Hilbert Space L2 Research Articles

New Properties of Frames

Let H be a finite-dimentional complex Hilbert space and l2(H) is the space of square summable sequences in H. We will give a new characterization of a frame for H, we give our definition of a frame for the Hilbert space l2(H), we also define and give the properties of the frame operator. We equally show that our definition is equivalent to the definition of a frame for the Hilbert space H. Finally, we give a way to construct frames for l2(Hn) from frames for l2(Hp) such that p<n via fusion frame theory.

From Schrödinger equation to tempered distribution space on Minkowski chronotope and Schwartz Linear Algebra

In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the SchrÅNodinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′ - represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.

Spectral structure of Moran Sierpinski-type measure on R2

Let Mn=diag[3pn,3qn] with pn,qn⩾1 for all n⩾1 and let D={(0,0)t,(1,0)t,(0,1)t} . One can generate a Borel probability measure μ{Mn},D=δM1−1D∗δ(M2M1)−1D∗δ(M3M2M1)−1D∗⋯. Such measure μ{Mn},D is called a Moran Sierpinski-type measure. It is known Deng et al (Acta Math. Sin. submitted) that the associated Hilbert space L2(μ{Mn},D) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L2(μ{Mn},D) . For such a maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L2(μ{Mn},D) or not.

Nonsmooth data error estimates for fully discrete finite element approximations of semilinear parabolic equations in Banach space

Implicit–explicit (IMEX) Euler formula together with finite element methods is proposed to solve numerically rather general semilinear parabolic differential equations with initial data in Banach space Xα, 0<α≤1. The time-space regularity of solutions to this class of equations is first investigated. Stability and error estimates are provided for this fully discrete scheme by discrete semigroup method. In the special case when X is the Hilbert space L2(Ω), these error estimates bridge the gap between the existing results on this scheme for semilinear parabolic problems with initial data in L2(Ω) and in H1(Ω) under general nonlinearity. Numerical experiments for nonlinear Allen–Cahn equations verify and complement our theoretical results.

Multi-Pursuer and One-Evader Evasion Differential Game with Integral Constraints for an Infinite System of Binary Differential Equations

In the Hilbert space l2, a differential evasion game involving multiple pursuers is considered. Integral constraints are imposed on player control functions. The pursuers are tasked with bringing the state of a system back to the origin of l2, while the evader simultaneously tries to avoid it. It is assumed that the energy of the evader is greater than the total energy of the pursuers. In this paper, we contribute to the solution of the differential evasion game with multiple pursuers by building an exact strategy for the evader.

A special fuzzy star-shaped number space with the sendograph metric

As a natural generalization of fuzzy numbers, fuzzy star-shaped numbers play a crucial role in fuzzy mathematics. Let Y be a non-degenerate compact convex subset of the n-dimensional Euclidean space, and let FSz0(Y) be the family consisting of all fuzzy star-shaped numbers with respect to z0 whose supports contained in Y. Using methods from infinite-dimensional topology, we mainly show that the space FSz0(Y) with the topology induced by the sendograph metric is homeomorphic to the separable Hilbert space ℓ2.

THE ASYMPTOTIC FORMULAS FOR THE SUM OF THE FOURTH DEGREES OF THE NEGATIVE EIGENVALUES OF A SINGULAR STURM-LIOUVILLE OPERATOR

Abstract. Let 𝐻 be a separable Hilbert space. In separable Hilbert space 𝐿2[0,∞) we consider the operator 𝐿 defined by the differential expression 𝑙(𝑦)=−𝑦′′(𝑥)+𝑞(𝑥)𝑦(𝑥) with the boundary condition 𝑦(0)+𝑦′(0) = 0. Suppose that the function 𝑞(𝑥) satisfies following conditions: 1) 𝑞(𝑥) is continuous, monotonous decreasing and positive valued function in [0,∞). 2) lim 𝑛→∞ 𝑞(𝑥)=0. It is known that the operator 𝐿 is semi bounded below and negative part of its spectrum is discrete [1]. Let t 𝜆1≤𝜆2≤⋯≤𝜆𝑛≤⋯ be negative eigenvalues of the operator 𝐿. In this work, we found the asymptotic formulas for the sum ∑ 𝜆𝑗 4 𝜆𝑗&lt;−𝜀 as 𝜀→+0 (𝜀&gt;0). The following theorem holds. Theorem. If the conditions 1)-2) are satisfied, then the asymptotic formula ∑𝜆𝑗 4 𝜆𝑗&lt;−𝜀 = 1 315𝜋[1+𝑂(𝜀𝑠0)]∫ √𝑞(𝑥)−𝜀 𝑞(𝑥)≥𝜀 [128𝑞4(𝑥)+64𝑞3(𝑥)𝜀+48𝑞2(𝑥)𝜀2+40𝑞(𝑥)𝜀3+35𝜀4]𝑑𝑥 holds for 𝜀→+0 where 𝑠0 is fixed positive number.

On a Linear Differential Game in the Hilbert Space ℓ2

Two player pursuit evasion differential game and time optimal zero control problem in &lt;i&gt;ℓ&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt; are considered. Optimal control for the corresponding zero control problem is found. A strategy for the pursuer that guarantees the solution for the pursuit problem is constructed.

Approximate Solution of Optimal Pulse Control Problem Associated with the Heat Conduction Process

Abstract We consider the approximate solution of the control problem with minimum energy for an object described by the heat equation, with the process described by the linear equation of parabolic type and the system controlled by impulsive external influences. Our optimal control problem deals with finding a control parameter belonging to the class of admissible controls that provides the desired temperature distribution in a finite time with minimal energy consumption (energy consumption is described by the quadratic functional). Previous works dedicated to optimal impulse control problems have mostly used the Pontryagin’s maximum principle. However, from a practical point of view, this approach does not lead to satisfactory results. This is due to the fact that the corresponding boundary value problems in this case have no solution in a traditional class of absolutely continuous trajectories. In this work, we propose a method based on the moment relations. We seek for the approximate solution of the corresponding boundary value problem in the form of finite Fourier sum and state our optimal control problem in a finite-dimensional phase space. As a result, we obtain an optimal impulse control problem in a finite-dimensional function space. Taking into account the given condition for a finite time, we reduce the obtained problem to the L-problem of moments. Thus, the problem of finding a control parameter is reduced to the solution of the system of Fredholm integral equations of the first kind, with the norm of the sought solution not exceeding a given number. By Levi’s theorem, every element of Hilbert space can be represented by the sum of the elements of two orthogonal subspaces. This assertion makes it possible to find control parameters in analytical form. We also establish the convergence of the chosen approximation.

Construction of Solutions and Study of Their Closeness in L2 for Two Boundary Value Problems for a Model of Multicomponent Suspension Transport in Coastal Systems

Three-dimensional models of suspension transport in coastal marine systems are considered. The associated processes have a number of characteristic features, such as high concentrations of suspensions (e.g., when soil is dumped on the bottom), much larger areas of suspension spread than the reservoir depth, complex granulometric (multifractional) content of suspensions, and mutual transitions between fractions. Suspension transport can be described using initial-boundary value diffusion–convection–reaction problems. According to the authors' idea, on a time grid constructed for the original continuous initial-boundary value problem, the right-hand sides are transformed with a “delay” so that the right-hand side concentrations of the components other than the underlying one (for which the initial-boundary value problem of diffusion–convection is formulated) are determined at the preceding time level. This approach simplifies the subsequent numerical implementation of each of the diffusion–convection equations. Additionally, if the number of fractions is three or more, the computation of each of the concentrations at every time step can be organized independently (in parallel). Previously, sufficient conditions for the existence and uniqueness of a solution to the initial-boundary value problem of suspension transport were determined, and a conservative stable difference scheme was constructed, studied, and numerically implemented for test and real-world problems. In this paper, the convergence of the solution of the delay-transformed problem to the solution of the original suspension transport problem is analyzed. It is proved that the differences between these solutions tends to zero at an O(τ) rate in the norm of the Hilbert space L2 as the time step t approaches zero.

On p-adic spectral measures

A Borel probability measure μ on a locally compact group is called a spectral measure if there exists a subset of continuous group characters which forms an orthogonal basis of the Hilbert space L2(μ). In this paper, we characterize all spectral measures in the field Qp of p-adic numbers.

Mean attractors of stochastic delay lattice [formula omitted]-Laplacian equations driven by superlinear noise in high-order product Bochner spaces

This paper is concerned with the existence and uniqueness of mean random attractors of a class non-autonomous stochastic delay p-Laplacian lattice systems defined on a high-dimensional integer set Zd driven by a family infinite-dimensional superlinear noise. We first establish the global-in-time existence and uniqueness of the solutions in C([τ,∞),L2k(Ω,ℓ2(Zd)))∩Lq(Ω,Llocq((τ,∞),ℓq(Zd))) for any k⩾1 when the draft term has an arbitrary polynomial growth rate q>2 and the coefficient of the noise admits a superlinear growth order q̃∈[2,q). We then show that the mean random dynamical system generated by the solution operators has a unique weakly compact and weakly attracting mean random attractor in the high-order product Bochner space L2k(Ω,F;ℓ2(Zd))×L2k(Ω,F;L2k((−ϱ,0),ℓ2(Zd))), where ϱ is the time delay parameter. The dissipative property of the draft term is employed to carefully controlling the superlinear growth diffusion term. When k=1, our results are new even in the product Hilbert space L2(Ω,F;ℓ2(Zd))×L2(Ω,F;L2((−ϱ,0),ℓ2(Zd))). This work can be regard as a further study of mean attractors of stochastic p-Laplacian lattice systems in the works of Wang and Wang (2020) and Chen et al. (2023).

Growing-dimensional partially functional linear models: non-asymptotic optimal prediction error

Under the reproducing kernel Hilbert spaces (RKHS), we focus on the penalized least-squares of the partially functional linear models (PFLM), whose predictor contains both functional and traditional multivariate parts, and the multivariate part allows a divergent number of parameters. From the non-asymptotic point of view, we study the rate-optimal upper and lower bounds of the prediction error. An exact upper bound for the excess prediction risk is shown in a non-asymptotic form under a more general assumption known as the effective dimension to the model, by which we also show the prediction consistency when the number of multivariate covariates p slightly increases with the sample size n. Our new finding implies a trade-off between the number of non-functional predictors and the effective dimension of the kernel principal components to ensure prediction consistency in the increasing-dimensional setting. The analysis in our proof hinges on the spectral condition of the sandwich operator of the covariance operator and the reproducing kernel, and on sub-Gaussian and Berstein concentration inequalities for the random elements in Hilbert space. Finally, we derive the non-asymptotic minimax lower bound under the regularity assumption of the Kullback-Leibler divergence of the models.

Fourier bases on general self-similar Sierpinski measures

Let |ρ|>1 be a real number and let Q be a 2 × 2 orthogonal and involutory matrix. Let R={0,u,v}⊂Z2 such that det(u,v)≠0 and 3∤det(u,v). In this paper, we prove that the Hilbert space L2(μρQ,R) generated by the general self-similar Sierpinski measure μρQ,R has an orthonormal basis of the form EΛ=e2πi〈λ,x〉:λ∈Λwith Λ⊂R2, i.e., μρQ,R is a spectral measure, if and only if there exist three integers d,m,n such that d2∣(m2+n2) and ρQ=3md3nd3nd−3md in the standard expression. This provides new and different characterization of spectrality of self-similar measures in the plane and gives a counterexample to Łaba–Wang’s conjecture in two dimension.

Guaranteed Pursuit and Evasion Times in a Differential Game for an Infinite System in Hilbert Space l2

The present paper is devoted to studying a pursuit differential game described by an infinite system of binary differential equations in Hilbert space l2. The control parameters of the players are subject to geometric constraints. The pursuer tries to bring the state of the system to the origin of the Hilbert space l2, and oppositely, the evader tries to avoid it. Our aim is to construct a strategy for the pursuer to complete a differential game and an evasion control. We obtain an equation for the guaranteed pursuit and evasion times.

Subsets of rectifiable curves in Banach spaces I: Sharp exponents in traveling salesman theorems

The analyst’s traveling salesman problem (TSP) is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space ℓ2 by Schul in 2007. In this paper, we establish sharp extensions of Schul’s necessary and sufficient conditions for a bounded set E⊂ℓp to be contained in a rectifiable curve from p=2 to 1<p<∞. While the necessary and sufficient conditions coincide when p=2, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when p≠2. We also identify and correct technical errors in the proof by Schul. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the analyst’s TSP in general metric spaces.

The cardinality of orthogonal exponentials of planar self-affine measures with two-element digit set

Abstract Let μ M , D {{\mu_{M,D}}} be the planar self-affine measure determined by an expanding integer matrix M ∈ M 2 ⁢ ( ℤ ) {M\in M_{2}(\mathbb{Z})} and a two-element digit set D ⊂ ℤ 2 {D\subset\mathbb{Z}^{2}} . It has been shown that the spectral or non-spectral problem on μ M , D {\mu_{M,D}} is only related to trace ⁡ ( M ) := r 1 {\operatorname{trace}(M):=r_{1}} and det ⁡ ( M ) := r 2 {\det(M):=r_{2}} . In the case when M ∈ M 2 ⁢ ( ℤ ) {M\in M_{2}(\mathbb{Z})} is an expanding matrix and r 1 2 = 3 ⁢ r 2 {r_{1}^{2}=3{r_{2}}} , r 2 ∈ 2 ⁢ ℤ + 1 ∖ { ± 1 } {{r_{2}}\in 2\mathbb{Z}+1\setminus\{\pm 1\}} , the Hilbert space L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} contains at most a finite number of orthogonal exponentials, and μ M , D {{\mu_{M,D}}} is a non-spectral measure. The remaining problem in this case is to determine the best upper bound on the cardinality of orthogonal exponentials in the Hilbert space L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} . In the present paper, we further the above research to show that there are at most 16 mutually orthogonal exponentials in the corresponding Hilbert space, and the number 16 is the best upper bound. This completes the non-spectrality of self-affine measures in the above case.

Scalar Variance and Scalar Correlation for Functional Data

In Functional Data Analysis (FDA), the existing summary statistics so far are elements in the Hilbert space L2 of square-integrable functions. These elements do not constitute an ordered set; therefore, they are not sufficient to solve problems related to comparability such as obtaining a correlation measurement or comparing the variability between two sets of curves, determining the efficiency and consistency of a functional estimator, among other things. Consequently, we present an approach of coherent redefinition of some common summary statistics such as sample variance, sample covariance and correlation in Functional Data Analysis (FDA). Regarding variance, covariance and correlation between functional data, our summary statistics lead to numbers instead of functions which is helpful for solving the aforementioned problems. Furthermore, we briefly discuss the functional forms coherence of some statistics already present in the FDA. We formally enumerate and demonstrate some properties of our functional summary statistics. Then, a simulation study is presented briefly, with evidence of the consistency of the proposed variance. Finally, we present the implementation of our statistics through two application examples.

Research of the Solutions Proximity of Linearized and Nonlinear Problems of the Biogeochemical Process Dynamics in Coastal Systems

The article considers a non-stationary three-dimensional spatial mathematical model of biological kinetics and geochemical processes with nonlinear coefficients and source functions. Often, the step of analytical study in models of this kind is skipped. The purpose of this work is to fill this gap, which will allow for the application of numerical modeling methods to a model of biogeochemical cycles and a computational experiment that adequately reflects reality. For this model, an initial-boundary value problem is posed and its linearization is carried out; for all the desired functions, their final spatial distributions for the previous time step are used. As a result, a chain of initial-boundary value problems is obtained, connected by initial–final data at each step of the time grid. To obtain inequalities that guarantee the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems, the energy method, Gauss’s theorem, Green’s formula, and Poincaré’s inequality are used. The scientific novelty of this work lies in the proof of the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems in the norm of the Hilbert space L2 as the time step τ tends to zero at the rate O(τ).

Weak mean attractors of stochastic p-Laplacian delay lattice systems driven by nonlinear noise

This paper deals with the existence and uniqueness of weak pullback mean random attractors for a class non-autonomous stochastic p-Laplacian delay lattice systems with local Lipschitz nonlinear terms driven by nonlinear noise defined on the entire integer set Z. We first establish the global well-posedness to non-autonomous stochastic p-Laplacian delay lattice system in C([τ,∞),L2(Ω,ℓ2)) when the nonlinear diffusion terms and drift terms are locally Lipschitz continuous functions, based on the uniform estimates of approximate solutions in a finite time-interval as well as an appropriate stopping time. We then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors in product Hilbert space L2(Ω,F;ℓ2)×L2(Ω,F;L2((−ρ,0),ℓ2)) under the condition that the constant λ in the system is relatively large.