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Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as $\mathrm{𝔰𝔩}(2,ℂ)$-modules. As finite-dimensional irreducible $\mathrm{𝔰𝔩}(2,ℂ)$-modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Gürlebeck in [3]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.

We shall show that every differential operator of 2-nd order in a real separable Hilbert space can be decomposed into a regular and an irregular operator. Then we shall characterize irregular operators and differential operators satisfying the maximum principle. Results obtained for the Lévy laplacian in [3] will be generalized for irregular differential operators satisfying the maximum principle.

We shall prove the following statements: Given a sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ in a Banach space $𝐗$ enjoying the weak Banach-Saks property, there is a subsequence (or a permutation) ${\left\{b_n\right\}}_{n=1}^{\infty }$ of the sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ such that $$\underset{n\to \infty }{lim}\frac{1}{n}\sum _{j=1}^n{b}_j=a$$ whenever $a$ belongs to the closed convex hull of the set of weak limit points of ${\left\{a_n\right\}}_{n=1}^{\infty }$. In case $𝐗$ has the Banach-Saks property and ${\left\{a_n\right\}}_{n=1}^{\infty }$ is bounded the converse assertion holds too. A characterization of reflexive spaces in terms of limit points and cores of bounded sequences is also given. The motivation for the...

Since 1970’s B. Fuglede and others have been studying finely holomorhic functions, i.e., ‘holomorphic’ functions defined on the so-called fine domains which are not necessarily open in the usual sense. This note is a survey of finely monogenic functions which were introduced in (Lávička, R., A generalisation of monogenic functions to fine domains, preprint.) like a higher dimensional analogue of finely holomorphic functions.