اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAn Interpolatory Estimate for the UMD-Valued Directional Haar Projection
75
0
0.0
(
0
)
تأليف
R. Lechner
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We establish an vector-valued interpolatory estimate between directional Haar projections and Riesz transforms.
قيم البحث
اقرأ أيضاً
The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting `a la Hedberg$&$Netrusov, which includes weighted anisotropic mixed-norm Besov and Lizorkin-Triebel space
s. In the special case of the classical Lizorkin-Triebel spaces, the intersection representation gives an improvement of the well-known Fubini property. The main result has applications in the weighted $L_{q}$-$L_{p}$-maximal regularity problem for parabolic boundary value problems, where weighted anisotropic mixed-norm Lizorkin-Triebel spaces occur as spaces of boundary data.
This paper presents a model reduction method for the class of linear quantum stochastic systems often encountered in quantum optics and their related fields. The approach is proposed on the basis of an interpolatory projection ensuring that specific
input-output responses of the original and the reduced-order systems are matched at multiple selected points (or frequencies). Importantly, the physical realizability property of the original quantum system imposed by the law of quantum mechanics is preserved under our tangential interpolatory projection. An error bound is established for the proposed model reduction method and an avenue to select interpolation points is proposed. A passivity preserving model reduction method is also presented. Examples of both active and passive systems are provided to illustrate the merits of our proposed approach.
We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function $sigma : mathbb{C} to mathbb{C}$ in which ea
ch neuron performs the operation $mathbb{C}^N to mathbb{C}, z mapsto sigma(b + w^T z)$ with weights $w in mathbb{C}^N$ and a bias $b in mathbb{C}$, and with $sigma$ applied componentwise. We completely characterize those activation functions $sigma$ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of $mathbb{C}^d$ arbitrarily well. Unlike the classical case of real networks, the set of good activation functions which give rise to networks with the universal approximation property differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as $sigma$ is neither a polynomial, a holomorphic function, or an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of $sigma$ is not a polyharmonic function.
The like-Lebesgue integral of real-valued measurable functions (abbreviated as textit{RVM-MI})is the most complete and appropriate integration Theory. Integrals are also defined in abstract spaces since Pettis (1938). In particular, Bochner integrals
received much interest with very recent researches. It is very commode to use the textit{RVM-MI} in constructing Bochner integral in Banach or in locally convex spaces. In this simple not, we prove that the Bochner integral and the textit{RVM-MI} with respect to a finite measure $m$ are the same on $mathbb{R}$. Applications of that equality may be useful in weak limits on Banach space.
We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends
to higher multiplicity (e.g., multiframes) their dilation approach. We prove several results for operator-valued frames concerning their parametrization, duality, disjointeness, complementarity, and composition and the relationship between the two types of similarity (left and right) of such frames. We then apply these notions to prove that the collection of multiframe generators for the action of a discrete group on a Hilbert space is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. The proof is obtained by parametrizing this collection by a class of partial isometries in a larger von Neumann algebra. In the multiplicity one case this class reduces to the unitary class which is path-connected in norm, but in the infinite multiplicity case this class is path connected only in the strong operator topology and the proof depends on properties of tensor product slice maps.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
