实交错 *-代数上的幂和球级数
Power and spherical series over real alternative *-algebras
摘要：
研究实交错$^*$ -代数$A$上的两类级数.第一种类型是$\sum_{n} (x-y)^{\punto n}a_n$形式的系列，其中$a_n$和$y$属于$A$，而$(x-y)^{\punto n}$表示$x-y$的$n$-次幂。通过要求不定$x$与$A$的元素的交换性而获得的通常乘积。在实数和复数情况下，幂级数和分别定义实解析函数和全纯函数。在四元数的情况下，这种类型的系列在其收敛集的内部产生属于最近引入的切片正则函数类的函数。我们还表明，在一般设置的替代代数$A$，幂级数的总和是一个切片正则函数。我们还考虑第二种类型的级数，即球面级数，其中幂被不同的切片正则多项式序列取代。众所周知，在四元数上，这些级数的收敛集是一个开集，这一性质在幂级数的情况下并不总是有效的。我们刻画了任意交错$^*$ -代数$A$的这类级数的收敛集。特别地，我们证明了这些集合在$A$的二次锥中总是开的。此外，我们证明了每个切片正则函数在每个点都有一个球级数展开。
英文摘要：
We study two types of series over a real alternative $^*$-algebra $A$. The first type are series of the form $\sum_{n} (x-y)^{\punto n}a_n$, where $a_n$ and $y$ belong to $A$ and $(x-y)^{\punto n}$ denotes the $n$--th power of $x-y$ w.r.t.\ the usual product obtained by requiring commutativity of the indeterminate $x$ with the elements of $A$. In the real and in the complex cases, the sums of power series define, respectively, the real analytic and the holomorphic functions. In the quaternionic case, a series of this type produces, in the interior of its set of convergence, a function belonging to the recently introduced class of slice regular functions. We show that also in the general setting of an alternative algebra $A$, the sum of a power series is a slice regular function. We consider also a second type of series, the spherical series, where the powers are replaced by a different sequence of slice regular polynomials. It is known that on the quaternions, the set of convergence of these series is an open set, a property not always valid in the case of power series. We characterize the sets of convergence of this type of series for an arbitrary alternative $^*$-algebra $A$. In particular, we prove that these sets are always open in the quadratic cone of $A$. Moreover, we show that every slice regular function has a spherical series expansion at every point.
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