Padé tables of entire functions of very slow and smooth growth

Letf(z)=σ _j?o ^∞ a _j z ^j be entire with $$|a_{j - 1} a_{j + 1} /a_j^2 | \leqslant \rho _0^2 ,j = 1,2,3, \ldots ,$$ whereρ ₀=0.4559... is the positive root of the equation $$2\sum\limits_{j = 1}^\infty {\rho ^{j^2 } = 1.}$$ . It is shown that the Padé table off is normal, and asL→∞, [L/M _L ](z) converges uniformly in compact subsets ofC tof, for any sequence of nonnegative integers {M _L } _L=1 ^∞. In particular, the diagonal sequence {[L/L]} converges uniformly in compact subsets ofC tof. Furthermore, the constantρ ₀ is shown to be best possible in a strong sense.     

Padé tables of entire functions of very slow and smooth growth,II

Letf(z):=Σ _j=0 ^∞ a _j z ^j , where a_j ≠ 0,j large enough, and for someq ε C such that ¦q¦ $$q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q,j \to \infty .$$ Define for m,n = 0,1,2,..., the Toeplitz determinant $$D(m/n): = \det (a_{m - j + k} )_{j,k = 1}^n .$$ Given ? > 0, we show that form large enough, and for everyn = 1,2,3,..., $$(1 - \varepsilon )^n \leqslant \left| {{{D(m/n)} \mathord{\left/ {\vphantom {{D(m/n)} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right| \leqslant (1 + \varepsilon )^n .$$ We apply this to show that any sequence of Padé approximants {[m _k /n _k ]} ₁ ^∞ tof, withm _k →∞ ask→ ∞, converges locally uniformly in C. In particular, the diagonal sequence {[n/n]} ₁ ^∞ converges throughout C. Further, under additional assumptions, we give sharper asymptotics forD(m/n).     

Vector spaces of delta-plurisubharmonic functions and extensions of the complex Monge–Ampère operator

In this paper we shall consider two types of vector ordering on the vector space of differences of negative plurisubharmonic functions, and the problem whether it is possible to construct supremum and infimum. Then we consider two different approaches to define the complex Monge–Ampère operator on these vector spaces, and we solve some Dirichlet problems. We end this paper by stating and discussing some open problems.     

Integral representations and holonomic q-difference structure of Askey–Wilson type functions

We derive in a unified way the difference equations for Askey–Wilson polynomials and their Stieltjes transforms, by using basic properties of the de Rham cohomology associated with q-integral representations (Jackson integrals of BC ₁ type) of these functions.     

Y spaces and global smooth solution of fractional Navier–Stokes equations with initial value in the critical oscillation spaces

For fractional Navier–Stokes equations and critical initial spaces X, one used to establish the well-posedness in the solution space which is contained in $C({\mathbb{R}}_{+},X)$. In this paper, for heat flow, we apply parameter Meyer wavelets to introduce Y spaces $Y^{m,\beta }$ where $Y^{m,\beta }$ is not contained in $C({\mathbb{R}}_{+},{\dot{B}}_{\infty }^{1?2\beta ,\infty })$. Consequently, for $\frac{1}{2}<\beta <1$, we establish the global well-posedness of fractional Navier–Stokes equations with small initial data in all the critical oscillation spaces. The critical oscillation spaces may be any Besov–Morrey spaces ${({\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\mathbb{R}}^n))}^n$ or any Triebel–Lizorkin–Morrey spaces ${({\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\mathbb{R}}^n))}^n$ where $1\le p,q\le \infty ,0\le {\gamma }_2\le \frac{n}{p},{\gamma }_1?{\gamma }_2=1?2\beta$. These critical spaces include many known spaces. For example, Besov spaces, Sobolev spaces, Bloch spaces, Q-spaces, Morrey spaces and Triebel–Lizorkin spaces etc.     

Approximation of partially smooth functions

In this paper we discuss approximation of partially smooth functions by smooth functions. This problem arises naturally in the study of laminated currents.     

Refinements of the Cauchy–Bunyakovsky–Schwarz inequality for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous functions of selfadjoint operators in Hilbert spaces that improve the Cauchy–Bunyakovsky–Schwarz inequality, are given.     

Multiplicative convolutions of functions from Lorentz spaces and convergence of series of Fourier–Vilenkin coefficients

Let f and g be functions from different Lorentz spaces L ^{p, q} [0, 1), h be theirmultiplicative convolution and xxxx be Fourier coefficients of h with respect to a multiplicative system with bounded generating sequence. We estimate the remainder of the series of xxxx with multiplicators of type k ^b in terms of the best approximations of f and g in the corresponding Lorentz spaces. We establish sharpness of this result and of its corollaries for the Lebesgue spaces.     

On the growth and range of functions in Möbius invariant spaces

This paper is a continuation of our earlier work and focuses on the structural and geometric properties of functions in analytic Besov spaces, primarily on univalent functions in such spaces and their image domains. We improve several earlier results.     

Some extensions of the Levi�CCivit�� functional equation and richly periodic spaces of functions

We study some classes of functional equations using geometric results on orbits for infinite-dimensional continuous representations of groups. One of the typical results is the following. Let G be a connected topological group. Suppose that continuous functions a _i , b _i , u _i , v _i on G satisfy the condition $$\sum_{i=1}^ma_i(g)b_i(hg) = \sum_{i=1}^n u_i(g)v_i(h), \qquad \forall g,h \in G.$$ If the functions a ₁, ... , a _m are linearly independent then all b _i are matrix elements of a continuous finite-dimensional representation of G. For ${G = \mathbb{R}^n}$ this means that b _i are quasipolynomials.     

On the theory of Besov–Herz spaces and Euler equations

Inspired by Kalton and Wood’s work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier–Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism T between the associated Fourier algebras (resp. Fourier–Stieltjes algebras) with completely bounded norm \({\left\| T \right\|_{cb}} < \sqrt {3/2} \left( {{\text{resp}}{\text{.}}{{\left\| T \right\|}_{cb}} < \sqrt {5/2} } \right)\). We show similar results involving the norm distortion ‖T‖‖T ^?1‖ with universal but non-explicit bound. Our results subsume Walter’s well-known structural theorems and also Lau’s theorem on the second conjugate of Fourier algebras.     

Bounded projections and duality on spaces of holomorphic functions in the unit ball ofℂ n

In this paper three Banach spacesA ₀(),A andA ¹() of functions holomorphic in the unit ballB of ⁿ are defined. We exhibit bounded projections fromC ₀(B) ontoA ₀(), fromL ¹(B) ontoA ¹(), and fromL(B) ontoA(). Using these projections, we show thatA ₀()* A ¹() andA ¹()* A().Supported in part by the National Natural Science Foundation of China.     

Coronae of product spaces and the coarse Baum–Connes conjecture

We study the coarse Baum–Connes conjecture for product spaces and product groups. We show that a product of CAT(0) groups, polycyclic groups and relatively hyperbolic groups which satisfy some assumptions on peripheral subgroups, satisfies the coarse Baum–Connes conjecture. For this purpose, we construct and analyze an appropriate compactification and its boundary, “corona”, of a product of proper metric spaces.     

A criterion for embedding of anisotropic Sobolev–Morrey spaces into the space of uniformly continuous functions

We prove the embedding theorems of the Sobolev–Morrey spaces into the space of uniformly continuous functions so extending the classical Sobolev Theorems.     

Uniformly summing sets of operators on spaces of vector–valued continuous functions

Let X, Y be Banach spaces. We say that a set is uniformly p–summing if the series is uniformly convergent for whenever (x_n) belongs to . We consider uniformly summing sets of operators defined on a -space and prove, in case X does not contain a copy of c₀, that is uniformly summing iff is, where T (φ x) = (T^#φ) x for all and x∈X. We also characterize the sets with the property that is uniformly summing viewed in . Received: 1 July 2005