A priori estimates for truncation error of continued fractionsK(1/b n )

A priori estimates are obtained for the truncation error of continued fractions of the formK(1/b _n ), with complex elementsb _n . The method employed is based on the calculation of bounds for successive diameters of a sequence of nested disks, where then-th approximant of the continued fraction is contained in then-th disk. Numerical examples are given to illustrate useful procedures and typical error estimates for continued fraction expansions of the complex logarithm and the ratio of consecutive Bessel functions.This research was supported by the National Science Foundation under Grant No. GP-9009 and by the United States Air Force through the Air Force Office of Scientific Research under Grant No. AFOSR-70-1888.     

Taylor Expansions of Distributions

Classical Taylor expansions of holomorphic functions in the complex plane are extended to distributions in Rⁿand in domains     

Existence of Universal Taylor Series for Nonsimply Connected Domains

It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that possess “universal” Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in ℂ\Ω that have connected complement. This paper shows, for nonsimply connected domains Ω, how issues of capacity, thinness and topology affect the existence of holomorphic functions on Ω that have universal Taylor series expansions about a given point.     

Hypercyclic convolution operators on Fréchet spaces of analytic functions

A result of Godefroy and Shapiro states that the convolution operators on the space of entire functions on Cn, which are not multiples of identity, are hypercyclic. Analogues of this result have appeared for some spaces of holomorphic functions on a Banach space. In this work, we define the space holomorphic functions associated to a sequence of spaces of polynomials and determine conditions on this sequence that assure hypercyclicity of convolution operators. Some known results come out as particular cases of this setting. We also consider holomorphic functions associated to minimal ideals of polynomials and to polynomials of the Schatten-von Neumann class.     

Gaps and Fatou theorems for series in Schauder basis of holomorphic functions

We prove a gap theorem and the “Fatou change-of-sign theorem” [Fatou, P., 1906, Sèries trigonométriques e séries de Taylor. Acta Mathematica, 39, 335–400] for expansions in common Schauder basis of holomorphic functions.     

Optimal convergence rate for Yosida approximations of bounded holomorphic semigroups

In this paper, we obtain optimal bounds for convergence rate for Yosida approximations of bounded holomorphic semigroups. We also provide asymptotic expansions for semigroups in terms of Yosida approximations and obtain optimal error bounds for these expansions.     

Tensor products,reproducing kernels,and power series

Tensor products of holomorphic discrete series representations in reproducing kernel Hilbert spaces are decomposed by considering power series expansions of functions in the direction perpendicular to the diagonal in $\text{D}$ × $\text{D}$.     

Fourier expansion of holomorphic modular forms on classical lie groups of tube type along the minimal parabolic subgroup

For holomorphic modular forms on tube domains, there are two types of known Fourier expansions, i.e. the classical Fourier expansion and the Fourier-Jacobi expansion. Either of them is along a maximal parabolic subgroup. In this paper, we discuss Fourier expansion of holomorphic modular forms on tube domains of classical type along the minimal parabolic subgroup. We also relate our Fourier expansion to the two known ones in terms of Fourier coefficients and theta series appearing in these expansions.     

Оценка емкости множества особенностей функций, заданных своим раэложением в непрерывную дробь

We obtain representations of Hankel’s determinants of functions defined by continued fraction expansions, via the parameters of the fraction. As a corollary of these representations, we prove that functions defined by continued fraction expansions of a certain type cannot be (uniquely) meromorphic continued beyond the convergence domain.     

Series expansions of analytic functions

Many of the classical polynomial expansions of analytic functions share a common property: the space of “expandable” functions is a Banach space isometrically isomorphic to the space of complex sequences with limit 0. Under the isometries, these polynomial expansions all correspond to essentially the same biorthogonal expansion in this sequence space. Sufficient conditions for such an isometry to exist are obtained, and convergence properties of the expansions are studied. The results obtained also apply to expansions other than polynomial expansions.     

On the size of balls covered by analytic transformations

Two quantitative forms of the inverse function theorem giving estimates on the size of balls covered biholomorphically are proved for holomorphic mappings of a ball in a Banach space into the space. Also, a Bloch theorem forK-quasiconformal mappings on the open unit ball of a Banach space is given and some mapping properties ofK-quasiconformal mappings are deduced.Research supported by N. S. F. Grant GP-33117A-2.     

POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS

We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter＇s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson＇s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson＇s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.     

Normality concerning the sequence of functions

Let {f_n} be a sequence of functions meromorphic in a domain D, let {h_n} be a sequence of holomorphic functions in D, such that that h(z)→h(z), where h.(z)→0 is holomorphic in D, and let k be a positive integer. If for each n∈N~+, f_n(z)≠0 and f_n~(k)(z)-h_n(z) has at most k distinct zeros(ignoring multiplicity) in D, then {f_n} is normal in D.     

Existence of the best <Emphasis Type="Italic">n</Emphasis>-term approximants for structured dictionaries

We extend the Pizzetti formulas, i.e., expansions of the solid and spherical means of a function in terms of the radius of the ball or sphere, to the case of real analytic functions and to functions of Laplacian growth. We also give characterizations of these functions. As an application we give a characterization of solutions analytic in time of the initial value problem for the heat equation ∂ _t u = Δu in terms of holomorphic properties of the solid and/or spherical means of the initial data.     

Holomorphic Cliffordian functions

The aim of this paper is to put the foundations of a new theory of functions, called holomorphic Cliffordian, which should play an essential role in the generalization of holomorphic functions to higher dimensions. Let ℝ_0,2m+1 be the Clifford algebra of ℝ^2m+1 with a quadratic form of negative signature, be the usual operator for monogenic functions and Δ the ordinary Laplacian. The holomorphic Cliffordian functions are functionsf: ℝ^2m+2 → ℝ_0,2m+1, which are solutions ofDδ ^m f = 0. Here, we will study polynomial and singular solutions of this equation, we will obtain integral representation formulas and deduce the analogous of the Taylor and Laurent expansions for holomorphic Cliffordian functions. In a following paper, we will put the foundations of the Cliffordian elliptic function theory.     

Matrix Continued Fractions

A matrix continued fraction is defined and used for the approximation of a function known as a power series in 1/zwith matrix coefficientsp×q, or equivalently by a matrix of functions holomorphic at infinity. It is a generalization of P-fractions, and the sequence of convergents converges to the given function. These convergents have as denominators a matrix, the columns of which are orthogonal with respect to the linear matrix functional associated to . The case where the algorithm breaks off is characterized in terms of .     

Polynomial approximation and maximal convergence on Faber sets

In this paper we study various problems concerning Faber sets and polynomial approximation on Faber sets. We give various conditions for a compact setK to be a Faber set and we characterize (for a certain class of Faber sets) the range of the Faber operator. Furthermore, we study the convergence behavior of Faber expansions and more general sequences of polynomials which approximate functions that are holomorphic onK and continuous on a level curve of the normalized conformal mapping from ......-...... onto ......-K.     

Some contributions to the asymptotic theory of Bayes solutions

Summary This paper deals with the asymptotic theory of Bayes solutions in (i) Estimation (ii) Testing when hypothesis and alternative are separated at least by an indifference region, under the assumption that the observations are independent and indentically distributed. The estimation results which are partial generalizations of results of LeCam begin with a proof of the convergence of the normalized posterior density to the appropriate normal density in a strong sense. From this result we derive the asymptotic efficiency of Bayes estimates obtained from smooth loss functions and in particular of the posterior mean. The last two theorems of this section deal with asymptotic expansions for the posterior risk in such estimation problems. The section on testing contains a limit theorem for the n-th root of the posterior risk under weak conditions on the prior and the loss function. Finally we discuss generalizations and some open problems.Part of this research was done while P. J. Bickel was on leave at Imperial College, London. — This research was partially supported by National Science Foundation Grant GP-5059.This research was partially supported by National Science Foundation Grant GP-5705. Part of this research was done while J. A. Yahav was visiting the department of Statistics at Stanford University.     

On some continued fraction expansions of the Rogers–Ramanujan type

By guessing the relative quantities and proving the recursive relation, we present some continued fraction expansions of the Rogers–Ramanujan type. Meanwhile, we also give some J-fraction expansions for the q-tangent and q-cotangent functions.     

Further results on the euler and Genocchi numbers

Summary We characterize the ordinary generating functions of the Genocchi and median Genocchi numbers as unique solutions of some functional equations and give a direct algebraic proof of several continued fraction expansions for these functions. New relations between these numbers are also obtained.