Riesz Theorem Research Articles

Notes on Entropic Uncertainty Relations Beyond the Scope of Riesz’s Theorem

We discuss those relations for entropies of two quantum measurements that do not follow from the Riesz theorem and its varieties. As measures of an uncertainty, both the Renyi entropies and Tsallis entropies are utilized. It is assumed that values of entropic parameters do not share the usual relation which is required for application of Riesz’s theorem. As is shown, the question is quite reduced to estimating from below the sum or the product of norm-like functions of two generated probability distributions. The considered approach is developed in details within the two examples. In the first example, entropic uncertainty relations are given for a pair of spin-1/2 components along two non-orthogonal axes. The second example deals with two measurements for state discrimination. The former projective measurement is assigned to the Helstrom scheme, the second POVM is related to the B92 protocol of quantum key distribution.

A note on Lorentz–Zygmund spaces

Abstract Let 1 ≤ q < p < ∞, and ℝ be the real line. Hörmander showed that any bounded linear translation invariant operator from Lp (ℝ) to Lq (ℝ) is trivial. Blozinski obtained an analogy to Hörmander in Lorentz spaces on the real line. In this paper, we generalize Blozinski's result in Lorentz–Zygmund spaces. Also, Bochkarev proved an inequality related to the Hausdorff–Young–Riesz theorem in Lorentz spaces, and the sharpness of the inequality. We improve Bochkarev's inequality in Lorentz–Zygmund spaces, and prove the sharpness of our inequality.

A Formal Proof Of The Riesz Representation Theorem

This paper presents a formal proof of the Riesz representation theorem in the PVS theorem prover. The Riemann Stieltjes integral was defined in PVS, and the theorem relies on this integral. In order to prove the Riesz representation theorem, it was necessary to prove that continuous functions on a closed interval are Riemann Stieltjes integrable with respect to any function of bounded variation. This result follows from the equivalence of the Riemann Stieltjes and Darboux Stieltjes integrals, which would have been a lengthy result to prove in PVS, so a simpler lemma was proved that captures the underlying concept of this integral equivalence. In order to prove the Riesz theorem, the Hahn Banach theorem was proved in the case where the normed linear spaces are the continuous and bounded functions on a closed interval. The proof of the Riesz theorem follows the proof in Haaser and Sullivan's book Real Analysis. The formal proof of this result in PVS revealed an error in textbook's proof. Indeed, the proof of the Riesz representation theorem is constructive, and the function contstructed in the textbook does not satsify a key property. This error illustrates the ability of formal verification to find logical errors. A specific counterexample is given to the proof in the textbook. Finally, a corrected proof of the Riesz representation theorem is presented.

Entropic uncertainty relations for extremal unravelings of super-operators

A way to pose the entropic uncertainty principle for trace-preserving super-operators is presented. It is based on the notion of extremal unraveling of a super-operator. For a given input state, different effects of each unraveling result in some probability distribution at the output. As is shown, all Tsallis’ entropies of positive order as well as some of Rényi's entropies of this distribution are minimized by the same unraveling of a super-operator. Entropic relations between a state ensemble and the generated density matrix are revisited in terms of both the adopted measures. Using Riesz's theorem, we obtain two uncertainty relations for any pair of generalized resolutions of the identity in terms of the Rényi and Tsallis entropies. The inequality with Rényi's entropies is an improvement of the previous one, whereas the inequality with Tsallis’ entropies is a new relation of a general form. The latter formulation is explicitly shown for a pair of complementary observables in a d-level system and for the angle and angular momentum. The derived general relations are immediately applied to extremal unravelings of two super-operators.

Archimedean operator-theoretic Positivstellensätze

We prove a general archimedean positivstellensatz for hermitian operator-valued polynomials and show that it implies the multivariate Fejer–Riesz theorem of Dritschel–Rovnyak and positivstellensätze of Ambrozie–Vasilescu and Scherer–Hol. We also obtain several generalizations of these and related results. The proof of the main result depends on an extension of the abstract archimedean positivstellensatz for ⁎-algebras that is interesting in its own right.

Banach spaces which embed into their dual

We use Birkhoff-James' orthogonality in Banach spaces to provide new conditions for the converse of the classical Riesz's representation theorem.

On the Regularity of Operators Near a Regular Operator

Using the Riesz theorem, we give a new proof that the linear operators near a regularoperator are regular.

A Helly theorem in metric spaces and majored operations

We present a generalization of a Helly type theorem given in [Ducho\v n,~M.---Mali\v ck\'y, P.: \textit{A Helly theorem for functions with values in metric spaces,} Tatra Mt. Math. Publ. \textbf{44} (2010), 159--168.] for sequences of functions with values in metric spaces and apply it to representations of some majored mappings on the space of continuous functions. A generalization of the Riesz theorem is formulated and proved. In particular, a representation of certain majored linear operators on the space of~continuous functions, into a Banach space.

Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

We show the David-Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require its surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As a consequence every Wolff snowflake has infinite surface measure.

Boundary-value problems for fourth-order equations of hyperbolic and composite types

Boundary-value problems for fourth-order linear partial differential equations of hyperbolic and composite types are studied. The method of energy inequalities and averaging operators with variable step is used to prove existence and uniqueness theorems for strong solutions. The Riesz theorem on the representation of linear continuous functionals in Hilbert spaces is used to prove the existence and uniqueness theorems for generalized solutions.

Why the theorem of Scheffé should be rather called a theorem of Riesz

In 1947 Henry Scheffe published a result which afterwards became known as Scheffe’s theorem, stating that the distributions of a sequence (f n ) of densities, which converge almost everywhere to a density f, converge uniformly to the distribution of f. But almost 20 years earlier Frigyes Riesz proved a sufficient condition for convergence in the p-th mean (p ≥ 1), wherefrom the Scheffe theorem is just a special case.

Semidefinite programming for optimizing convex bodies under width constraints

We consider the problem of minimizing a functional (such as the area, perimeter and surface) within the class of convex bodies whose support functions are trigonometric polynomials. The convexity constraint is transformed via the Fejér–Riesz theorem on positive trigonometric polynomials into a semidefinite programming problem. Several problems such as the minimization of the area in the class of constant-width planar bodies, rotors and space bodies of revolution are revisited. The approach seems promising to investigate more difficult optimization problems in the class of three-dimensional convex bodies.

A Helly theorem for functions with values in metric spaces

Abstract We present a Helly type theorem for sequences of functions with values in metric spaces and apply it to representations of some mappings on the space of continuous functions. A generalization of the Riesz theorem is formulated and proved. More concretely, a representation of certain majored linear operators on the space of continuous functions into a complete metric space.

Characterizations on 2-isometries

We consider 2-isometries, weak 2-isometries and 2-continuous mappings and investigate the relation between three mappings in linear 2-normed spaces. Also we prove that the Riesz theorem holds when X is a linear 2-normed spaces.

Riesz theorem in probabilistic inner product spaces

Riesz theorem in probabilistic inner product spaces

Statistical convergence and ideal convergence for sequences of functions

Let I ⊂ P ( N ) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I -convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space ( X , M , μ ) , we obtain a statistical version of the Egorov theorem (when μ ( X ) < ∞ ). We show that, in its assertion, equi-statistical convergence on a big set cannot be replaced by uniform statistical convergence. Also, we consider statistical convergence in measure and I -convergence in measure, with some consequences of the Riesz theorem. We prove that outer and inner statistical convergences in measure (for sequences of measurable functions) are equivalent if the measure is finite.

An F. and M. Riesz Theorem on the Complex Sphere

Let μ be a measure on the complex sphere. Denote by μpq the projection of μ on H(p, q), the space of complex spherical harmonics. Assume that μpq = 0 if (p − 1)q ≠ 0, and ∥μ1q∥∞ → 0 as q → ∞. Then μ is absolutely continuous with respect to Lebesgue measure on the sphere.

Singularity of Vector Valued Measures in Terms of Fourier Transform

We study how the singularity (in the sense of Hausdorff dimension) of a vector valued measure can be affected by certain restrictions imposed on its Fourier transform. The restrictions, we are interested in, concern the direction of the (vector) values of the Fourier transform. The results obtained could be considered as a generalizations of F. and M. Riesz theorem, however a phenomenon, which have no analogy in the scalar case, arise in the vector valued case. As an example of application, we show that every measure from \(\mu=(\mu_1,\dots,\mu_d)\in M({\mathbb R}^d,{\mathbb R}^{d})\) annihilating gradients of \(C^{(1)}_0({\mathbb R}^d)\) embedded in the natural way into \(C_0({\mathbb R}^d,{\mathbb R}^{d}),\) i.e., such that \(\sum_i\int\partial^if\,d\mu_i=0\,\) for \(f\in C^{(1)}_0({\mathbb R}^d)\), has Hausdorff dimension at least one. We provide examples which show both completeness and incompleteness of our results.

Noncommutative Riesz theorem and weak Burnside type theorem on twisted conjugacy

The paper consists of two parts. In the first part, we prove a noncommutative analog of the Riesz(— Markov—Kakutani) theorem on representation of functionals on an algebra of continuous functions by regular measures on the underlying space.

Modules over subalgebras of the disk algebra

This paper deals with the problem of characterizing sub- modules of C(T) over certain subalgebras of the disk algebra A. We obtain results that are analogues of the classical characterizations of sub- spaces of C(T) that are invariant under multiplication by z, i.e., that are submodules over A. These characterizations yield generalizations of Wermer's maximality theorem applicable to these subalgebras. We also present an invariant subspace theorem that seems to be of independent interest and show the equivalence of the classical theorem of the brothers Riesz on analytic measures with the theorem of Fatou.