On the Distributional Index 2F1-transform

In this paper the index ₂F₁-transform is defined on a space of generalized functions by using the method of adjoints. We prove that smooth functions having suitable asymptotic behaviors around zero and infinite are ₂F₁-transforms of generalized functions.     

Application of bases to the study of measures and operators on measure spaces

Summary Under investigation are measures m defined on a σ-algebraA with range in a Banach space X having a Schauder basis {x_n}. Utilizing the corresponding coefficient functionals and coefficient measures, we study the interaction between properties for these measures and properties for the existing measure m. For the spaceCA(A, X) of all countably additive measures of finite variation fromA into X, we show that certain separable subspaces of it are isomorphic-isometric with certain separable subspaces of functions of bounded variation on the interval [0, 1]. For {v_n} inCA(A, X) such that {v_n(A)} converges to v(A) for all A εA a certain ? control measure ? is constructed. An integral for set functions relative to a measure is defined and necessary and sufficient conditions are given for weak convergence. Bounded operators on the space FAC(A,R) of finitelly additive scalar valued set functions onA which are absolutely continuous with respect to λ are considered. Necessary and sufficient conditions for continuity and for compactness are given for Banach space valued operators T defined on such spaces. Weak compactness of T is also studied. Finally, a different kind of representation is given for Banach space valued operators defined on the space FA_v(A, X) of finitely additive set functions fromA into X. Entrata in Redazione il 26 novembre 1976. The first author expresses his gratitude to the Italian Consiglio Nazionale delle Ricerche and the Università degli Studi di Param for the support of his work during his tenure as Visiting Professor of Mathematics. This work was supported by NATO Research Grant No. 835.     

Compact and Weakly Compact Homomorphisms on Fréchet Algebras of Holomorphic Functions

We study homomorphisms between Fréchet algebras of holomorphic functions of bounded type. In this setting we prove that any pointwise bounded homomorphism into the space of entire functions of bounded type is rank one. We characterize up to the approximation property of the underlying Banach space, the weakly compact composition operators on H_b(V), V absolutely convex open set.     

When separability of the space of ultra-extremal functions is preserved

Abstract

The ultrametrically injective hull T_X of an ultrametric space (X, d) is investigated by viewing it as the space of ultra-extremal functions over X. It turns out that the ultra-extremal functions are also ultra-Ka?etov functions, satisfying two inequalities derived from the strong triangle inequality. We shall compare the ultra-extremal functions with some classes of functions defined with the help of one of the two inequalities from the definition of ultra-Kat?tov functions. We shall consider the question of when separability of the space of ultra-extremal functions is preserved.     

Entropy,Approximation Quantities and the Asymptotics of the Modulus of Continuity

The paper deals with the approximation of bounded real functions f on a compact metric space (X, d) by so-called controllable step functions in continuation of [Ri/Ste]. These step functions are connected with controllable coverings, that are finite coverings of compact metric spaces by subsets whose sizes fulfil a uniformity condition depending on the entropy numbers ε_n(X) of the space X. We show that a strong form of local finiteness holds for these coverings on compact metric subspaces of IR^m and S^m. This leads to a Bernstein type theorem if the space is of finite convex information. In this case the corresponding approximation numbers ε_n(f) have the same asymptotics its ω(f, ε_n(X)) for f ε C(X). Finally, the results concerning functions f ε M(X) and f ε C(X) are transferred to operators with values in M(X) and C(X), respectively.     

Sets which are dense in the space of all Pettis integrable functions

Using the examples given by S.J. Dilworth and M. Girardi, we prove that the set of all nowhere Pettis differentiable functions is a G _δ -dense set in the space of all X-valued Pettis integrable functions on [0, 1]. Also an another example of a dense set in the space of all Pettis integrable functions is given.     

Study of generalized eigenfunctions of a perturbed isotropic elastic half‐space

We consider the self‐adjoint operator governing the propagation of elastic waves in a perturbed isotropic half‐space (perturbation with compact support of a homogeneous isotropic half‐space) with a free boundary condition. We propose a method to obtain, numerical values included, a complete set of generalized eigenfunctions that diagonalize this operator. The first step gives an explicit representation of these functions using a perturbative method. The unbounded boundary is a new difficulty compared with the method used by Wilcox [25], who set the problem in the complement of bounded open set. The second step is based on a boundary integral equations method which allows us to compute these functions. For this, we need to determine explicitly the Green's function of (A₀ – ω²), where A₀ is the self‐adjoint operator describing elastic waves in a homogeneous isotropic half‐space. Copyright © 2000 John Wiley & Sons, Ltd.     

EMBEDDING C0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract

We show that the normed space of μ-measurable Pettis integrable functions on a probability space with values in a Banach space X contains a copy of the sequence space c₀ if and only if X contains a copy of c₀. In this case, if the probability μ has infinite range, a copy of c₀ consisting of μ-measurable functions can be found, such that it is complemented in the bigger space of all weakly μ-measurable Pettis integrable functions.     

Linear methods for approximation of some classes of holomorphic functions from the Bergman space

We construct a linear method {ie910-01} for the approximation (in the unit disk) of classes of holomorphic functions {ie910-02} that are the Hadamard convolutions of the unit balls of the Bergman space A _p with reproducing kernels {ie910-03}. We give conditions for ψ under which the method {ie910-04} approximates the class {ie910-05} in the metrics of the Hardy space H _s and the Bergman space A _s , 1 ≤ s ≤ p, with an error that coincides in order with the value of the best approximation by algebraic polynomials. Translated from in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 783–795, June, 2008.     

Reflexive operators with domain in Köthe spaces

It is proved that if a K?the space λ₁(A) is distinguished and E is an arbitrary Fréchet space then every reflexive map T: λ₁(A)→E (i.e., T maps bounded sets into relatively weakly compact ones) factorizes through a reflexive Fréchet space. An analogous result is proved for Montel maps (i.e., which map bounded sets into relatively compact ones). The result is a consequence of the fact proved also in this paper that, for a distinguished λ₁(A) space, the spaces of reflexive maps R(λ₁(A), C(K)) and of Montel maps M(λ₁(A), C(K)) are the Mackey completions of the spaces of weakly compact and compact maps, respectively. Consequences for spaces of vector-valued (weakly) continuous functions are also obtained. Received: 24 November 1997 / Revised version: 14 May 1998     

Functions of the first Baire class with values in metrizable spaces

We show that every mapping of the first functional Lebesgue class that acts from a topological space into a separable metrizable space that is linearly connected and locally linearly connected belongs to the first Baire class. We prove that the uniform limit of functions of the first Baire class ƒ _n: X → Y belongs to the first Baire class if X is a topological space and Y is a metric space that is linearly connected and locally linearly connected. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 568–572, April, 2006.     

On an optimal quadrature formula in the sense of Sard

In this paper we construct an optimal quadrature formula in the sense of Sard in the Hilbert space K ₂(P ₂). Using S.L. Sobolev’s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L₂⁽²⁾(0,1)L_2^{(2)}(0,1). The obtained optimal quadrature formula is exact for the trigonometric functions sinx and cosx. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.     

On the coincidence of the canonical embeddings of a metric space into a Banach space

Recall the two classical canonical isometric embeddings of a finite metric space X into a Banach space. That is, the Hausdorff–Kuratowsky embedding x → ρ(x, ⋅) into the space of continuous functions on X with the max-norm, and the Kantorovich–Rubinshtein embedding x → δ _x (where δ _x , is the δ-measure concentrated at x) with the transportation norm. We prove that these embeddings are not equivalent if |X| > 4. Bibliography: 2 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 153–161.     

Clifford-Hermite-monogenic operators

In this paper we consider operators acting on a subspace ℳ of the space L ₂ (ℝ^m; ℂ_m) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ℳ is defined as the orthogonal sum of spaces ℳ_s,k of specific Clifford basis functions of L ₂(ℝ^m; ℂm). Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they transform a space ℳ_s,k into a similar space ℳ_s′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known. Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail.     

Compact convex sets and complex convexity

We construct a quasi-Banach space which cannot be given an equivalent plurisubharmonic quasi-norm, but such that it has a quotient by a one-dimensional space which is a Banach space. We then use this example to construct a compact convex set in a quasi-Banach space which cannot be affinely embedded into the spaceL ₀ of all measurable functions.     

On the Fourier-Coefficients of Vector-Valued Functions

We study the decay of the Fourier-coefficients of vector-valued functions F : T → X, X a Banach space. Differentiable functions f generally have absolutely summable Fourier-coefficients, f (n) <, iff X is K-convex. More precise statements on the decay of f (n) for regular functions f can be given if X has Fourier-type p. If f belongs to the Besov space, the sequence (||f(n)||) belongs to the Lorentz sequence space l_t,v with 1/t = λ + 1/max (u′, p′). This result is the best possible in the vector-valued case and generalizes the well-known scalar results.     

Composition operators on a Banach space of analytic functions

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL _α ¹ (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range. Supported by NNSFC No.19671036     

Weighted Composition Operators between Different Hardy Spaces

Let and be two analytic functions defined on such that. The operator given by is called a weighted composition operator. In this paper we deal with the boundedness, compactness, weak compactness, and complete continuity of weighted composition operators from a Hardy space H _p into another Hardy space H _q . We apply these results to study composition operators on Hardy spaces of a half-plane. Submitted: November 20, 2001.     

AdS3/CFT2 on a Torus in the Sum over Geometries

We investigate the AdS₃/CFT₂ correspondence for the Euclidean AdS₃ space compactified on a solid torus with the CFT field on the regularizing boundary surface in the bulk. Correlation functions corresponding to the bulk theory at a finite temperature tend to the standard CFT correlation functions in the limit of removed regularization. In the sum over geometries in both the regular and the ℤ _N orbifold cases, the two-point correlation function for massless modes transforms into a finite sum of products of the conformal-anticonformal CFT Green's functions up to divergent terms proportional to the volume of the SL(2, ℤ)/ℤ group. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 17–30, January, 2006.     

Factorization of the r-matrix for the quantum algebra <Emphasis Type="Italic">Uq</Emphasis>(<Emphasis Type="Italic">sl</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>)

We propose a method for construction of the general solution of the Yang–Baxter equation with the U _q (sℓ _n ) symmetry algebra. This method is based on the factorization property of the corresponding L-operator. We present a closed-form expression for the universal R-matrix in the form of a difference operator acting on the space of functions of n(n − 1) variables. Bibliography: 16 titles.