论文标题
降低球对球上的基础发育功能
Reduced-quaternion inframonogenic functions on the ball
论文作者
论文摘要
如果$ \ edlline {\ partial} \，f \ f \ overline {\ partial} = 0 $，$ \ mathbb {r}^3 $从$ \ mathbb {r}^3 $中的一个函数$ f $被认为是基础的。 x_1）e_1+（\ partial/\ partial x_2）e_2 $。所有基础发育功能都是双骑的。在功能的上下文中，$ f = f_0+f_1e_1+f_2e_2 $在减少的四元中采用值，我们表明，$ n $的均质多项式$ n $形成尺寸的子空间$ 6N+3 $。我们使用它们来构建一个明确的，可计算的正交基础，用于在$ \ mathbb {r}^3 $中定义在球中定义的正方形的基础式基础发育功能的希尔伯特空间。
A function $f$ from a domain in $\mathbb{R}^3$ to the quaternions is said to be inframonogenic if $\overline{\partial}\, f\overline{\partial} =0$, where $\overline{\partial} = \partial/\partial x_0+ (\partial/\partial x_1)e_1+(\partial/\partial x_2) e_2$. All inframonogenic functions are biharmonic. In the context of functions $f=f_0+f_1e_1+f_2e_2$ taking values in the reduced quaternions, we show that the homogeneous polynomials of degree $n$ form a subspace of dimension $6n+3$. We use them to construct an explicit, computable orthogonal basis for the Hilbert space of square-integrable inframonogenic functions defined in the ball in $\mathbb{R}^3$.
