Space Of Functions Research Articles

Partial Smoothing of the Stochastic Wave Equation and Regularization by Noise Phenomena

AbstractWe establish partial smoothing properties of the transition semigroup $$(P_t)$$ ( P t ) associated to the linear stochastic wave equation driven by a cylindrical Wiener noise on a separable Hilbert space. These new results allow the study of related vector-valued infinite-dimensional PDEs in spaces of functions which are Hölder continuous along special directions. As an application we prove strong uniqueness for semilinear stochastic wave equations involving nonlinearities of Hölder type. We stress that we are able to prove well-posedness although the Markov semigroup $$(P_t)$$ ( P t ) is not strong Feller.

Characterization of continuous homomorphisms on entire slice monogenic functions

Abstract This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.

The continuous quaternion wavelet transform on function spaces

In this paper, boundedness results for the continuous quaternion wavelet transform on Besov, $BMO$ and Hardy $H^{p}$ spaces are established. Furthermore, the continuous quaternion wavelet transform is also studied on the weighted Besov, $BMO_k$ and $H^{p}_{k}$ spaces associated with a tempered weighted function.

Boundedness and compactness of Hausdorff operators on Fock spaces

We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space F α p F^p_\alpha and taking its values into a larger one F α q , 0 > p ≤ q ≤ ∞ F^q_\alpha ,\ 0 > p \leq q \leq \infty , as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. Some results are written in the context of mixed norm Fock spaces. Also the compactness of Hausdorff operators on a Fock space is characterized. The compactness result for Hausdorff operators on the Fock space F α ∞ F^\infty _\alpha is extended to more general Banach spaces of entire functions with weighted sup norms defined in terms of a radial weight and conditions for the Hausdorff operators to become p p -summing are also included.

Recombination of stable sampling sets and stable interpolation sets in functional quasinormed spaces

This contribution is aimed in obtaining new results in combining stable sampling sets (respectively, stable interpolation sets) of a given quasinormed space in order to obtain other new ones. We apply these results to Paley-Wiener spaces and. In addition, we study the problem of obtaining a generator system of a given quasinormed space, and obtain conditions for a finite product of subsets of a given quasinormed space to be a generator system, using the interpolation and sampling theory for quasinormed spaces of functions.

On the smoothness of slowly varying functions

Abstract In this paper, we consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show that every slowly varying function of this type is equivalent to a slowly varying function that has continuous classical derivatives of all orders.

Three-Body 3D-Kepler Electromagnetic Problem—Existence of Periodic Solutions

The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence–uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld’s hypothesis for the He atom.

Some measure-theoretic aspects of fractal functions: invariant measures and function spaces

Some measure-theoretic aspects of fractal functions: invariant measures and function spaces

Regularization of Solutions of One Class of Systems of Linear Volterra Integral Equations of the First Kind in the Space of Differentiable Vector Functions on the Axis

Regularizing operators according to Lavrentiev are constructed and sufficient conditions for the uniqueness of solutions to systems of linear Volterra integral equations of the first kind on the axis are established.

Nonuniform Sampling in Lp-Subspaces Associated with the Multi-Dimensional Special Affine Fourier Transform

In this paper, the sampling and reconstruction problems in function subspaces of Lp(Rn) associated with the multi-dimensional special affine Fourier transform (SAFT) are discussed. First, we give the definition of the multi-dimensional SAFT and study its properties including the Parseval’s relation, the canonical convolution theorems and the chirp-modulation periodicity. Then, a kind of function spaces are defined by the canonical convolution in the multi-dimensional SAFT domain, the existence and the properties of the dual basis functions are demonstrated, and the Lp-stability of the basis functions is established. Finally, based on the nonuniform samples taken on a dense set, we propose an iterative reconstruction algorithm with exponential convergence to recover the signals in a Lp-subspace associated with the multi-dimensional SAFT, and the validity of the algorithm is demonstrated via simulations.

On the Properties of Quasi-Banach Function Spaces

In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they possess a generalised version of Riesz–Fischer property, that embeddings between them are always continuous, and that the dilation operator is bounded on them. We also provide a characterisation of separability for quasi-Banach function spaces over the Euclidean space. Furthermore, we extend the classical Riesz–Fischer theorem to the context of quasinormed spaces and, as a consequence, obtain an alternative proof of completeness of quasi-Banach function spaces.

Equilibrium States of Endomorphisms of $\mathbb{P}^{k}$: Spectral Stability and Limit Theorems

AbstractWe establish the existence of a spectral gap for the transfer operator induced on $\mathbb{P}^{k} = \mathbb{P}^{k} (\mathbb{C})$ by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: CLT, Berry-Esseen Theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).

Completeness of the systems of Bessel functions of index $-5/2$

Let $L^2((0;1);x^4 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)\rightarrow\mathbb C$, satisfying $\int_{0}^1 t^4 |f(t)|^2\, dt<+\infty$. Let $J_{-5/2}$ be the Bessel function of the first kind of index $-5/2$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system $\big\{\rho_k^2\sqrt{x\rho_k}J_{-5/2}(x\rho_k):k\in\mathbb N\big\}$ in the space $L^2((0;1);x^4 dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$. In this case, we study an integral representation of some class $E_{4,+}$ of even entire functions of exponential type $\sigma\le 1$. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than $-1$, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.

Relativity of Reductive Chain Complexes of Non-abelian Simplexes

Chain total double complexes with reductive differentials for non-abelian simplexes with associated spaces are considered. It is conjectured that corresponding relative cohomology is equivalent to the coset space of vanishing functionals over non-vanishing functionals related to differentials of complexes. The conjecture is supported by the theorem for the case of spaces of correlation functions and generalized connections on vertex operator algebra bundles.

Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis

We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication $-$ a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication.

The uniqueness and existence of solutions in a new complex function space for Kannan nonlinear dynamical systems

The uniqueness and existence of solutions in a new complex function space for Kannan nonlinear dynamical systems

Growth of hypercyclic functions: a continuous path between -frequent hypercyclicity and hypercyclicity

Abstract We are interested in the optimal growth in terms of Lp-averages of hypercyclic and $\mathcal{U}$ -frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between $\mathcal{U}$ -frequent hypercyclicity and hypercyclicity.

Through their eyes: Multi-subject brain decoding with simple alignment techniques

Abstract To-date, brain decoding literature has focused on single-subject studies, that is, reconstructing stimuli presented to a subject under fMRI acquisition from the fMRI activity of the same subject. The objective of this study is to introduce a generalization technique that enables the decoding of a subject’s brain based on fMRI activity of another subject, that is, cross-subject brain decoding. To this end, we also explore cross-subject data alignment techniques. Data alignment is the attempt to register different subjects in a common anatomical or functional space for further and more general analysis. We utilized the Natural Scenes Dataset, a comprehensive 7T fMRI experiment focused on vision of natural images. The dataset contains fMRI data from multiple subjects exposed to 9,841 images, where 982 images have been viewed by all subjects. Our method involved training a decoding model on one subject’s data, aligning new data from other subjects to this space, and testing the decoding on the second subject based on information aligned to the first subject. We also compared different techniques for fMRI data alignment, specifically ridge regression, hyper alignment, and anatomical alignment. We found that cross-subject brain decoding is possible, even with a small subset of the dataset, specifically, using the common data, which are around 10% of the total data, namely 982 images, with performances in decoding comparable to the ones achieved by single-subject decoding. Cross-subject decoding is still feasible using half or a quarter of this number of images with slightly lower performances. Ridge regression emerged as the best method for functional alignment in fine-grained information decoding, outperforming all other techniques. By aligning multiple subjects, we achieved high-quality brain decoding and a potential reduction in scan time by 90%. This substantial decrease in scan time could open up unprecedented opportunities for more efficient experiment execution and further advancements in the field, which commonly requires prohibitive (20 hours) scan time per subject.

A two-weight boundedness criterion and its applications

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, [Formula: see text], on [Formula: see text] in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley [Formula: see text]-functions, Lusin area functions, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on [Formula: see text] in the aforementioned function spaces.

Design logic of post-industrial landscape structure based on Metaphor-Deconstruction-Empowerment: A case study of the Truss Red Gallery in Hebei

Design logic of post-industrial landscape structure based on Metaphor-Deconstruction-Empowerment: A case study of the Truss Red Gallery in Hebei