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On the zeros of a quaternionic polynomial: An extension of the Eneström-Kakeya theorem

Abdullah Mir (2023)

Czechoslovak Mathematical Journal

We present some results on the location of zeros of regular polynomials of a quaternionic variable. We derive new bounds of Eneström-Kakeya type for the zeros of these polynomials by virtue of a maximum modulus theorem and the structure of the zero sets of a regular product established in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our results extend some classical results from complex to the quaternionic setting as well.

Pervasive algebras on planar compacts

Jan Čerych (1999)

Commentationes Mathematicae Universitatis Carolinae

We characterize compact sets X in the Riemann sphere 𝕊 not separating 𝕊 for which the algebra A ( X ) of all functions continuous on 𝕊 and holomorphic on 𝕊 ∖ X , restricted to the set X , is pervasive on X .

Pointwise and locally uniform convergence of holomorphic and harmonic functions

Libuše Štěpničková (1999)

Commentationes Mathematicae Universitatis Carolinae

We shall characterize the sets of locally uniform convergence of pointwise convergent sequences. Results obtained for sequences of holomorphic functions by Hartogs and Rosenthal in 1928 will be generalized for many other sheaves of functions. In particular, our Hartogs-Rosenthal type theorem holds for the sheaf of solutions to the second order elliptic PDE's as well as it has applications to the theory of harmonic spaces.

Polyanalytic Besov spaces and approximation by dilatations

Ali Abkar (2024)

Czechoslovak Mathematical Journal

Using partial derivatives ∂ f / ∂ z and ∂ f / ∂ z ¯ , we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree q can be approximated in norm by polyanalytic polynomials of degree at most q .

Polynomial approximations and universality

A. Mouze (2010)

Studia Mathematica

We give another version of the recently developed abstract theory of universal series to exhibit a necessary and sufficient condition of polynomial approximation type for the existence of universal elements. This certainly covers the case of simultaneous approximation with a sequence of continuous linear mappings. In the case of a sequence of unbounded operators the same condition ensures existence and density of universal elements. Several known results, stronger statements or new results can be...

Pseudo shift operators with large images

M. C. Calderón-Moreno (2002)

Colloquium Mathematicae

We give suitable conditions for the existence of many holomorphic functions f on a disc such that the image of any nonempty open subset under the action of pseudo shift operators on f is arbitrarily large. This generalizes an earlier result about images of derivatives and completes another one on infinite order differential operators.

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