Riesz Theorem Research Articles

On the Cauchy transform vanishing outside a compact

Motivated by a problem in holomorphic dynamics, we present a certain generalization of the celebrated F. and M. Riesz theorem.

An analogue of the Riesz theorem in Hardy-Morrey classes

An analogue of the Riesz theorem in Hardy-Morrey classes

The Cyclical Compactness in Banach C∞(Q)-Modules

In this paper, we study the class of laterally complete commutative unital regular algebras A over arbitrary fields. We introduce a notion of passport Γ(X) for a faithful regular laterally complete A- modules X, which consist of uniquely defined partition of unity in the Boolean algebra of all idempotents in A and of the set of pairwise different cardinal numbers. We prove that A-modules X and Y are isomorphic if and only if Γ(X)= Γ(Y ). Further we study Banach A-modules in the case A = C∞(Q) or A = C∞(Q)+ i · C∞(Q). We establish the equivalence of all norms in a finite-dimensional (respectively, σ-finite-dimensional) A-module and prove an A-version of Riesz Theorem, which gives the criterion of a finite-dimensionality (respectively, σ-finite-dimensionality) of a Banach A-module.

Extensions of the Lax–Milgram theorem to Hilbert $$C^*$$-modules

We present three versions of the Lax-Milgram theorem in the framework of Hilbert C*-modules, two for those over W*-algebras and one for those over C*-algebras of compact operators. It is remarkable that while the Riesz theorem is not valid for certain Hilbert C*-modules over C*-algebras of compact operators, our Lax-Milgram theorem turns out to be valid for all of them. We also give several examples to illustrate our results, in particular, we show that the main theorem is not true for Hilbert modules over arbitrary C*-algebras.

On the geometrically nonlinear analysis of sandwich shells with viscoelastic core: A layerwise dynamic finite element formulation

The objective of this work is to present a finite element formulation for dynamic analysis of sandwich shells with viscoelastic core under large deformation. The present study is based on an incremental updated Lagrangian approach together with the Newmark integration scheme. The viscoelastic constitutive model which is used to define the behavior of the core, comes from the Riesz theorem and the corresponding creep functions are estimated using Dirichlet-Prony series. Also, the viscoelastic deferred strain is derived in an appropriate incremental form using the state variables. The employed layerwise shell element which is based on zig-zag theory has eight nodes on its mid layer. What’s more, in addition to three translational and two rotational degrees of freedom per node, the upper and lower layers are allowed to rotate independently relative to their neighbor layers. Thus, the damping effect of the viscoelastic core could be well described within the shear deformable displacement field. The presented nonlinear formulation is then implemented to a nonlinear finite element program to be appraised by solving different kinds of problems. The obtained numerical results correlate well with those available in the literature.

The Kolmogorov–Riesz theorem and some compactness criterions of bounded subsets in weighted variable exponent amalgam and Sobolev spaces

We study totally bounded subsets in weighted variable exponent amalgam and Sobolev spaces. Moreover, this paper includes several detailed generalized results of some compactness criterions in these spaces.

Fatou and brothers Riesz theorems in the infinite-dimensional polydisc

We study the boundary behavior of functions in the Hardy spaces on the infinite-dimensional polydisc. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz- Zygmund type theorems for radial convergence of functions with Fourier spectrum supported on $$\mathbb{N}_0^\infty\cup(-\mathbb{N}_0^\infty)$$ . As a consequence one obtains easy new proofs of the brothers F. and M. Riesz Theorems in infinite dimensions, as well as being able to extend a result of Rudin concerning which functions are equal to the modulus of an H1 function almost everywhere to $$\mathbb{T}^\infty$$ . Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.

The Boundary Behavior of Self-Conjugate Differential Forms in $$C^1$$ C 1 -domains

This paper deals with the boundary behavior of self-conjugate differential forms. In particular, we state and prove trace theorems and Plemelj-type formulas in $$C^1$$ -domains; some remarkable consequences of these results are discussed too. As a further goal, we obtain boundary inequalities extending the classical M. Riesz theorem for conjugate functions.

Positive semidefinite univariate matrix polynomials

We study sum-of-squares representations of symmetric univariate real matrix polynomials that are positive semidefinite along the real line. We give a new proof of the fact that every positive semidefinite univariate matrix polynomial of size $$n\times n$$ can be written as a sum of squares $$M=Q^TQ$$ , where Q has size $$(n+1)\times n$$ , which was recently proved by Blekherman–Plaumann–Sinn–Vinzant. Our new approach using the theory of quadratic forms allows us to prove the conjecture made by these authors that these minimal representations $$M=Q^TQ$$ are generically in one-to-one correspondence with the representations of the nonnegative univariate polynomial $$\det (M)$$ as sums of two squares. In parallel, we will use our methods to prove the more elementary hermitian analogue that every hermitian univariate matrix polynomial M that is positive semidefinite along the real line, is a square, which is known as the matrix Fejer–Riesz Theorem.

Classes of analytic functions related to Blaschke products

Basing on the well known Riesz Theorem on the canonical factorization of bounded analytic functions, we distinguished the subfamilies of the Carath?odory class of functions. The basic properties of introduced classes have been proved. Related classes of analytic functions are discussed also.

Hardy spaces of vector-valued Dirichlet series

Given a Banach space $X$ and $1 \leq p \leq \infty$, it is well known that the two Hardy spaces $H_p(\mathbb{T},X)$ ($\mathbb{T}$ the torus) and $H_p(\mathbb{D},X)$ ($\mathbb{D}$ the disk) have to be distinguished carefully. This motivates us to define and study two different types of Hardy spaces $\mathcal{H}_p(X)$ and $\mathcal{H}^+_p(X)$ of Dirichlet series $\sum_n a_n n^{-s}$ with coefficients in $X$. We characterize them in terms of summing operators as well as holomorphic functions in infinitely many variables, and prove that they coincide whenever $X$ has the analytic Radon-Nikodým Property. Consequences are, among others, a vector-valued version of the Brother's Riesz Theorem in the infinite-dimensional torus, and an answer to the question when $\mathcal{H}_1(X^{\ast})$ is a dual space.

F. Riesz Theorem

Summary In this article, we formalize in the Mizar system [1, 4] the F. Riesz theorem. In the first section, we defined Mizar functor ClstoCmp, compact topological spaces as closed interval subset of real numbers. Then using the former definition and referring to the article [10] and the article [5], we defined the normed spaces of continuous functions on closed interval subset of real numbers, and defined the normed spaces of bounded functions on closed interval subset of real numbers. We also proved some related properties. In Sec.2, we proved some lemmas for the proof of F. Riesz theorem. In Sec.3, we proved F. Riesz theorem, about the dual space of the space of continuous functions on closed interval subset of real numbers, finally. We applied Hahn-Banach theorem (36) in [7], to the proof of the last theorem. For the description of theorems of this section, we also referred to the article [8] and the article [6]. These formalizations are based on [2], [3], [9], and [11].

Rényi and Tsallis entropies related to eigenfunctions of quantum graphs

For certain families of finite quantum graphs, we study the question of how eigenfunctions are distributed over the graph. To characterize properties of the distribution, generalized entropies of the Rényi and Tsallis types are considered. The presented approach is similar to entropic uncertainty relations of the Maassen–Uffink type. Using the Riesz theorem, we derive lower bounds on symmetrized generalized entropies of eigenfunctions. A quality of such estimates will depend on boundary conditions used at vertices of the given graph. Rényi and Tsallis entropies of eigenfunctions of star graphs are separately examined. Relations between generalized entropies and variances of eigenfunctions are considered as well. When such relations remain valid on average, they may be used in studies of quantum ergodicity.

Existence of undercompressive weak solutions to the Euler equations

AbstractWe prove the local existence of an undercompressive hydrodynamical shock to the isothermal Euler equations with a non‐monotone pressure function. This nonlinear problem will be formulated as an abstract hyperbolic initial boundary value problem. The existence of a weak solution to a linearized version of the problem is shown with the use of Riesz theorem. Using the results of the linear system yields by an iteration scheme (local in time) well‐posedness of the nonlinear problem. The system of equations is obtained by modeling the motion of sharp liquid‐vapor interfaces including configurational forces as well as surface tension. The considered non‐viscous Van der Waals fluid is compressible and allows phase transitions. The propagating phase boundary is controlled by a modified version of the Rankine‐Hugoniot jump condition obtained by the Young‐Laplace law. Entropy dissipation at the interface is precisely described by a kinetic relation. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $\Omega:= \mathbb{R}^{n+1}\setminus E$ satisfy Carleson measure estimates, and are "$\varepsilon$-approximable". Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure, and surface measure.

Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets

This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set DL, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) L(X)=A0⊗I+∑j=1gAj⊗Xj⪰0, where Aj are self-adjoint linear operators on a separable Hilbert space, Xj matrices and I is an identity matrix. If Aj are matrices, then L(X)⪰0 is called a linear matrix inequality (LMI) and DL a free spectrahedron. For monic LMIs, i.e., A0=I, and nc matrix-valued polynomials the certificates of positivity were established by Helton, Klep and McCullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry. Since the full strength of the theory of complete positivity is not restricted to finite dimensions, but works well also in the infinite-dimensional setting, we use it to tackle our problems. First we extend the characterization of the inclusion DL1⊆DL2 from monic LMIs to monic LOIsL1 and L2. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron DL and its projection, called a free Hilbert spectrahedrop. Further on, using this characterization in a separation argument, we obtain a certificate for multivariate matrix-valued nc polynomials F positive semidefinite on a free Hilbert spectrahedron defined by a monic LOI. Replacing the separation argument by an operator Fejér–Riesz theorem enables us to extend this certificate, in the univariate case, to operator-valued polynomials F. Finally, focusing on the algebraic description of the equality DL1=DL2, we remove the assumption of boundedness from the description in the LMIs case by an extended analysis. However, the description does not extend to LOIs case by counterexamples.

Singular Sets for Harmonic Measure on Locally Flat Domains with Locally Finite Surface Measure

A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each $d>1$, there exists a Reifenberg flat domain $\Omega\subset \mathbb{R}^{d+1}$ with $\mathcal{H}^{d}(\partial\Omega)<\infty$ and a subset $E\subset \partial \Omega$ with positive harmonic measure yet zero $\mathcal{H}^{d}$-measure. In particular, this implies that a classical theorem of F. and M. Riesz theorem fails in higher dimensions for this type of domains.

Time-Dependent Response of Mass Continuous Solid Phase Media by Integral Form of Constitutive Equations - Mathematical Modeling

Mathematical modeling of boundary value problems in linear theory of viscoelasticity. Definitions and basic principles in the mathematical modeling theory. Constitutive functional and its transformation into a form of Stieltjes integral. Application of theory of algebraic sets and corresponding subsets. Riesz theorem of representation and its application for derivation of constitutive equations. Integral operator forms of stress-strain relationships for a solid-phase continuous media.

On n-collinear elements and Riesz theorem

On n-collinear elements and Riesz theorem

M. Riesz Theorem on Conjugate Harmonic Functions for Octonion-Valued Monogenic Functions

The classical theorem of M. Riesz about the conjugate harmonic functions is extended onto octonion-valued monogenic functions.