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Register a new userBoundary Julia theory for slice regular functions
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The theory of slice regular functions is nowadays widely studied and has found its elegant applications to a functional calculus for quaternionic linear operators and Schur analysis. However, much less is known about their boundary behaviors. In this paper, we initiate the study of the boundary Julia theory for quaternions. More precisely, we establish the quaternion
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Slice regular functions have been extensively studied over the past decade, but much less is known about their boundary behavior. In this paper, we initiate the study of Julia theory for slice regular functions. More specifically, we establish the quaternion
In this paper, we study the (possible) solutions of the equation $exp_{*}(f)=g$, where $g$ is a slice regular never vanishing function on a circular domain of the quaternions $mathbb{H}$ and $exp_{*}$ is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function $f$ which satisfies $exp_{*}(f)=g$ is called a $*$-logarithm of $g$. We provide necessary and sufficient conditions, expressed in terms of the zero set of the ``vector part $g_{v}$ of $g$, for the existence of a $*$-logarithm of $g$, under a natural topological condition on the domain $Omega$. By the way, we prove an existence result if $g_{v}$ has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of $g_{v}$ are finite, the domain is either the unit ball, or $mathbb{H}$, or $mathbb{D}$ and a further condition on the ``real part $g_{0}$ of $g$ is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of $g_{v}$, in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.
Given a quaternionic slice regular function $f$, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of $f$ with those of its slice derivative $partial_{c}f$ obtaining a countable family of differential equations satisfied by any slice regular function. The results are proved in all details and are accompanied to several examples. For some of the results, we also give alternative proofs.
A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $kappa_1, ldots, kappa_N$, quaternions $p_1, ldots, p_N$ all of modulus $1$, so that the $2$-spheres determined by each point do not intersect and $p_u eq 1$ for $u = 1,ldots, N$, and quaternions $s_1, ldots, s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$lim_{substack{rrightarrow 1 rin(0,1)}} s(r p_u) = s_uquad {rm for} quad u=1,ldots, N,$$ and $$lim_{substack{rrightarrow 1 rin(0,1)}}frac{1-s(rp_u)overline{s_u}}{1-r}lekappa_u,quad {rm for} quad u=1,ldots, N.$$ Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.
In this paper we prove two Bloch type theorems for quaternionic slice regular functions. We first discuss the injective and covering properties of some classes of slice regular functions from slice regular Bloch spaces and slice regular Bergman spaces, respectively. And then we show that there exits a universal ball contained in the image of the open unit ball $mathbb{B}$ in quaternions $mathbb{H}$ through the slice regular rotation $widetilde{f}_{u}$ of each slice regular function $f:overline{mathbb{B}}rightarrow mathbb{H}$ with $f(0)=1$ for some $uin partialmathbb{B}$.
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