Bounds for the number of points on curves over finite fields

Let U be a bounded open subset of ?^d, d ≥ 2 and f ∈ C(?U). The Dirichlet solution f^CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure ū of U in general if U is not regular but it is always Baire-one.Let H(U) be the space of all functions continuous on the closure ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f^CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f^CU belongs to the subclass B_1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].     

On Lattice Structure of the Space of Pointwise Limits of Harmonic Functions

Let H¹(U) denote the space of all pointwise limits of bounded sequences from H(U), where H(U) consists of all continuous functions on the closure [`(U)]\overline{U} of a bounded open set U⊂ℝ^m that are harmonic on U. It is shown that the space H¹(U) is a lattice in the natural ordering if and only if the set ∂_regU of all regular points of U is an F_σ-set.     

On a Bornological Structure in Infinite-Dimensional Holomorphy

The bornology (b) of bounded subsets with respect to continuous convergence is used on spaces of holomorphic functions. It is shown that Hom_coH_b(U) ? U for a circled convex open subset U of a complete nuclear space. Exponential laws for spaces of holomorphic functions with bornological structures are proved and the connection with Colombeau's Silva holomorphic functions is established.     

M-structure in dual Banach spaces

A result previously known only for certain ordered Banach spaces is generalized to arbitrary real Banach spaces. Let ℒ be the Banach algebra of operators generated by theL-projections of a real Banach spaceU, and let ℳ (U ^* be the bounded operators on the dual spaceU ^* with adjoint in ℒ(U ^**. Then the adjoint operation maps ℒ (U) onto ℳ (U ^*). In particular, anyM-projection ofU ^* is weak^* continuous. Supported in part by the National Science Foundation.     

Pointwise convergence and radial limits of harmonic functions

This paper characterizes those real-valued functions on a compact setK in ℝ ⁿ that can be expressed as the pointwise limit of a sequence (h _m ), where each functionh _m is harmonic on some neighbourhood ofK. It also characterizes those functions on the unit sphere that can arise as the radial limit function at infinity of an entire harmonic function. Both results rely on important recent work of Lukeš et al. concerning approximation of affine Baire-one functions.     

On the Riemann Summability of Fourier Integrals and Real Hardy Spaces

We consider the Riemann means of single and multiple Fourier integrals of functions belonging to L¹ or the real Hardy spaces defined on ℝⁿ, where n ≥ 1 is an integer. We prove that the maximal Riemann operator is bounded both from H¹(ℝ) into L¹(ℝ) and from L¹(ℝ) into weak –L¹(ℝ). We also prove that the double maximal Riemann operator is bounded from the hybrid Hardy spaces H^(1,0)(ℝIsup2), H^(0,1)(ℝ²⁾ into weak –L¹(ℝ²). Hence pointwise Riemann summability of Fourier integrals of functions in H^(1,0) ∪ H^(0,1)(ℝ²) follows almost everywhere.The maximal conjugate Riemann operators as well as the pointwise convergence of the conjugate Riemann means are also dealt with.     

Analytic functions of bounded mean oscillation and the Bloch space

In this paper we show that the closure of the space BMOA of analytic functions of bounded mean oscillation in the Bloch spaceB is the image P(U) of space of all continuous functions on the maximal ideal space ofH under the Bergman projection P. It is proved that the radial growth of functions in P(U) is slower than the iterated logarithm studied by Makarov. So some geometric conditions are given for functions in P(U), which we can easily use to construct many Bloch functions not in P(U).     

Complementability of Spaces of Harmonic Functions

Let U be a bounded open set of the Euclidean space ℝ ^d and let H(U) denote the space of all real-valued continuous functions on that are harmonic on U. We present a sufficient condition on the set ∂ _reg U of all regular points of U that ensures that H(U) is complemented in . We also present examples showing that this condition is not necessary. The proof of the positive result is based upon a general result on complementability of a simplicial function space in a space. Research was supported in part by the grants GA ČR GACR 201/06/0018 and in part by the Research Project MSM 0021620839 from the Czech Ministry of Education.     

Pointwise bounded approximation and Dirichlet algebras

Those open sets U of S² for which A(U) is pointwise boundedly dense in H^∞(U) are characterized in terms of analytic capacity. It is also shown that the real parts of the functions in A(U) are uniformly dense in C_R(∂U) if and only if each component of U is simply connected and A(U) is pointwise boundedly dense in H^∞(U).     

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     

Holomorphic functions on strict inductive limits

We study strict inductive limits of Fréchet Montel (FM) spaces and reflexive Fréchet (RF) spaces and we obtain some interesting examples in the theory of infinite dimensional holomorphy. P_M(^kE′) and P_HY(^kE′) will denote respectively the set of all k-homogeneous polynomials on E′ that are bounded on bounded sets and the set of all k-homogeneous polynomials on E′ that are continuous on compact sets. ?_SM(^kE′) is the space of all symetric k -multilinear mappings from E′ × ... × E′ into C that are bounded on bounded sets. H_HY(E′) will denote the set of all G-analytic functions on E′ that are continuous on the compact subsets of E′.     

A Class of Operators Similar to the Shift on H 2(G)

For a compact subset K in the complex plane, let A(K) denote the algebra of all functions continuous on K and analytic on K° and let R(K) denote the uniform closure of the rational functions with poles off K. Let G is a bounded open subset whose complement in the plane has a finite number of components. Suppose that and every function in H^∞(G) is the pointwise limit of a bounded sequence of functions in . The purpose of this paper is to characterize all subnormal operators similar to M_z, the operator of multiplication by the independent variable z on the Hardy space H²(G). We also characterize all bounded linear operators that are unitarily equivalent to M_z in the case when each of the components of G is simply connected. In particular, our similarity result extends a well-known result of W. Clary on the unit disk to multiply connected domains.     

Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum

Let G/H be a semisimple symmetric space. Then the space L²(G/H) can be decomposed into a finite sum of series of representations induced from parabolic subgroups of G. The most continuous part of the spectrum of L²(G/H) is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions on a bounded symmetric domain of tube type, containing G/H as a boundary component. A boundary value map is constructed and we show that it induces a G-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part L_mc²(G/H) of L²(G/H). We also relate our Hardy space to the classical Hardy space on the bounded symmetric domain.Supported in part by NSF-grant DMS-0070816 and the MSRISupported in part by NSF-grant DMS-0097314 and the MSRISupported in part by NSF-grant DMS-0070607 and the MSRI     

Groups of homeomorphisms on C(X) and pointwise limits

This study looks at some subgroups of the group H(C(X)) of homeomorphisms on the space C(X) of continuous real-valued functions on a topological space X, where C(X) has the compact-open topology. The main result shows that, for certain spaces X, the subgroup of H(C(X)) generated by the algebraic and vertical homeomorphisms on C(X) is dense in H(C(X)) with the pointwise topology. Also, for X equal to the unit interval, a subgroup of H(C(X)) is developed using integration of the members of C(X), and this subgroup is used as an example and to illustrate certain properties that subgroups of H(C(X)) can have.     

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.     

The approximation property for spaces of holomorphic functions on infinite dimensional spaces II

Let H(U) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τω and τδ respectively denote the compact-ported topology and the bornological topology on H(U). We show that if E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on E is weakly continuous on bounded sets, then (H(U),τω) and (H(U),τδ) have the approximation property for every open subset U of E. The classical space c₀, the original Tsirelson space T^∗ and the Tsirelson^∗-James space are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains.     

Optimal Error of Analytic Continuation from a Finite Set with Inaccurate Data in Hilbert Spaces of Holomorphic Functions

We consider the problem of analytic continuation with inaccurate data from a finite subset U of a domain D of C ⁿ to a point z ₀D\U for the functions f belonging to a bounded correctness set V in a Hilbert space H(D) of analytic functions in D. In the case when H(D) is a Hilbert space with a reproducing kernel, we find constructive formulas for calculating the optimal error, the optimal function, and the optimal linear algorithm for extrapolation to a point z ₀ for functions in V whose approximate values are given on a set U. Moreover, we study the asymptotics of the optimal error in the case when the errors of initial data vanish.     

The eigen functions of the complex projective space

In the complexn-dimensional projective spaceCP ⁿ , let λ _p (=4p(p+n)) be the eigen value of the Laplace-Beltrami operator andH _p be the space of all eigen functions of eigen value λ _p . The reproducing kernelh _p (z, w) ofH _p is constructed explicitly in this paper, and a system of complete orthogohal functions ofH _p is constructed fromh _p (z,w)(p=1,2, …). Partially supported by NSF of China     

On pointwise convergence of the family of Urysohn‐type integral operators

In this paper, we consider a family of nonlinear integral operators of Urysohn‐type and study the pointwise convergence of the family at characteristic points of L₁?function. The kernel K_λ(x,t,u(t)) depends on the positive parameter λ changing on the set of numbers with the accumulation point at infinity and K_λ(x,t,u(t)) is an entire analytic function of variable u, which is a bounded function belonging to L₁(R).