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 $σ: \mathbb{C} \to \mathbb{C}$ in which each neuron performs the operation $\mathbb{C}^N \to \mathbb{C}, z \mapsto σ(b + w^T z)$ with weights $w \in \mathbb{C}^N$ and a bias $b \in \mathbb{C}$, and with $σ$ applied componentwise. We completely characterize those activation functions $σ$ 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 $σ$ 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 $σ$ is not a polyharmonic function.
Universal approximation with complex-valued deep narrow neural networks
Mathematical Institute for Machine Learning and Data Science (MIDS), Catholic University of Eichstätt–Ingolstadt (KU)（艾希施泰特-英戈尔施塔特天主教大学 机器学习与数据科学数学研究所）
We present a characterization for a positive definite operator valued kernel to be universal or $C_{0}$-universal, and apply these characterizations to a family of operator valued kernels that are shown to be well behaved. Later, we obtain a characterization for an operator valued differentiable kernel to be $C^{q}$-universal and $C_{0}^{q}$-universal. In order to obtain such characterization and examples we generalize some well known results concerning the structure of differentiable kernels to the operator valued context. On the examples is given an emphasis on the radial kernels on Euclidea…
We prove martingale-ergodic and ergodic-martingale theorems for vector valued Bochner integrable functions. We obtain dominant and maximal inequalities. We also prove weighted and multiparameter martingale-ergodic and ergodic martingale theorems.
We give two different definitions of what it means for a matrix-valued function to be log concave, guided by similar notions in complex differential geometry. After discussing a few simple examples, we proceed to develop some of the basic properties associated with these new concepts. Finally, we prove a matrix-valued Prékopa theorem using a weighted, vector-valued Paley-Wiener theorem, and positivity properties of direct image bundles.
We prove a Montel theorem for Hilbert space valued functions, and a non-commutative version of this theorem, by composing with unitaries to achieve convergence.
In this paper, we study the fixed point theory for multi-valued mappings on partial cone metric spaces. We prove an analogous to the well-known Kannan$'s$ fixed point theorem and Chatterjea$'s$ fixed point theorem for multi-valued mappings on partial cone metric spaces.
We investigate some main properties of lattice group-valued k-triangular set functions and prove some Brooks-Jewett, Nikodym, Vitali-Hahn-Saks and Schur-type theorems and their equivalence. A Drewnowski-type theorem on existence of continuous restrictions of (s)-bounded set functions is given.
The aim of this paper is to discuss Penot's problem on a generalization of Caristi's fixed point theorem. We settle this problem in the negative and we present some new theorems on the existence of fixed points of set-valued mappings in ordered metric spaces.
In this paper, we establish coincidence fixed point and common fixed point theorems for two mapping in complete $C^*$-algebra-valued metric spaces which satisfy new contractive conditions. Some applications of our obtained results are given.
We establish the first common fixed point theorem for commutative set-valued mappings. This may help to generalize common fixed point theorems in single-valued setting to those in set-valued. We also prove the existence of a fixed point in a continuously expanding set under a none convex upper semicontinuous set-valued mapping; as a result we answer positivly to a question of Lau and Yao.
We obtain a fundamental inequality for a contraction with respect to a $C^*$-algebra valued metric space. As an application of this inequality a simple proof is given for the fixed point theorem in $C^*$-algebra valued metric space.
We prove a vector-valued non-homogeneous Tb theorem on certain quasimetric spaces equipped with what we call an upper doubling measure. Essentially, we merge recent techniques from the domain and range side of things, achieving a Tb theorem which is quite general with respect to both of them.
In this paper we discuss on the vector valued version of Caristi's theorem. We show that the regularity of the cone is an essential condition to reach the vector valued version of Caristi's theorem in vector valued metric spaces. It is shown that how the absence of the regularity causes some previous Caristi type theorems and results fail to be correct. Our main result give a vector valued version of Caristi's theorem for correspondences with weakened hypotheses in comparison with previous results and generalize them.
The Egoroff theorem for measurable $\bold X$-valued functions and operator-valued measures $\bold m: Σ\to L(\bold X, \bold Y)$, where $Σ$ is a $σ$-algebra of subsets of $T \neq \emptyset$ and $\bold X$, $\bold Y$ are both locally convex spaces, is proved. The measure is supposed to be atomic and the convergence of functions is net.
In this paper we extend dyadic shifts and the dyadic representation theorem to an operator-valued setting: We first define operator-valued dyadic shifts and prove that they are bounded. We then extend the dyadic representation theorem, which states that every scalar-valued Calderón-Zygmund operator can be represented as a series of dyadic shifts and paraproducts averaged over randomized dyadic systems, to operator-valued Calderón-Zygmund operators. As a corollary, we obtain another proof of the operator-valued, global T(1) theorem. We work in the setting of integral operators that have R-bound…
A Vitali-type theorem for vector lattice-valued modulars with respect to filter convergence is proved. Some applications are given to modular convergence theorems for moment operatorsin the vector lattice setting, and also for the Brownian motion and related stochastic processes.
We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, `vector-valued' means `taking values in a function lattice with the UMD (unconditional martingale differences) property'. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local…
Let X be a UMD space with type t and cotype q, and let T be a Fourier multiplier operator with a scalar-valued symbol m. If the Mihlin multiplier estimate holds for all partial derivatives of m up to the order n/max(t,q')+1, then T is bounded on the X-valued Bochner spaces. For scalar-valued multipliers, this improves the theorem of Girardi and Weis (J. Funct. Anal., 2003) who required similar assumptions for derivatives up to the order n/r+1, where r is a Fourier-type of X. However, the present method does not apply to operator-valued multipliers, which are also covered by the Girardi-Weis th…
We prove a fixed point theorem for closed-graphed, decomposable-valued correspondences whose domain and range is a decomposable set of functions from an atomless measure space to a topological space. One consequence is an improvement of the fixed point theorem in Cellina, Colombo, and Fonda (1986).
In this paper, we equip a C*-algebra-valued b-metric spaces with a graph G = (V,E) and establish some common fixed point theorems. Also, some examples in support of our main results are provided. Finally, as applications, existence and uniqueness of solution for a type of operator equation and an integral equation are discussed.
The entropy numbers of certain finite-dimensional operators acting between vector-valued sequence spaces are estimated, thus providing a generalization of the famous result of Schutt. In addition, two-sided estimates of the entropy numbers of some diagonal operators are obtained.
In this paper, we prove some common coupled fixed point theorems for mappings satisfying different contractive conditions in the context of complete $C^*$-algebra-valued metric spaces. Moreover, the paper provides an application to prove the existence and uniqueness of a solution for Fredholm nonlinear integral equations.
For a non-simply connected domain Omega in C and f a holomorphic function on Omega we prove that f admits one-valued primitives of any order in Omega, if and only if, it extends holomorphically in the simply connected envelope of Omega . This leads to a generalization of the F. and M. Riesz theorem.
We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak*-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a se…
We prove a version of the Lebesgue Differentiation Theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon-Nikodým property.
The paper studies semi-almost periodic holomorphic functions on a polydisk which have, in a sense, the weakest possible discontinuities on the boundary torus. The basic result used in the proofs is an extension of the classical Bohr approximation theorem for almost periodic holomorphic functions on a strip to the case of Banach-valued almost periodic holomorphic functions.
Fixed Point Theorems for Set-Valued Maps with Contractive Orbits
Faculty of Mathematics and Informatics, Sofia University（索非亚大学数学与信息学院）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息研究所）
Some topics concerning the Gould integral are presented here: new results of integrability on finite measure spaces with values in an M-space are given, together with a Radon-Nikodym theorem relative to a Gould-type integral of real functions with respect to a multisubmeasure.
We prove that the operator norm of every Banach space valued Calderon-Zygmund operator T on the weighted Lebesgue-Bochner space depends linearly on the Muckenhoupt A_2 characteristic of the weight. In parallel with the proof of the real-valued case, the proof is based on pointwise dominating every Banach space valued Calderon-Zygmund operator by a series of positive dyadic shifts. In common with the real-valued case, the pointwise dyadic domination relies on Lerner's local oscillation decomposition formula, which we extend from real-valued functions to Banach space valued functions. The extens…
We show a Dvoretsky-Rogers type Theorem for the adapted version of the $q$-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guaranty that the space has to be finite dimensional, contrarily to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, sinc…
In this paper, we obtain a version of Ekeland's variational principle for interval-value functions by means of the Dancs-Hegedus-Medvegyev theorem [14]. We also derive two versions of Ekeland's variational principle involving the generalized Hukuhara Gateaux differentiability of interval-valued functions as well as a version of Ekeland's variational principle for interval-valued bifunctions. Finally, we apply these new versions of Ekeland's variational principle to fixed point theorems, to interval-valued optimization problems, to the interval-valued Mountain Pass Theorem, to noncooperative in…
This is a survey of some recent applications of Boolean valued analysis to operator theory and harmonic analysis. Under consideration are pseudoembedding operators, the noncommutative Wickstead problem, the Radon-Nikodym Theorem for JB-algebras, and the Bochner Theorem for lattice-valued positive definite mappings on locally compact groups.
We introduce the metric space valued in partially ordered groups, and define the convergence of sequences and the multi-valued weak contractions, etc., on the space. We then establish endpoint theorems for the defined maps. Our contributions extend the theory of cone metric space constructed by Huang and Zhang (2007) and some recent results on the fixed point and endpoint theory, such as the endpoint theorem given by Amini-Harandi (2010).
We study the Lusin approximation problem for real-valued measurable functions on Carnot groups. We prove that k-approximate differentiability almost everywhere is equivalent to admitting a Lusin approximation by $C^{k}_{\mathbb{G}}$ maps. We also prove that existence of an approximate (k-1)-Taylor polynomial almost everywhere is equivalent to admitting a Lusin approximation by maps in a suitable Lipschitz function space.
Let $(Ω,{\cal F},P)$ be a probability space and $L^{0}({\cal F},R)$ the algebra of equivalence classes of real-valued random variables on $(Ω,{\cal F},P)$. When $L^{0}({\cal F},R)$ is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from $L^{0}({\cal F},R)$ to $L^{0}({\cal F},R)$. As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module $(S,\|\cdot\|)$ is random uniformly convex iff $L^{p}(S)$ is uniformly convex for…
We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of $f$ is given by the integral transform $M^{f}_φ(x,y)=(f\astφ_{y})(x),$ $(x,y)\in\mathbb{R}^{n}\times\mathbb{R}_{+}$, with kernel $φ_{y}(t)=y^{-n}φ(t/y)$. We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on $\left\{x_0\right\}\t…
We present a natural and simple proof of the Radon - Nikodym theorem for measures with values in the space of bounded linear operators on a separable Hilbert space. This space is not separable, that is why it is essential to assume in the theorem that the range of the measure is separable. We also discuss distinctions between uniform and strong operator measures (Remark 3.3 to Lemma 3.2, Theorem 3.5 and its Corollary 3.6). Proof of this version of the Radon - Nikodym theorem either is not presented in known literature, or given for special cases and with some inaccuracies (see Comment at the e…
This paper deals with subnormality of Toeplitz operators with matrix-valued symbols and, in particular, with an appropriate reformulation of Halmos's Problem 5: Which subnormal Toeplitz operators with matrix-valued symbols are either normal or analytic? In 1976, M. Abrahamse showed that if $φ\in L^\infty$ is such that $φ$ or $\overlineφ$ is of bounded type and if $T_φ$ is subnormal, then $T_φ$ is either normal or analytic. In this paper we establish a matrix-valued version of Abrahamse's Theorem and then apply this result to solve the following Toeplitz completion problem: Find the unspecified…
We define a scalar valued Fourier transform for functions on the Heisenberg group and establish some of its basic properties like inversion formula, Plancherel theorem and Riemann-Lebesgue lemma. We also restate certain well known theorems for the group Fourier transform in terms of the new transform which we would like to call Strichartz Fourier transform.
Suppose $α$ is a rotationally symmetric norm on $L^{\infty}\left(\mathbb{T}\right) $ and $β$ is a "nice" norm on $L^{\infty}\left(Ω,μ\right) $ where $μ$ is a $σ$-finite measure on $Ω$. We prove a version of Beurling's invariant subspace theorem for the space $L^β\left(μ,H^α\right) .$ Our proof uses the recent version of Beurling's theorem on $H^α\left(\mathbb{T}\right) $ proved by the first author and measurable cross-section techniques. Our result significantly extends a result of H. Rezaei, S. Talebzadeh, and D. Y. Shin.
In this paper we study the set of functions $\GG$-valued which can be approximated by $\GG$-valued continuous functions in the norm $L^\infty_{\GG}(I,w)$, where $I$ is a compact interval, $\GG$ is a real and separable Hilbert space and $w$ is certain $\GG$-valued weakly measurable weight. Thus, we obtain a new extension of celebrated Weierstrass approximation theorem.
The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces of functions with values in a Banach space with the unconditionality property of martingale differences (UMD). The new proof deals directly with all Lebesgue exponents p in the range 1<p<infinity, and relies on delicate estimates for the non-h…
We develop a general framework for the analysis of operator-valued multilinear multipliers acting on Banach-valued functions. Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces. A concrete case of our theorem is a multilinear generalization of Weis' operator-valued Hörmander-Mihlin linear multiplier theorem. Furthermore, we derive from our main result a wide range of mixed $L^p$-norm estimates for multi-parameter multilinear paraproducts, leading to a novel mixed norm version of the partial fractional Leibniz rule…
Operations on $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathrm{Alt}^{\bullet}(U,M)$. Stokes theorem for locally convex linear space valued forms. Divergence theorem for locally convex linear space valued vector fields
In this paper, we generalize the $A_\fz$ extrapolation theorem in \cite{cmp} and the $A_p$ extrapolation theorem of Rubio de Francia to Schrödinger settings. In addition, we also establish the weighted vector-valued inequalities for Schrödinger type maximal operators by using weights belonging to $ A_p^{ρ,\tz}$ which includes $A_p$. As their applications, we establish the weighted vector-valued inequalities for some Schödinger type operators and pseudo-differential operators.
The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical ``$T(1)\in BMO$'' assumptions that are not easily checkable. Recently, Hong, Liu and Mei (J. Funct. Anal. 2020) observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical $T(1)$ criterion still guarantees their $L^2$-boundedness on Hilbert space …
In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holo…
