Riesz Property Research Articles

Fredholm Theory Relative to Any Algebra Homomorphisms

In this paper, we give another definition of Ruston elements and almost Ruston elements, which is equivalent to the definitions given by Mouton and Raubenheimer in the case that the homomorphism has a closed range and Riesz property. For two homomorphisms, we consider the preserver problems of Fredholm theory and Fredholm spectrum theory. In addition, we study the spectral mapping theorems of Fredholm (Weyl, Browder, Ruston, and almost Ruston) elements relative to a homomorphism. Last but not least, the dependence of Fredholm theory on three homomorphisms is considered, and meanwhile, the transitivity of Fredholm theory relative to three homomorphisms is illustrated. Furthermore, we consider the Fredholm theory relative to more homomorphisms.

On Riesz property and equivalent basis property of the system of root vector functions of Dirac-type operator

On Riesz property and equivalent basis property of the system of root vector functions of Dirac-type operator

On Fractional Musielak–Sobolev Spaces and Applications to Nonlocal Problems

In this work, we establish some abstract results on the perspective of the fractional Musielak–Sobolev spaces, such as: uniform convexity, Radon–Riesz property with respect to the modular function, $$(S_{+})$$ -property, Brezis–Lieb type Lemma to the modular function and monotonicity results. Moreover, we apply the theory developed to study the existence of solutions to the following class of nonlocal problems $$\begin{aligned} \left\{ \begin{array}{ll} (-\varDelta )_{\varPhi _{x,y}}^s u = f(x,u),&{} \text{ in } \varOmega ,\\ u=0,&{} \text{ on } {\mathbb {R}}^N\setminus \varOmega , \end{array} \right. \end{aligned}$$ where $$N\ge 2$$ , $$\varOmega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz boundary $$\partial \varOmega $$ and $$f:\varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$$ is a Carathéodory function not necessarily satisfying the Ambrosetti–Rabinowitz condition. Such class of problems enables the presence of many particular operators, for instance, the fractional operator with variable exponent, double-phase and double-phase with variable exponent operators, anisotropic fractional p-Laplacian, among others.

The zero set $$\Lambda $$ of exponential polynomials and the Riesz property of its exponential system $$E_{\Lambda }$$ in $$L^2(0,T)$$

The zero set $$\Lambda $$ of exponential polynomials and the Riesz property of its exponential system $$E_{\Lambda }$$ in $$L^2(0,T)$$

Topological properties of some multiplication operators on L(X)

A pair (u, ?) in X ? X?, where X is an infinite dimensional Banach space and X? its topological dual space, induces in a natural way two multiplication operators L?,u and R?,u on the Banach space L(X), defined by L?,u (T) (x) = ? (T(x)) u, and R?,u (T) (x) = ?(x)T(u), for all T in L(X) and x in X. In this paper, we present necessary and sufficient conditions for the compactness, demicompactness, stongly demicompactess, power compactness and Riesz property of this family of operators. We also establish sufficient conditions for the quasi-compactness and weak compactness of these operators. Finally, we show that the Dunford-Pettis property fails for the Banach space L(X) whenever either X or L(X) is reflexive.

Extremal mappings of finite distortion and the Radon–Riesz property

We consider Sobolev mappings f\in W^{1,q}(\Omega,\mathbb{C}) , 1<q<\infty , between planar domains \Omega\subset \mathbb{C} . We analyse the Radon–Riesz property for polyconvex functionals of the form f\mapsto \int_\Omega \Phi(|Df(z)|,J(z,f)) \, dz and show that under certain criteria, which hold in important cases, weak convergence in W_{\mathrm{loc}}^{1,q}(\Omega) of (for instance) a minimising sequence can be improved to strong convergence. This finds important applications in the minimisation problems for mappings of finite distortion and the L^p and Exp-Teichmüller theories.

Riesz Inequality for the System of Root Functions of Second Order Ordinary Differential Operator

An ordinary differential operator of second order with coefficients is considered. The Riesz property of the system of root functions of the given operator is studied. The criterion of Bessel property in 2 L , of root functions system is established and use it to obtain sufficient conditions for the Riesz property of a system of normalized root functions of this operator in p L .

Boundary behavior of generalized analytic functions

We study the boundary properties of the solutions of the elliptic equation a(z)∂z¯u+b(z)∂zu+A(z)u+B(z)u¯=0 under the assumption that a and b are Hölder continuous and A and B are in Lp for some p>2. These properties include the Hp property, the F. and M. Riesz property and the Rudin–Carleson property.

On the vaguelet and Riesz properties of [formula omitted]-unbounded transformations of orthogonal wavelet bases

In this work, we prove that certain L2-unbounded transformations of orthogonal wavelet bases generate vaguelets. The L2-unbounded functions involved in the transformations are assumed to be quasi-homogeneous at high frequencies. We provide natural examples of functions which are not quasi-homogeneous and for which the resulting transformations are not vaguelets. We also address the related question of whether the considered family of functions is a Riesz basis in L2(R). The Riesz property could be deduced directly from the results available in the literature or, as we outline, by using the vaguelet property in the context of this work. The considered families of functions arise in wavelet-based decompositions of stochastic processes with uncorrelated coefficients.

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.

Riesz inequality for systems of root vector functions of the Dirac operator

We establish a criterion for the Riesz property of systems of root vector functions of the one-dimensional Dirac operator.

Representation of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers

Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator’s best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator’s best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.

Wavelets from the Loop Scheme

Anewwavelet-based geometric mesh compression algorithm was developed recently in the area of computer graphics by Khodakovsky, Schroder, and Sweldens in their interesting article [23]. The new wavelets used in [23] were designed from the Loop scheme by using ideas and methods of [26, 27], where orthogonal wavelets with exponential decay and pre-wavelets with compact support were constructed. The wavelets have the same smoothness order as that of the basis function of the Loop scheme around the regular vertices which has a continuous second derivative; the wavelets also have smaller supports than those wavelets obtained by constructions in [26, 27] or any other compactly supported biorthogonal wavelets derived from the Loop scheme (e.g., [11, 12]). Hence, the wavelets used in [23] have a good time frequency localization. This leads to a very efficient geometric mesh compression algorithm as proposed in [23]. As a result, the algorithm in [23] outperforms several available geometric mesh compression schemes used in the area of computer graphics. However, it remains open whether the shifts and dilations of the wavelets form a Riesz basis of L2(ℝ2). Riesz property plays an important role in any wavelet-based compression algorithm and is critical for the stability of any wavelet-based numerical algorithms. We confirm here that the shifts and dilations of the wavelets used in [23] for the regular mesh, as expected, do indeed form a Riesz basis of L2(ℝ2) by applying the more general theory established in this article.

Archimedeanness and the MacNeille completion of pseudoeffect algebras and po-groups

Pseudoeffect (PE-) algebras are partial algebras differing from effect algebras in that they need not satisfy the commutativity assumption. PE-algebras typically arise from intervals of po-groups; this applies in particular to all those which satisfy a certain Riesz property.

An F. and M. Riesz theorem for planar vector fields

We prove that solutions of the homogeneous equation Lu=0, where L is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if $\Omega$ is an open subset of the plane with smooth boundary, $u\in C^1(\Omega)$ satisfiesLu=0 on $\Omega$ , has tempered growth at the boundary, and its weak boundary value is a measure $\mu$ , then $\mu$ is absolutely continuous with respect to Lebesgue measure on the noncharacteristic portion of $\partial\Omega$ .

Spaces from pieces IV

We are going to investigate simultaneous extensions of various topological structures (i.e. traces on several subsets at the same time are prescribed), also with separation axioms T0, T1, symmetry (in the sense of Part I, § 3), Riesz property, Lodato property. The following questions will be considered: (i) Under what conditions is there an extension? (ii) How can the finest extension be described? (iii) Is there a coarsest extension? (iv) Can we say more about extensions of two structures than in the general case? (v) Assume that certain subfamilies (e.g. the finite ones) can be extended; does the whole family have an extension, too? The general categorial results from Part I will be applied whenever possible (even they are not really needed).

On the approximation of measurable linear functionals

Every Gaussian Radon measure (nonsymmetric) has the Riesz property. On R ∞, a shift μ 0 ∗ σ b of the law μ 0 of the Bernoulli sequence has the Riesz property iff b ∉ l 2 or b ϵ R 0 ∞ .