sparse signals, namely $O\left(s\log(en) + \log(δ^{-1})\right) \cdot \log^2(es) \cdot \log(en/s).$ Moreover, the corresponding near-minimax estimator is tractable, and it can be used to build a test statistic with a near-minimax detection threshold in the associated detection problem. These statistical results rest upon an approximation-theoretic one: we show that finite-dimensional shift-invariant subspaces admit compactly supported reproducing kernels whose Fourier spectra have nearly the smallest possible $\ell_p$-norms, for all $p \in [1,+\infty]$ at once.

It is well-known that to establish the almost everywhere convergence of a sequence of operators on $L_1$-space, it is sufficient to obtain a weak $(1,1)$-type inequality for the maximal operator corresponding to the sequence of operators. However, in practical applications, the establishment of the mentioned inequality for the maximal operators is very tricky and difficult job. In the present paper, the main aim is a novel outlook at above mentioned inequality for the tensor product of two weighted one-dimensional Walsh-Fourier series. Namely, our main idea is naturally to consider uniformly boundedness conditions for the sequences which imply the weak type estimation for the maximal operator of the tensor product. More precisely, we are going to establish weak type of inequality for the tensor product of two dimensional maximal operators while having the uniform boundedness of the corresponding one dimensional operators. Moreover, the convergence of the tensor product of operators is established at the two dimensional Walsh-Lebesgue points.

We generalize some of the well-known results of Karlin and James to Banach spaces with besselian Schauder frames (then in particular with unconditional Schauder frames). It is well-known that each Banach space with an unconditional Schauder basis has the Pełczyński's property (u). Also, that all of the classical Banach spaces ($C(ω^ω)$, $C([0,1])$, $L_{1}([0,1])$, the James's space $\mathcal{J}$, and for each $p\geq q>1$: $K(l_{p},l_{q})$, $l_{p}\otimes_πl_{q^{*}}$, and $l_{p^{*}}\otimes_{\varepsilon}l_{q}$) have no unconditional Schauder bases. In this paper, we extend these results to Banach spaces with besselian Schauder frames (then in particular with unconditional Schauder frames). Finally, for each Banach space $E$ with a finite dimensional decomposition, we give an explicit method to construct a Schauder frame for $E$. Therefore, the Szarek's famous example becomes an example of a Banach space with a Schauder frame which fails to have a Schauder basis.

The concept of a bi-g-fusion frame for a Hilbert space, which is a generalizations of a controlled g-fusion frame, is introduced and an example is given. Finally, bi-g-fusion frame in tensor product of Hilbert spaces is considered.

The Minkowski problem for torsional rigidity ($2$-torsional rigidity) was firstly studied by Colesanti and Fimiani \cite{CA} using variational method. Moreover, Hu \cite{HJ00} also studied this problem by the method of curvature flows and obtained the existence of smooth even solutions. In addition, the smooth non-even solutions to the Orlicz Minkowski problem $w. r. t$ $q$-torsional rigidity were given by Zhao et al. \cite{ZX} through a Gauss curvature flow. The dual curvature measure and the dual Minkowski problem were first posed and considered by Huang, Lutwak, Yang and Zhang in \cite{HY}. The dual Minkowski problem is a very important problem, which has greatly contributed to the development of the dual Brunn-Minkowski theory and extended the other types dual Minkowski problem. To the best of our knowledge, the dual Minkowski problem $w. r. t$ ($q$) torsional rigidity is still open because the dual ($q$) torsional measure is blank. Thus, it is a natural problem to consider the dual Minkowski problem for ($q$) torsional rigidity. In this paper, we introduce the $p$-th dual $q$-torsional measure by the variational method and propose the $p$-th dual Minkowski problem for $q$-torsional rigidity with $q>1$. Then we confirm the existence of smooth even solutions for $p<n$ ($p\neq 0$) to the $p$-th dual Minkowski problem for $q$-torsional rigidity by method of a Gauss curvature flow. Specially, we also obtain the smooth non-even solutions with $p<0$ to this problem.

Motivated from Deutsch entropic uncertainty principle and several product uncertainty principles, we derive an uncertainty principle for the product of entropies using functions.

In this paper, we study a theoretical math problem of game theory and calculus of variations in which we minimize a functional involving two players. A general relationship between the optimal strategies for both players is presented, followed by computer analysis as well as polynomial approximation. Nash equilibrium strategies are determined through algebraic manipulation and linear programming. Lastly, a variation of the game is also investigated.

In this short note we present a far generalization of the following very well-known assertion: assume that we have two orthonormal sequences in a Hilbert space and these sequences are quadratically close to each other. Then if one of these sequences is a basis in the Hilbert space then so is the other one.

In the theory of matrix-valued orthogonal polynomials, there exists a longstanding problem known as the Matrix Bochner Problem: the classification of all $N \times N$ weight matrices $W(x)$ such that the associated orthogonal polynomials are eigenfunctions of a second-order differential operator. In [4], Casper and Yakimov made an important breakthrough in this area, proving that, under certain hypotheses, every solution to this problem can be obtained as a bispectral Darboux transformation of a direct sum of classical scalar weights. In the present paper, we construct three families of weight matrices $W(x)$ of size $N \times N$, associated with Hermite, Laguerre, and Jacobi weights, which can be considered 'singular' solutions to the Matrix Bochner Problem because they cannot be obtained as a Darboux transformation of classical scalar weights.

We review the so-called Nikiforov-Uvarov method along with some basic results about classical orthogonal polynomials and hypergeometric functions related to the hypergeometric differential equation. The method is employed to address certain eigenvalue problems that appear in quantum mechanics, namely, time-independent Schrödinger equation, paying special attention to properly and completely characterising its spectrum. Finally, some potentials are discussed and solved.

This paper investigates the fundamental connections between linear super-commuting maps, super-biderivations, and centroids in Hom-Lie superalgebras under certain conditions. Our work generalizes the results of Bresar and Zhao on Lie algebras.

Let $K$ be a field of characteristic zero and let $\mathfrak{sl}_2 (K)$ be the 3-dimensional simple Lie algebra over $K$. In this paper we describe a finite basis for the $\mathbb{Z}_2$-graded identities of the adjoint representation of $\mathfrak{sl}_2 (K)$, or equivalently, the $\mathbb{Z}_2$-graded identities for the pair $(M_3(K), \mathfrak{sl}_2 (K))$. We work with the canonical grading on $\mathfrak{sl}_2 (K)$ and the only nontrivial $\mathbb{Z}_2$-grading of the associative algebra $M_3(K)$ induced by that on $\mathfrak{sl}_2(K)$.