The Taylor expansion of weighted monogenic functions

Abstract

In this paper, we will express weighted monogenic functions as series composed of weighted monogenic polynomials. Firstly, the definition of p order homogeneous weighted monogenic polynomials is given. In order to obtain the basis of the set composed of the above polynomials, the hypercomplex variables are introduced. Secondly, we prove the relationship between the analytic as well as weighted monogenic functions and the p order homogeneous weighted monogenic polynomials. By the relationship, the Taylor expansion of the weighted monogenic functions at a certain point is given. Then, the uniform convergence of the Taylor expansion of $E_{\omega }(x,\xi )={\displaystyle \frac{1}{\det (B{)}^{\frac{1}{2}}{\omega }_n}\frac{1}{{\rho }^n}\sum _{i,j=1}^n\overline{{\psi }_i}{A}_{\mathrm{ij}}(x_j-{\xi }_j)}$ on every compact subset of a certain domain is proved. From the above results, the uniform convergence of the Taylor expansion of arbitrary weighted monogenic function f on every compact subset of the above domain is further obtained, and the inverse theorem of Taylor expansion is also obtained. Finally, the uniqueness theorem is obtained by the Taylor expansion and the connectivity of Ω.

Keywords:

• Weighted monogenic functions
• p order homogeneous weighted monogenic polynomials
• hypercomplex variables
• Taylor expansion
• uniqueness theorem

AMS Subject Classifications:

• 30B10
• 30G35
• 32A05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Natural Science Foundation of Hebei Province [grant numbers A2020205008, A2015205012], Key Foundation of Hebei Normal University [grant number L2021Z01] and the National Natural Science Foundation of China [grant numbers 11401162,11871191,11571089].