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We link together three themes which had remained separated so far: the Hilbert space properties of the Riemann zeros, the ``dual Poisson formula'' of Duffin-Weinberger (also named by us co-Poisson formula), and the ``Sonine spaces'' of…
We simplify the proof of some widely used theoretical theorems, extending their applicability, while correcting some erroneous results. We also generalize key results and present new results that contribute to the development of the theory.…
This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to…
We study the Stokes phenomenon via hyperfunctions for the solutions of the 1-dimensional complex heat equation under the condition that the Cauchy data are holomorphic on $\mathbb{C}$ but a finitely many singular or branching points with…
This thesis is concerned with the behavior of random analytic functions. In particular, we are interested in the value distribution of Taylor series with independent random coefficients. We begin with a study of the properties of Fourier…
We establish common fixed point theorems for two pairs of weakly compatible self-mappings using an auxiliary function of two variables. Unlike classical results, our theorems do not assume continuity of the mappings and require completeness…
We determine all entire functions $f$ such that for nonzero complex values $a\neq b$ the implications $f=a \Rightarrow f' =a$ and $f' =b \Rightarrow f=b$ hold. This solves an open problem in uniqueness theory. In this context we give a…
A method of constructing an entire function with given zeros and estimates of growth is suggested. It gives a possibility to describe zero sets of certain classes of entire functions of one and several variables in terms of growth of volume…
Functions with fixed initial coefficient have been widely studied. A new methodology is proposed in this paper by making appropriate modifications and improvements to the theory of second-order differential subordination. Several…
In this paper, we employ the theory of normal families in several complex variables to obtain some uniqueness theorems for entire functions. These results extend the related works of Li and Yi [11], and Lu et al. [18] to the setting of…
We develop classical balayage (sweeping) measures and subharmonic functions on the ray system $S$ with a general origin on the complex plane $\mathbb C$. This allows for a subharmonic function $v$ on $\mathbb C$ to construct also a…
The dominant theme of this thesis is that random matrix valued analytic functions, generalizing both random matrices and random analytic functions, for many purposes can (and perhaps should) be effectively studied in that level of…
Given a finite and non-empty set $X$ and randomly selected specific functions and relations on $X$, we investigate the existence and non-existence of fixed points and reflexive points, respectively. First, we consider the class of…
In this talk, I will discuss the use of harmonic functions to study the geometry and topology of complete manifolds. In my previous joint work with Luen-fai Tam, we discovered that the number of infinities of a complete manifold can be…
Monotonicity properties of the ratio $$ \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, $$ where $f$ is an entire function are investigated. Earlier results for Euler's gamma function and other entire functions of genus 1 are…
We expand the classical balayage of measures and subharmonic functions on a system of rays $S$ with a common origin on the complex plane $\mathbb C$. This allows for an arbitrary subharmonic function $v$ of finite order on $\mathbb C$ build…
We study the distribution of zeros of general solutions of the Airy and Bessel equations in the complex plane. Our results characterize the patterns followed by the zeros for any solution, in such a way that if one zero is known it is…
In this paper we study the existence and uniqueness of fixed points of a class of mappings defined on complete, (sequentially compact) cone metric spaces, without continuity conditions and depending on another function.
Applying techniques originally developed for systems lacking a variational structure, we establish conditions for the existence of solutions in systems that possess this property but their energy functional is unbounded both lower and…
We consider the statistical distribution of zeros of random meromorphic functions whose poles are independent random variables. It is demonstrated that correlation functions of these zeros can be computed analytically and explicit…