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Register a new userPolynomial approximation in slice regular Fock spaces
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The main purpose of this paper is to prove some density results of polynomials in Fock spaces of slice regular functions. The spaces can be of two different kinds since they are equipped with different inner products and contain different functions. We treat both the cases, providing several results, some of them based on constructive methods which make use of the Taylor expansion and of the convolution polynomials. We also prove quantitative estimates in terms of higher order moduli of smoothness and in terms of the best approximation quantities.
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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.
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
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.
Inspired from the Cholewinski approach see [5], we investigate a family of Fock spaces in the quaternionic slice hyperholomorphic setting as well as some associated quaternionic linear operators. In a particular case, we reobtain the slice hyperholomorphic Fock space introduced and studied in [2].
In this article we introduce Variable exponent Fock spaces and study some of their basic properties such as the boundedness of evaluation functionals, density of polynomials, boundedness of a Bergman-type projection and duality.
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