Unconditional Basis Research Articles

Spectral Properties of Ordinary Differential Operators with Involution

Let P and Q be ordinary differential operators of order n and m generated by s boundary conditions (where s = max{n, m}) on a bounded interval [a, b]. We study operators of the form L = JP + Q, where J is an involution operator in the space L2[a, b]. Three cases are considered, namely, n > m, n < m, and n = m, for which the concepts of regular, almost regular, and normal boundary conditions are defined. Theorems on an unconditional basis property and the completeness of the root functions of the operator L depending on the type of boundary conditions from the chosen classes are announced.

Wavelets in weighted norm spaces

We give a complete characterization of the classes of weight functions for which the higher rank Haar wavelet systems are unconditional bases in weighted norm Lebesgue spaces. Particulary it follows that higher rank Haar wavelets are unconditional bases in the weighted norm spaces with weights which have strong zeros at some points. This shows that the class of weight functions for which higher rank Haar wavelets are unconditional bases is much richer than it was supposed.

On Bohr radii of finite dimensional complex Banach spaces

We study the Bohr radius of the unit ball of a~complex $n$-dimensional Banach space with an $1$-unconditional basis in terms of its lattice convexity/concavity constants. As an application we give asymptotic estimates of the Bohr radius of the unit ball of the $n$-th section of Lorentz and Marcinkiewicz sequence spaces.

Polyhedrality and Decomposition

Abstract The aim of this note is to present two results that make the task of finding equivalent polyhedral norms on certain Banach spaces, having either a Schauder basis or an uncountable unconditional basis, easier and more transparent. The hypotheses of both results are based on decomposing the unit sphere of a Banach space into countably many pieces, such that each one satisfies certain properties. Some examples of spaces having equivalent polyhedral norms are given.

Thin sets of constant width

We prove that every Banach space which admits an unconditional basis can be renormed to contain a constant width set with empty interior, thus guaranteeing, for the first time, existence of such sets in a reflexive space. In the isometric case we prove that normal structure is characterized by the property that the class of diametrically complete sets and the class of sets with constant radius from the boundary coincide.

The universal Banach space with a $K$-suppression unconditional basis

Using the technique of Fraisse theory, for every constant $K\ge 1$ we consruct a universal object in the class of Banach spaces with normalized $K$-suppression unconditional Schauder bases.

Convergence of Eigenfunction Expansions of a Differential Operator with Integral Boundary Conditions

For a second-order ordinary differential operator on an interval of the real line with integral boundary conditions, conditions for the unconditional basis property and uniform convergence of the expansion of a function in terms of the eigen- and associated functions of this operator are established. The convergence and equiconvergence rates of this expansion and the equiconvergence rate of the trigonometric Fourier expansion of this function are estimated. The uniform convergence of its expansion in the adjoint system is studied.

Solid hulls and cores of weighted $$H^\infty $$ H ∞ -spaces

We determine the solid hull and solid core of weighted Banach spaces $$H_v^\infty $$ of analytic functions functions f such that v|f| is bounded, both in the case of the holomorphic functions on the disc and on the whole complex plane, for a very general class of radial weights v. Precise results are presented for concrete weights on the disc that could not be treated before. It is also shown that if $$H_v^\infty $$ is solid, then the monomials are an (unconditional) basis of the closure of the polynomials in $$H_v^\infty $$ . As a consequence $$H_v^\infty $$ does not coincide with its solid hull and core in the case of the disc. An example shows that this does not hold for weighted spaces of entire functions.

The principles of the dependent and independent nationality of spouses: advantages and disadvantages

On marriage issues, many countries have espoused the independence of nationality policy, that is, they accept neutral effects of marriage on nationality. We don’t see the point of bestowing nationality on an alien woman who has married a national man but lives abroad. The ratio of countries in favor of dependent nationality to those in favor of independent nationality is one to three. So, there are only a few countries left still pursuing a policy of forcing husbands’ nationality upon alien women on an unconditional basis. The main question in this paper is: Should the nationality of one spouse be imposed on the other one, making them both subjects of one State? After an introduction (chapter Ⅰ), we analyze the theory of the unity of nationality and the theory of independent nationality (chapter Ⅱ). In chapter Ⅲ we see international documents on the theories of dependent and independent nationality. Finally, we take care of the present situation of the world in respect to nationality laws and then we resume some conclusions; the main one is that some political approaches seems to discriminates between national and foreign women.&#x0D;

On spreading sequences and asymptotic structures

In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibits a striking resemblance to the geometry of James space. Further, we show that the averaging projections onto subspaces spanned by constant coefficient blocks with no gaps between supports are bounded. As a consequence, every Banach space with a spreading basis contains a complemented subspace with an unconditional basis. This gives an affirmative answer to a question of H. Rosenthal. The second part contains two results on Banach spaces X X whose asymptotic structures are closely related to c 0 c_0 and do not contain a copy of ℓ 1 \ell _1 : i) Suppose X X has a normalized weakly null basis ( x i ) (x_i) and every spreading model ( e i ) (e_i) of a normalized weakly null block basis satisfies ‖ e 1 − e 2 ‖ = 1 \|e_1-e_2\|=1 . Then some subsequence of ( x i ) (x_i) is equivalent to the unit vector basis of c 0 c_0 . This generalizes a similar theorem of Odell and Schlumprecht and yields a new proof of the Elton–Odell theorem on the existence of infinite ( 1 + ε ) (1+\varepsilon ) -separated sequences in the unit sphere of an arbitrary infinite dimensional Banach space. ii) Suppose that all asymptotic models of X X generated by weakly null arrays are equivalent to the unit vector basis of c 0 c_0 . Then X ∗ X^* is separable and X X is asymptotic- c 0 c_0 with respect to a shrinking basis ( y i ) (y_i) of Y ⊇ X Y\supseteq X .

Affine Walsh-type systems of functions in symmetric spaces

Affine Walsh-type systems of functions in symmetric spaces are investigated. It is shown that such a system can only be an unconditional basis in . On the other hand the Besselian affine system generated by a function in the Zygmund-Orlicz space , , is an -system in a symmetric space if and only if , where is the closure of in and . Bibliography: 20 titles.

Factorization of the identity through operators with large diagonal

Given a Banach space X with an unconditional basis, we consider the following question: does the identity operator on X factor through every operator on X with large diagonal relative to the unconditional basis? We show that on Gowers' unconditional Banach space, there exists an operator for which the answer to the question is negative. By contrast, for any operator on the mixed-norm Hardy spaces Hp(Hq), where 1≤p,q<∞, with the bi-parameter Haar system, this problem always has a positive solution. The spaces Lp,1<p<∞, were treated first by Andrew (1979) [2].

Unconditional wavelet bases in Lebesgue spaces

In this paper, we prove that an orthonormal wavelet basis associated with a general isotropic expansive matrix must be an unconditional basis for all $L^{p}$(ℝ${}^{d})$ with $1$<$p$<$\infty$, provided the wavelet functions satisfy some usual conditions.

НЕПРЕОДОЛИМЫЕ ОБСТОЯТЕЛЬСТВА КАК ОСНОВАНИЕ ОТВЕТСТВЕННОСТИ ЭНЕРГОСНАБЖАЮЩЕЙ ОРГАНИЗАЦИИ ПО ПРИНЦИПУ ВИНЫ

In the article, the authors distinguish between “force majeure” and “force majeure” caused by an obstacle beyond human control and having an objective character. Special attention is paid to the failure of the voltage of electric energy as an insurmountable circumstance. The correlation of the concept of “break in the supply of energy” with the concept of “voltage dips” is shown. It is proved that the voltage dips in the supply of electric energy are random and unpredictable events, the occurrence of which can not be completely avoided. On the basis of the conducted research the conclusion is made: the insurmountability of other uncontrollable circumstances is not an unconditional basis for the release of the power supply organization from responsibility. It is offered: to establish responsibility of the power supplying organization for failures (differences) of tension, in the presence of fault, by introduction of additions in point 2 of article 547 of the Civil code of the Russian Federation.

Basis property of eigen- and associated functions of an operator with nondense domain of definition in the example of the Orr–Sommerfeld problem

In the paper, we propose a method for proving the unconditional basis property of eigen- and associated functions of an integro-differential operator defined on a nondense domain of definition. In particular, we obtain a new simpler proof of the unconditional basis property of eigenand associated functions of the spectral Orr–Sommerfeld problem, well-known in hydromechanics, which reduces to the eigenvalue problem for the operator under study.

On unconditional bases of reproducing kernels in Fock-type spaces

The existence of unconditional bases of reproducing kernels in the Fock-type spaces F φ with radial weights φ is studied. It is shown that there exist functions φ(r) of arbitrarily slow growth for which ln r = o(φ(r)) as r → ∞ and there are no unconditional bases of reproducing kernels in the space F φ . Thus, a criterion for the existence of unconditional bases cannot be given only in terms of the growth of the weight function.

Superconcentration, and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in Rnin the ℓ-position, and such that the space (Rn,‖⋅‖B) admits a 1-unconditional basis. Then for any ε∈(0,1/2], and for random cεlog⁡n/log⁡1ε-dimensional subspace E distributed according to the rotation-invariant (Haar) measure, the section B∩E is (1+ε)-Euclidean with probability close to one. This shows that the “worst-case” dependence on ε in the randomized Dvoretzky theorem in the ℓ-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let B be as before and assume additionally that B has a smooth boundary and Eγn‖⋅‖B≤ncEγn‖gradB(⋅)‖2 for a small universal constant c>0, where gradB(⋅) is the gradient of ‖⋅‖B and γn is the standard Gaussian measure in Rn. Then for any p∈[1,clog⁡n] the p-th power of the norm ‖⋅‖Bp is Clog⁡n-superconcentrated in the Gauss space.

Bases in the space of regular multilinear operators on Banach lattices

For Banach lattices E1,…,Em and F with 1-unconditional bases, we show that the monomial sequence forms a 1-unconditional basis of Lr(E1,…,Em;F), the Banach lattice of all regular m-linear operators from E1×⋯×Em to F, if and only if each basis of E1,…,Em is shrinking and every positive m-linear operator from E1×⋯×Em to F is weakly sequentially continuous. As a consequence, we obtain necessary and sufficient conditions for which the m-fold Fremlin projective tensor product E1⊗ˆ|π|⋯⊗ˆ|π|Em (resp. the m-fold positive injective tensor product E1⊗ˇ|ϵ|⋯⊗ˇ|ϵ|Em) has a shrinking basis or a boundedly complete basis.

Sums of asymptotically midpoint uniformly convex spaces

We study the property of asymptotic midpoint uniform convexity for infinite direct sums of Banach spaces, where the norm of the sum is defined by a Banach space $E$ with a 1-unconditional basis. We show that a sum $(\sum_{n=1}^\infty X_n)_E$ is asymptotically midpoint uniformly convex (AMUC) if and only if the spaces $X_n$ are uniformly AMUC and $E$ is uniformly monotone. We also show that $L_p(X)$ is AMUC if and only if $X$ is uniformly convex.

On Unconditional Bases of the Kernels Generated by Differential Equations of the Second Order

We establish necessary and sufficient conditions for a system of functions generated by differential equations of the second order to be a basis. Our method is based on the application of the Muckenhoupt condition.