Riesz Theorem Research Articles

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.

On a Theorem of F. Riesz

We prove a theorem of Riesz characterizing the convergence in spaces of functions N ɸ generated by concave functions ɸ.

Matrix Fejér–Riesz theorem with gaps

The matrix Fejér–Riesz theorem characterizes positive semidefinite matrix polynomials on the real line R. We extend a characterization to arbitrary closed semialgebraic sets K⊆R by the use of matrix preorderings from real algebraic geometry. In the compact case a denominator-free characterization exists, while in the non-compact case there are counterexamples. However, there is a weaker characterization with denominators in the non-compact case. At the end we extend the results to algebraic curves.

A Decomposition Method for a Unilateral Contact Problem with Tresca Friction Arising in Electro-elastostatics

This article is concerned with the numerical modeling of unilateral contact problems in an electro-elastic material with Tresca friction law and electrical conductivity condition. First, we prove the existence and uniqueness of the weak solution of the model. Rather than deriving a solution method for the full coupled problem, we present and study a successive iterative (decomposition) method. The idea is to solve successively a displacement subproblem and an electric potential subproblem in block Gauss-Seidel fashion. The displacement subproblem leads to a constraint non-differentiable (convex) minimization problem for which we propose an augmented Lagrangian algorithm. The electric potential unknown is computed explicitly using the Riesz's representation theorem. The convergence of the iterative decomposition method is proved. Some numerical experiments are carried out to illustrate the performances of the proposed algorithm.

On the first degree Fejér–Riesz factorization and its applications to X + A⁎X−1A = Q

Given a Laurent polynomial with matrix coefficients that is positive semi-definite over the unit circle in the complex plane, the Fejér–Riesz theorem asserts that it can always be factorized as the product of a polynomial with matrix coefficients and its adjoint. This paper exploits such a factorization in its simplest form of degree one and its relationship with the nonlinear matrix equation X+A⁎X−1A=Q. In particular, the nonlinear equation can be recast as a linear Sylvester equation subject to unitary constraint. The Sylvester equation is readily obtainable from hermitian eigenvalue computation. The unitary constraint can be enforced by a hybrid of a straightforward alternating projection for low precision estimation and a coordinate-free Newton iteration for high precision calculation. This approach offers a complete parametrization of all solutions and, in contrast to most existent algorithms, makes it possible to find all solutions if so desired.

Uncertainty Relations for General Canonically Conjugate Observables in Terms of Unified Entropies

We study uncertainty relations for a general class of canonically conjugate observables. It is known that such variables can be approached within a limiting procedure of the Pegg–Barnett type. We show that uncertainty relations for conjugate observables in terms of generalized entropies can be obtained on the base of genuine finite-dimensional consideration. Due to the Riesz theorem, there exists an inequality between norm-like functionals of two probability distributions in finite dimensions. Using a limiting procedure of the Pegg–Barnett type, we take the infinite-dimensional limit right for this inequality. Hence, uncertainty relations are derived in terms of generalized entropies. In particular, the case of measurements with a finite precision is addressed. It takes into account that in any experiment an accuracy of performed measurements is always limited.

An inequality of Hardy–Littlewood type for Dirichlet polynomials

The Lq norm of a Dirichlet polynomial F(s)=∑n=1Nann−s is defined as‖F‖q:=(limT→∞⁡1T∫0T|F(it)|qdt)1/q for 0<q<∞. It is shown that(∑n=1N|an|2|μ(n)|[d(n)]log⁡qlog⁡2−1)1/2≤‖F‖q when 0<q<2; here μ is the Möbius function and d the divisor function. This result is used to prove that the Lq norm of DN(s):=∑n=1Nn−1/2−s satisfies ‖DN‖q≫(log⁡N)q/4 for 0<q<∞. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the reverse inequality ‖DN‖q≪(log⁡N)q/4 is shown to be valid in the range 1<q<∞. Similar bounds are found for a fairly large class of Dirichlet series including, on one of Selberg's conjectures, the Selberg class of L-functions.

On the mathematics underlying dispersion relations

The history of mathematical methods underlying the study of dispersion relations in physics is discussed. In particular, some misconceptions connected with a theorem known in the physics literature as Titchmarsh’s Theorem are addressed. It is pointed out that the aforementioned theorem is a compilation of two well-known theorems in mathematics, the Paley-Wiener theorem and the Marcel Riesz theorem.

Noncommutative orlicz-hardy spaces

Let (Φ, Ψ) be a pair of complementary N-functions and HΦ(A) and HΨ(A) be the associated noncommutative Orlicz-Hardy spaces. We extend the Riesz, Szegö and inner-outer type factorization theorems of Hp(A) to this case.

Subharmonicity and a Version of Riesz Theorem on Harmonic Conjugates

The aim of the paper is to prove a monogenic version of classical M. Riesz theorem on harmonic conjugates in the framework of quaternionic analysis in \({\mathbb{R}^{4}}\). Our proof is subharmonic and somewhat simpler than that for less general Riesz-Stein-Weiss systems of harmonic conjugate functions.

The F. and M. Riesz theorem for nonelliptic Vekua’s equations

The F. and M. Riesz theorem for nonelliptic Vekua’s equations

Pseudo-holomorphic functions at the critical exponent

We study Hardy classes on the disk associated to the equation \bar \partial w= \alpha \bar w for \alpha \in L^r with 2 \leq r &lt; \infty . The paper seems to be the first to deal with the case r=2 . We prove an analog of the M. Riesz theorem and a topological converse to the Bers similarity principle. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth domains of the Dirichlet problem with weighted L^p boundary data for 2D isotropic conductivity equations whose coefficients have logarithm in W^{1,2} . In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for W^{1,2}_0 -functions.

БАЗИСНЫЕ ФУНКЦИИ МЕТОДА ДВУСТОРОННИХОЦЕНОК В ЗАДАЧАХ УСТОЙЧИВОСТИ УПРУГИХ НЕОДНОРОДНОСЖАТЫХ СТЕРЖНЕЙ

The author considers the method of two-sided evaluations in the problems of stability of a one-span elastic non-uniformly compressed rod under various conditions of fixation of its ends.The required minimum critical value of the loading parameter for the rod is the minimum value of the functional equal to the ratio of the norms of Hilbert space elements squared. Using the inequalities following from the problem of the best approximation of a Hilbert space element through the basic functions, it is possible to construct two sequences of functionals, the minimum values of which are the lower evaluations and the upper ones. The basic functions here are the orthonormal forms of the stability loss for a rod with constant cross-section, compressed by longitudinal forces at the ends, which are fixed just so like the ends of the non-uniformly compressed rod.Having used the Riesz theorem about the representation of a bounded linear functional in the Hilbert space, the author obtains the additional functions from the domain of definition of the initial functional, which correspond to the basic functions. Using these additional functions, the calculation of the lower bounds is reduced to the determination of the maximum eigenvalues of the matrices represented in the form of second order modular matrices with the elements expressed in the form of integrals of basic and additional functions. The calculation of the upper bound value is reduced to the determination of the maximum eigenvalue of the matrix, which almost coincides with one of the modular matrices. It is noted that the obtained upper bound evaluations are not worse than the evaluations obtained through the Ritz method with the use of the same basic functions.

Cones in the space of block-matrices

In the space $$\mathcal{M}$$ $$\mathcal{m}$$ ( $$\mathcal{M}$$ $$\mathcal{n}$$ ) of $$\mathcal{m}$$ X $$\mathcal{m}$$ block-matrices with entries in Mn three natural cones related to positivity are introduced. In this paper several known results related to positivity: dilation theorem, truncated moment theorems and the Fejér–Riesz theorem on factorization of positive polynomials, are applied to derive interrelationships among those three cones. At the end of the paper, some form of matrix Schwarz inequalities is presented. The paper is largely of expository character.

Set-Valued Lebesgue and Riesz Type Theorems

Abstract In this paper, we continue a previous study concerning different types of pseudo-convergences of sequences of measurable functions with respect to set-valued nonadditive monotonic set functions and we establish some pseudo-versions of Lebesgue and Riesz theorems in the set-valued case. We also characterize some important structural properties of fuzzy multimeasures.

Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

An ordinary differential operator of arbitrary order is considered. We find necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of a system of root functions of this operator in Lp.

Decomposition of ℓ-group-valued measures

We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete ℓ-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete ℓ-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for ℓ-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.

Number-phase uncertainty relations in terms of generalized entropies

Number-phase uncertainty relations are formulated in terms of unified entropies which form a family of two-parametric extensions of the Shannon entropy. For two generalized measurements, unified-entropy uncertainty relations are given in both the state-dependent and state-independent forms. A few examples are discussed as well. Using the Pegg--Barnett formalism and the Riesz theorem, we obtain a nontrivial inequality between norm-like functionals of generated probability distributions in finite dimensions. The principal point is that we take the infinite-dimensional limit right for this inequality. Hence number-phase uncertainty relations with finite phase resolutions are expressed in terms of the unified entropies. Especially important case of multiphoton coherent states is separately considered. We also give some entropic bounds in which the corresponding integrals of probability density functions are involved.

Riesz theorem for positive multilinear functions

Riesz theorem for positive multilinear functions

EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS II: HOOKEAN-TYPE MODELS

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum ṵ0for the Navier–Stokes equation and a non-negative initial probability density function ψ0for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution t ↦ (ṵ(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ṵ(0), ψ(0)) = (ṵ0, ψ0), such that t ↦ ṵ(t) belongs to the classical Leray space and t ↦ ψ(t) has bounded relative entropy with respect to M and t ↦ ψ(t)/M has integrable Fisher information (with respect to the measure [Formula: see text]) over any time interval [0, T], T&gt;0. If the density of body forces [Formula: see text] on the right-hand side of the Navier–Stokes momentum equation vanishes, then a weak solution constructed as above is such that t ↦ (ṵ(t), ψ(t)) decays exponentially in time to [Formula: see text] in the [Formula: see text]-norm, at a rate that is independent of (ṵ0, ψ0) and of the center-of-mass diffusion coefficient. Our arguments rely on new compact embedding theorems in Maxwellian-weighted Sobolev spaces and a new extension of the Kolmogorov–Riesz theorem to Banach-space-valued Sobolev spaces.