The main objective of this article is a constructive generalization of the holomorphic power and Laurent series expansions in C to dimension 3 using the framework of hypercomplex function theory. For this reason, deals the first part of this article with generalized Fourier & Taylor series expansions in the space of square integrable quaternion-valued functions which possess peculiar properties regarding the hypercomplex derivative and primitive. In analogy to the complex one-dimensional case, both series expansions are orthogonal series with respect to the unit ball in R^3 and their series coefficients can be explicitly (one-to-one) linked with each other. Furthermore, very compact and efficient representation formulae (recurrence, closed-form) for the elements of the orthogonal bases are presented. The latter results are then used to construct a new orthonormal bases of outer solid spherical monogenics in the space of square integrable quaternion-valued functions. This finally leads to the definition of a generalized Laurent series expansion for the spherical shell.
We establish basic results of complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds (such as algebras of Bohr's holomorphic almost periodic functions on tube domains or algebras of all fibrewise bounded holomorphic functions arising, e.g., in the corona problem for $H^\infty$). In particular, in this context we obtain results on holomorphic extension from complex submanifolds, properties of divisors, corona type theorems, holomorphic analogues of the Peter-Weyl approximation theorem, Hartogs type theorems, characterizations of uniqueness sets,…
We prove a Zalcman-Pang lemma in several complex variables and apply it to obtain several complex variables analogues of the known normality criteria like Lappan's five-point theorem and Schwick's theorem.
If f: C -> P^n is a holomorphic curve of hyper-order less than one for which 2n + 1 hyperplanes in general position have forward invariant preimages with respect to the translation t(z)=z+c, then f is periodic with period c. This result, which can be described as a difference analogue of M. Green's Picard-type theorem for holomorphic curves, follows from a more general result presented in this paper. The proof relies on a new version of Cartan's second main theorem for the Casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives, both …
This paper re-develops the Nevanlinna theory for meromorphic functions on $\mathbb C$ in the viewpoint of holomorphic forms. According to our observation, Nevanlinna's functions can be formulated by a holomorphic form. Applying this thought to Riemann surfaces, one then extends the definition of Nevanlinna's functions using a holomorphic form $\mathscr S$. With the new settings, an analogue of Nevanlinna theory on \emph{weak $\mathscr S$-exhausted Riemann surfaces} is obtained, which is viewed as a generalization of the classical Nevanlinna theory on $\mathbb C$ and $\mathbb D.$
In this paper we prove a metric version of Hartogs' theorem where the holomorphic function is replaced by a locally symmetric Hermitian metric. As an application, we prove that if the Kobayashi metric on a strongly pseudoconvex domain with $\mathcal{C}^2$ smooth boundary is a Kähler metric, then the universal cover of the domain is the unit ball.
We prove an analogue of Sadullaev's theorem concerning the size of the set where a maximal totally real manifold can meet a pluripolar set. The manifold has to be of class C-1 only. This readily leads to a version of Shcherbina's theorem for C-1 functions f that are defined in a neighborhood of certain compact sets K in the complex plane. If the graph of f on K is pluripolar, then f satisfies the Cauchy Riemann equations in the closure of the fine interior of K.
The present paper is a continuation of our work [11], where we introduced a fractional operator calculus related to a fractional $ψ-$Fueter operator in the one-dimensional Riemann-Liouville derivative sense in each direction of the quaternionic structure, that depends on an additional vector of complex parameters with fractional real parts. This allowed us also to study a pair of lower order fractional operators and prove the associated analogues of both Stokes and Borel-Pompieu formulas for holomorphic functions in two complex variables.
We show that each proper holomorphic self map of a symmetric power of the unit ball is an automorphism naturally induced by an automorphism of the unit ball, provided the ball is of dimension at least two.
In the study of holomorphic functions of one complex variable, one well-known theory is that of elliptic functions and it is possible to take the zeta-function of Weierstrass as a building stone of this vast theory. We are working the analogue theory in the natural context of higher dimensional spaces : holomorphic and elliptic Cliffordian functions.
We study the asymptotic behavior of the discrete analogue of the holomorphic map $z^a$. The analysis is based on the use of the Riemann-Hilbert approach. Specifically, using the Deift-Zhou nonlinear steepest descent method we prove the asymptotic formulae which was conjectured in 2000 by the first co-author and S.I.~Agafonov.
Let $w$ be an unbounded radial weight on the complex plane. We study the following approximation problem: find a proper holomorphic map $f: \mathbb{C}\to\mathbb{C}^n$ such that $|f|$ is equivalent to $w$. We give several characterizations of those $w$ for which the problem is solvable. In particular, a constructive characterization is given in terms of tropical power series. Moreover, the following natural objects and properties are involved: essential weights on the complex plane, approximation by power series with positive coefficients, approximation by the maximum of a holomorphic function …
We consider convergence sets of formal power series of the form $f(z,t)=\sum_{n=0}^{\infty} f_n(z)t^n$, where $f_n(z)$ are holomorphic functions on a domain $Ω$ in $\mathbb{C}$. A subset $E$ of $Ω$ is said to be a convergence set in $Ω$ if there is a series $f(z,t)$ such that $E$ is exactly the set of points $z$ for which $f(z,t)$ converges as a power series in a single variable $t$ in some neighborhood of the origin. A $σ$-convex set is defined to be the union of a countable collection of polynomially convex compact subsets. We prove that a subset of $\mathbb{C}$ is a convergence set if and o…
We prove an analogue of Alexander's Theorem for holomorphic mappings of the unit ball in a complex Hilbert space: Every holomorphic mapping which takes a piece of the boundary of the unit ball into the boundary of the unit ball and whose differential at some point of this boundary is onto is the restriction of an automorphism of the ball. We also show that it is enough to assume that the mapping is only Gateaux-holomorphic.
Real analytic functions on the boundary of the sphere which have separate holomorphic extension along the complex lines through a boundary point have holomorphic extension to the ball. This was proved in a previous preprint by an argument of CR geometry. We give here an elementary proof based on the expansion in holomorphic and antiholomorphic powers.
In this paper we obtain the full asymptotic expansion of the Bergman-Hodge kernel associated to a high power of a holomorphic line bundle with non-degenerate curvature. We also explore some relations with asymptotic holomorphic sections on symplectic manifolds
In their recent work, Gentili and Struppa proposed a different quaternionic analogue of the notion of holomorphic functions in the complex plane, called \textit{slice regular functions}, which has led to several analogues of classical theorems in complex function theory in the quaternionic setting. The quaternionic analogue of meromorphic functions is called \textit{semiregular functions}. Such a function is said to be \textit{slice preserving} if it maps each domain in a complex line to a domain in the same complex line. In this work, we deal with the relation between analytic and algebraic p…
We give a criterium of holomorphy for some type formal power series. This gives a stronger form of a Rothstein's type extension theorem for a particular ring of holomorphic functions.
We obtain an analogue of Nevanlinna theory of holomorphic mappings from a complete and stochastically complete Kähler manifold into a complex projective manifold. When certain curvature conditions are imposed, the Nevanlinna-type defect relation based on heat diffusion is derived.
Let $f$ be a continuous function on the unit circle $Γ$, whose Fourier series is $ω$-absolutely convergent for some weight $ω$ on the set of integers $\mathcal{Z}$. If $f$ is nowhere vanishing on $Γ$, then there exists a weight $ν$ on $\mathcal{Z}$ such that $1/f$ had $ν$-absolutely convergent Fourier series. This includes Wiener's classical theorem. As a corollary, it follows that if $ϕ$ is holomorphic on a neighbourhood of the range of $f$, then there exists a weight $χ$ on $\mathcal{Z}$ such that \hbox{$ϕ\circ f$} has $χ$-absolutely convergent Fourier series. This is a weighted analogue of …
In a recent work of J. Peetre and M. Englius explicit formulae were obtained for Green functions of the powers of the Möbius-invariant Laplace operator in the unit disc. In the present work their q-analogues for the first and the second powers are obtained. By the way a q-analogue of the dilogarithm in Rogers' form arises.
We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.
We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the averaged system disappears, due to the cooperation of the generators. We investigate the iteration and spectral properties of transition operators. We show that under certain conditions, in the limit stage, "singular functions on the complex plane" appear. In particular, we consider the functions $T$ which represent the probability of tending to infinity wit…
