Analysis and Mathematical Physics最新文献

We consider the class of standard weighted Bergman spaces (A^2_{alpha }(mathbb {D})) and the set (SF^N(mathbb {T})) of simple partial fractions of degree N with poles on the unit circle. We prove that under certain conditions, the simple partial fractions of order N, with n poles on the unit circle attain minimal norm if and only if the points are equidistributed on the unit circle. We show that this is not the case if the conditions we impose are not met, exhibiting a new interesting phenomenon. We find sharp asymptotics for these norms. Additionally we describe the closure of these fractions in the standard weighted Bergman spaces.

This paper establishes an extended version of Heisenberg’s uncertainty principle for the Symplectic Wigner distribution via linear canonical transform (SWL), which generalizes existing Symplectic Wigner distributions. Furthermore, the properties of SWL are enumerated, and a comprehensive analysis of its Heisenberg uncertainty relation and special cases is fully elucidated. Finally, a numerical example is presented to demonstrate the efficacy of this novel distribution in detecting single-component linear frequency modulated (LFM) signals.

We construct an analogue of Dyson Brownian motion in the Siegel half-space (mathcal {H}) that we term Siegel Brownian motion. Given (beta in (0,infty ]), a stochastic flow for (Z_tin mathcal {H}) is introduced so that the law of the eigenvalues (lambda _t) of the cross ratio matrix ({mathfrak {R}}(Z_t,varvec{i}I_n)) is determined, after a change of variables to (sigma (lambda ) in (0,infty )^n), by the Itô differential equation

where (W_t) is a standard Wiener process in (mathbb {R}^n). This interacting particle system corresponds to stochastic gradient ascent

where (S(sigma )= log textrm{vol},mathcal {O}_{lambda (sigma )}) is a Boltzmann entropy that enumerates the microstates in the group orbit (mathcal {O}_lambda = {Z in mathcal {H}left| textrm{eig}left( {mathfrak {R}}(Z,varvec{i}I_n)right) =lambda right. }). In the limit (beta =infty ), the group orbits (mathcal {O}_{lambda _t}) evolve by motion by minus a half times mean curvature.

For the Reshetnyak-class homeomorphisms (varphi :Omega rightarrow Y), where (Omega ) is a domain in some Carnot group and Y is a metric space, we obtain an equivalent description as the homeomorphisms which induce the bounded composition operator

where (1le qle infty ), as (varphi ^*u=ucirc varphi ) for (uin textrm{Lip}(Y)). We demonstrate the utility of our approach by characterizing the homeomorphisms (varphi :Omega rightarrow Omega ') of domains in some Carnot group ({mathbb {G}}) which induce the bounded composition operator

on homogeneous Sobolev spaces. The new proof of this known criterion is much shorter than the one already available, requires a minimum of tools, and enables us to obtain new properties of the homeomorphisms in question.

It is known that the Funk transform (FT) is invertible in the class of even (symmetric) continuous functions defined on the unit 2-sphere ({textbf{S}}^{2}). In this article, for the reconstruction of (fin { {mathcal {C}}}^{1}({textbf{S}}^{2})) (can be non-even), an additional condition is found, which is a weighted Funk transform (to reconstruct an odd function), and the injectivity of the so-called two data Funk transform is considered. The transform consists of the classical FT and the weighted FT. An iterative inversion formula of the transform is presented. Such inversions have theoretical significance in convexity theory, integral geometry and spherical tomography.

In this article we investigate the property of complete monotonicity within a special family (mathcal {F}_s) of functions in s variables involving logarithms. The main result of this work provides a linear isomorphism between (mathcal {F}_s) and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in (mathcal {F}_s) is semi-algebraic. This gives a finite time algorithm to decide whether a function in (mathcal {F}_s) is completely monotone.

Let (-Delta _{mathcal {S}}) be the Laplace operator in (mathcal{S} subset mathbb {R}^3) on a waveguide shaped surfaces, i.e., ({mathcal {S}}) is built by translating a closed curve in a constant direction along an unbounded spatial curve. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of (-Delta _{mathcal {S}}) and discuss conditions under which discrete eigenvalues emerge. Furthermore, we analyze the Laplacian in the case of a broken sheared waveguide shaped surface.

This paper studies the degenerate kernel approximation theory for the second-kind Fredholm integral equations in (L^p) spaces ((1 leqslant p leqslant infty )). By introducing the mixed norm of kernel functions to construct an adaptive analytical tool, the research framework is extended from the classical (L^2) space to the more general (L^p) spaces. Based on the degenerate kernel approximation method, through refined norm estimation of iterated kernels and convergence analysis of the resolvent kernel series, the resolvent kernel representation theory for the solution of integral equations is established. This weakens the strong constraints on the growth of kernel functions and improves the applicability of the theory to non-compact intervals and weakly decaying kernel scenarios. On this basis, two types of error estimations are proposed: dual resolvent kernel error estimate and single resolvent kernel error estimate, which clearly characterize the error relationship between the approximate solution and the exact solution. This research provides a unified framework for the analysis of solutions to integral equations with different regularity characteristics and improves the system of integral equation approximation theory. Furthermore, the degenerate kernel approximation theory and error estimation results established in this paper can be directly extended to integral equations in high-dimensional Euclidean spaces, and their analytical framework and conclusions remain valid in high-dimensional cases.