Space Of Square Integrable Functions Research Articles

Time-Slicing Approximation of Feynman Path Integrals on Compact Manifolds

We construct fundamental solutions to the time-dependent Schr\"odinger equations on compact manifolds by the time-slicing approximation of the Feynman path integral. We show that the iteration of short-time approximate solutions converges to the fundamental solutions to the Schr\"odinger equations modified by the scalar curvature in the uniform operator topology from the Sobolev space to the space of square integrable functions. In order to construct the time-slicing approximation by our method, we only need to consider broken paths consisting of sufficiently short classical paths. We prove the convergence to fundamental solutions by proving two important properties of the short-time approximate solution, the stability and the consistency.

Average sampling in certain subspaces of Hilbert–Schmidt operators on $$L^2(\mathbb {R}^d)$$

The concept of translation of an operator allows to consider the analogue of shift-invariant subspaces in the class of Hilbert–Schmidt operators. Thus, we extend the concept of average sampling to this new setting, and we obtain the corresponding sampling formulas. The key point here is the use of the Weyl transform, a unitary mapping between the space of square integrable functions in the phase space \(\mathbb {R}^d\times \widehat{\mathbb {R}}^d\) and the Hilbert space of Hilbert–Schmidt operators on \(L^2(\mathbb {R}^d)\), which permits to take advantage of some well established sampling results.

Hermite Functions and Fourier Series

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

A New Upper Bound for Sampling Numbers

We provide a new upper bound for sampling numbers (g_n)_{nin mathbb {N}} associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants C,c>0 (which are specified in the paper) such that gn2≤Clog(n)n∑k≥⌊cn⌋σk2,n≥2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} g^2_n \\le \\frac{C\\log (n)}{n}\\sum \\limits _{k\\ge \\lfloor cn \\rfloor } \\sigma _k^2,\\quad n\\ge 2, \\end{aligned}$$\\end{document}where (sigma _k)_{kin mathbb {N}} is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding mathrm {Id}:H(K) rightarrow L_2(D,varrho _D). The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of H^s_{text {mix}}(mathbb {T}^d) in L_2(mathbb {T}^d) with s>1/2. We obtain the asymptotic bound gn≤Cs,dn-slog(n)(d-1)s+1/2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} g_n \\le C_{s,d}n^{-s}\\log (n)^{(d-1)s+1/2}, \\end{aligned}$$\\end{document}which improves on very recent results by shortening the gap between upper and lower bound to sqrt{log (n)}. The result implies that for dimensions d>2 any sparse grid sampling recovery method does not perform asymptotically optimal.

A Second Regularized Trace Formula for a Fourth Order Differential Operator

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.

Octonion Fourier transform of Lipschitz real-valued functions of three variables on the octonion algebra

In this paper, we examine the order of magnitude of the octonion Fourier transform (OFT) for real-valued functions of three variables and satisfiying certain Lipschitz conditions. In addition, using the analog of the operator Steklov, we construct the generalized modulus of smoothness, and also using the Laplacian operator we define the K-functional. We use the octonion Fourier transform (OFT) of real-valued functions of three variables to prove the equivalence between K-functionals and modulus of smoothness in the space of square-integrable functions (in Lebesgue sense).

Statistical solutions of the incompressible Euler equations

We propose and study the framework of dissipative statistical solutions for the incompressible Euler equations. Statistical solutions are time-parameterized probability measures on the space of square-integrable functions, whose time-evolution is determined from the underlying Euler equations. We prove partial well-posedness results for dissipative statistical solutions and propose a Monte Carlo type algorithm, based on spectral viscosity spatial discretizations, to approximate them. Under verifiable hypotheses on the computations, we prove that the approximations converge to a statistical solution in a suitable topology. In particular, multi-point statistical quantities of interest converge on increasing resolution. We present several numerical experiments to illustrate the theory.

Spectral analysis of the multidimensional diffusion operator with random jumps from the boundary

We develop a Hilbert-space approach to the diffusion process of the Brownian motion in a bounded domain with random jumps from the boundary introduced by Ben-Ari and Pinsky in 2007. The generator of the process is introduced by a diffusion elliptic differential operator in the space of square-integrable functions, subject to non-self-adjoint and non-local boundary conditions expressed through a probability measure on the domain. We obtain an expression for the difference between the resolvent of the operator and that of its Dirichlet realization. We prove that the numerical range is the whole complex plane, despite the fact that the spectrum is purely discrete and is contained in a half-plane. Furthermore, for the class of absolutely continuous probability measures with square-integrable densities we characterise the adjoint operator and prove that the system of root vectors is complete. Finally, under certain assumptions on the densities, we obtain enclosures for the non-real spectrum and find a sufficient condition for the non-zero eigenvalue with the smallest real part to be real. The latter supports the conjecture of Ben-Ari and Pinsky that this eigenvalue is always real.

Metatimes, random measures and cylindrical random variables

Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra of a metatime with a measure and defining the composition measure given by the metatime on the domain σ-algebra, we identify metatimes with bounded linear operators between spaces of square integrable functions. We also analyse the possibility to define a metatime from a given bounded linear operator between Hilbert spaces, which we show is possible for invertible operators. Next we establish a link between orthogonal random measures and cylindrical random variables following a classical construction. This enables us to view metatime-changed orthogonal random measures as cylindrical random variables composed with linear operators, where the linear operators are induced by metatimes. In the paper we also provide several results on the basic properties of metatimes as well as some applications towards trawl processes.

МЕТОД ПОСТРОЕНИЯ НЕЛИНЕЙНОГО ВЕЙВЛЕТНОГО КОДА ДЛЯ ОБЕСПЕЧЕНИЯ ЦЕЛОСТНОСТИ ДАННЫХ В КАНАЛАХ СВЯЗИ

A method for constructing a nonlinear wavelet code (NVC) to ensure data integrity in communication channels, taking into account current threats to information security in a modern dynamic stochastic environment, is proposed. A special place among the methods of combating threats to the integrity of information is occupied by noise-resistant encoding. The article presents a computationally effective method for ensuring data integrity in communication channels by using nonlinear transformations and wavelets. The approximation of the wavelet transform refers to the division of the signal into approximating and detailing components. Continuous and discrete wavelet transforms are widely used [2] for signal analysis in modern communication channels. The set of functions defining the wavelet transform belongs to the space of square-integrable functions on a straight line and provides a necessary condition for constructing constructions of nonlinear codes based on the theory of wavelet decomposition. As is known, in the process of wavelet analysis, the signal is decomposed along the orthogonal basis formed by shifts of the wavelet function. A distinctive feature of this approach is that convolution of the signal with wavelets allows us to identify the characteristic features of the signal in the area of localization of these wavelets. To perform computational calculations, you need a set of scaling function coefficients and a wavelet. The wavelet transform matrix depends on the coefficients of the scaling function. The results presented in the article describe a new approach to ensuring data integrity in communication channels using nvcs. A computational example is presented.

The modified fuzzy mortality model based on the algebra of ordered fuzzy numbers.

The new trends in fuzzy analysis are based on the algebraic approach to fuzzy numbers. The essential idea in such an approach is representing the membership function of a fuzzy number as an element of any square-integrable function space. As a starting point, we consider the Koissi-Shapiro model known as a fuzzy version of the Lee-Carter mortality model, in which triangular membership functions of fuzzy variables are assumed. In our approach, the algebra of ordered fuzzy numbers is used to reformulate the fuzzy Lee-Carter mortality model and to improve the prediction accuracy.

Robustness of constant-delay predictor feedback for in-domain stabilization of reaction–diffusion PDEs with time- and spatially-varying input delays

This paper discusses the in-domain feedback stabilization of reaction–diffusion PDEs with Robin boundary conditions in the presence of an uncertain time- and spatially-varying delay in the distributed actuation. The proposed control design strategy consists of a constant-delay predictor feedback designed based on the known nominal value of the control input delay and is synthesized on a finite-dimensional truncated model capturing the unstable modes of the original infinite-dimensional system. By using a small-gain argument, we show that the resulting closed-loop system is exponentially stable provided that the variations of the delay around its nominal value are small enough. The proposed proof actually applies to any distributed-parameter system associated with an unbounded operator that 1) generates a C0-semigroup on a weighted space of square integrable functions over a compact interval; and 2) is self-adjoint with compact resolvent.

Tractability for Volterra problems of the second kind with convolution kernels

We study ε-approximation of the solution of the d-variateVolterra problem of the second kind, the Volterra operator having a convolution kernel. The Volterra operator is defined on an arbitrary normed function space Fd that is continuously embedded in the space of square integrable functions defined on the unit d-cube.Admissible information is given by continuous linear functionals on Fd. These functionals might be arbitrarily chosen; alternatively, we may allow only function values. (In the latter case, we can only consider spaces Fd for which function values are well-defined continuous linear functionals.) The error and cost are measured in the worst case or randomized setting in the L2-norm.Our first result is that lower and upper bounds on the information complexity of the Volterra problem and of d-variate L2-approximation differ by at most a constant factor. Using this result, we then show that necessary and sufficient conditions characterizing a given kind of tractability for multivariate approximation also characterize that same kind of tractability for the Volterra problem. We consider different kinds of algebraic tractabilities (in which we compare the information complexity to d and ε−1) and exponential tractability (in which we compare the information complexity to d and 1+lnε−1).However, our comparison of Volterra to approximation falls short in one respect. Multivariate approximation is linear; for many spaces (e.g., Hilbert spaces), the combinatory cost of multivariate approximation is roughly the same as its information complexity. But since the Volterra problem is nonlinear, it is unclear what the combinatory cost will be for the Volterra problem. This means that we do not know the extent to which the combinatory cost will exceed the information complexity. We partially address this issue by seeing whether the Picard iteration can give us an approximation without too great a penalty when we include the combinatory cost; this penalty is measured by the normalized combinatory cost, defined as the ratio of the combinatory cost to the cost of one admissible operation.In particular, suppose that we agree to compute an ε-approximation to the Volterra problem by (randomized) Monte Carlo. Suppose further that the convolution kernels are uniformly bounded in the L∞-norm. We then obtain an upper bound on the normalized combinatory cost of the Monte Carlo algorithm in the randomized setting. This upper bound is larger than the information complexity by roughly a factor of ε−2. This factor does not change positive results for algebraic tractability, but it does affect some of the positive results for exponential tractability.We also describe a deterministic algorithm that implements the Picard iteration. Assume that the approximation problem can be solved by linear algorithms that lie in a space of dimension proportional to the information complexity. We then find that the combinatory cost of the Picard iteration in the worst case setting is at most a power of the information complexity of d-variate approximation. This power is a constant if d−1lnε−1 is uniformly bounded. However, if d−1lnε−1 goes to infinity, then this power is of order (d−1lnε−1)∕ln(d−1lnε−1). Hence if d is large relative to lnε−1, this result is quite positive. On the other hand, for general d and ε, only some kinds of algebraic and exponential tractabilities hold when the normalized combinatory cost is included.

On Multiple Completeness of Root Functions for Ordinary Differential Polynomial Pencils with Constant Coefficients

In the space of square integrable functions on a finite segment, we consider a class of polynomial pencils of nth-order ordinary differential operators with constant coefficients and two-point boundary conditions (at the edges of the segment). We suppose that all roots of the characteristic equations of pencils of the said class are simple and nonzero. We find sufficient conditions for the m-multiple completeness (1 ≤ m ≤ n) of the system of root functions of pencils from the specified class in the space of square integrable functions on the said segment.

Weighted L^2 Version of Mergelyan and Carleman Approximation

We study the density of polynomials in H^2(E,varphi ), the space of square integrable functions with respect to mathrm{e}^{-varphi }mathrm{d}m and holomorphic on the interior of E in {mathbb {C}}, where varphi is a subharmonic function and dm is a measure on E. We give a result where E is the union of a Lipschitz graph and a Carathéodory domain, which we state as a weighted L^2-version of the Mergelyan theorem. We also prove a weighted L^2-version of the Carleman theorem.

Extremal Shift in a Problem of Tracking a Solution of an Operator Differential Equation

A control problem for an operator differential equation in a Hilbert space is considered. The problem consists in constructing an algorithm generating a feedback control and guaranteeing that the solution of the equation follows a solution of another equation, which is subject to an unknown disturbance. We assume that both equations are given on an infinite time interval and the unknown disturbance is an element of the space of square integrable functions; i.e., the perturbation may be unbounded. We construct two algorithms based on elements of the theory of ill-posed problems and the extremal shift method known in the theory of positional differential games. The algorithms are stable with respect to information noises and calculation errors. The first and second algorithms can be used in the cases of continuous and discrete measurement of solutions, respectively.

Poisson stochastic master equation unravelings and the measurement problem: A quantum stochastic calculus perspective

This paper studies a class of quantum stochastic differential equations, modeling an interaction of a system with its environment in the quantum noise approximation. The space representing quantum noise is the symmetric Fock space over L2R+. Using the isomorphism of this space with the space of square-integrable functionals of the Poisson process, the equations can be represented as classical stochastic differential equations, driven by Poisson processes. This leads to a discontinuous dynamical state reduction which we compare to the Ghirardi-–Rimini–Weber model. A purely quantum object, the norm process, is found, which plays the role of an observer {in the sense of Everett [H. Everett III, Rev. Mod. Phys. 29(3), 454 (1957)]}, encoding all events occurring in the system space. An algorithm introduced by Dalibard et al. [Phys. Rev. Lett. 68(5), 580 (1992)] to numerically solve quantum master equations is interpreted in the context of unraveling, and the trajectories of expected values of system observables are calculated.

On Regularity of the Logarithmic Forward Map of Electrical Impedance Tomography

This work considers properties of the logarithm of the Neumann-to-Dirichlet boundary map for the conductivity equation in a Lipschitz domain. It is shown that the mapping from the (logarithm of) the conductivity, i.e. the (logarithm of) the coefficient in the divergence term of the studied elliptic partial differential equation, to the logarithm of the Neumann-to-Dirichlet map is continuously Fr\'echet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Fr\'echet derivative defines a bounded linear operator on the space of square integrable functions living on the domain boundary, although the logarithm of the Neumann-to-Dirichlet map itself is unbounded in that topology. In particular, it follows from the fundamental theorem of calculus that the difference between the logarithms of any two Neumann-to-Dirichlet maps is always bounded on the space of square integrable functions. All aforementioned results also hold if the Neumann-to-Dirichlet boundary map is replaced by its inverse, i.e. the Dirichlet-to-Neumann map.

К вопросу о регуляризации классических условий оптимальности в выпуклой задаче оптимального управления c фазовыми ограничениями

We consider the regularization of classical optimality conditions in a convex optimal control problem for a linear system of ordinary differential equations with pointwise state constraints such as equality and inequality, understood as constraints in the Hilbert space of square-integrable functions. The set of admissible task controls is traditionally embedded in the space of square-integrable functions. However, the target functional of the optimization problem is not, generally speaking, strongly convex. Obtaining regularized classical optimality conditions is based on a technique involving the use of two regularization parameters. One of them is used for the regularization of the dual problem, while the other is contained in a strongly convex regularizing addition to the target functional of the original problem. The main purpose of the obtained regularized Lagrange principle and the Pontryagin maximum principle is the stable generation of minimizing approximate solutions in the sense of J. Varga for the purpose of practical solving the considered optimal control problem with pointwise state constraints.

On the convergence of stationary solutions in the Smoluchowski-Kramers approximation of infinite dimensional systems

We prove the convergence, in the small mass limit, of statistically invariant states for a class of semi-linear damped wave equations, perturbed by an additive Gaussian noise, both with Lipschitz-continuous and with polynomial non-linearities. In particular, we prove that the first marginals of any sequence of invariant measures for the stochastic wave equation converge in a suitable Wasserstein metric to the unique invariant measure of the limiting stochastic semi-linear parabolic equation obtained in the Smoluchowski-Kramers approximation. The Wasserstein metric is associated to a suitable distance on the space of square integrable functions, that is chosen in such a way that the dynamics of the limiting stochastic parabolic equation is contractive with respect to such a Wasserstein metric. This implies that the limiting result is a consequence of the validity of a generalized Smoluchowski-Kramers limit at fixed times. The proof of such a generalized limit requires new delicate bounds for the solutions of the stochastic wave equation, that must be uniform with respect to the size of the mass.