On the order of the difference and sum bases of polynomials in Clifford setting. (English) Zbl 1233.30022

Let \(D\) be the generalized Cauchy-Riemann operator. A polynomial \(P(x)\) is said to be special monogenic if \(DP(x)=0\) (so, \(P(x)\) is monogenic) and there exist constants \(a_{i,j}\) that belong to the real Clifford algebra \({\mathcal A}_m\) for which \[ P(x)=\sum_{i,j}^{\text{finite}} {\bar x}^i\,x^j a_{i,j}. \] Denote by \(\{P_n(x)=\sum_k z_k(x)\,P_{n,k}\}\) the base of \(P(x),\) where \((P_{n,k})\) is the Clifford matrix of coefficients. Then we have the representation \[ z_n(x)=\sum_{k} P_k(x)\overline{P}_{n,k}\,\; \overline{P}_{n,k}\in{\mathcal A}_m, \] where \(({\overline P}_{n,k})\) is the inverse matrix of the matrix \((P_{n,k})\) and \(({\overline P}_{n,k})\) is called the Clifford matrix of operators.
In this paper the authors obtain best possible bounds for orders of two kinds of special monogenic basic sets of polynomials, the so called simple difference and simple sum bases when the associated bases have given orders. The authors prove interesting results concerning relations between Cannon’s function and Whittaker’s order by discussing the \(T_{\rho}\) property of both difference simple base and sum simple base in the Clifford setting.


30G35 Functions of hypercomplex variables and generalized variables
41A10 Approximation by polynomials
Full Text: DOI


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