On the global operator and Fueter mapping theorem for slice polyanalytic functions

Abstract

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.

Approximation by Convolution Polyanalytic Operators in the Complex and Quaternionic Compact Unit Balls

“…Finally, we note that, in (9) we have to take z = 0 because z = 0 cannot be represented as function of ϕ. The theorem is proved.…”

“…The quaternionic cases in Sects. 4 and 5 are motivated by the recent introduction of the class of polyanalytic functions in the quater-nionic framework, see Alpay-Diki-Sabadini [7][8][9], Alpay-Colombo-Diki-Sabadini [6].…”

On a Polyanalytic Approach to Noncommutative de Branges–Rovnyak Spaces and Schur Analysis

“…x∈ H with all the components f k which are slice regular functions and n is the order of poly-analyticity. The third approach consists in considering slice polyanalytic functions as subclass of the null solutions of the n-th power of a global operator with non-constant coefficients, see [13]. The study in this paper is in the quaternionic context and it is based on some polynomials (P n (x)) n≥0 where…”

Poly slice monogenic functions, Cauchy formulas and the PS-functional calculus

“…with two functions F 0 , F 1 : U → R n+1 that satisfy (12). If in addition F 0 and F 1 are in C M (U ) and satisfy the poly Cauchy-Riemann equations of order M ∈ N…”

“…In fact, here we extend the slice monogenic function theory and its S-functional calculus to the poly slice monogenic setting. The quaternionic counterpart of the function theory started with the recent works [11,12], while the functional calculus in this setting is introduced in this paper for the first time. The poly S-functional calculus, that in the following will be called the P S-functional calculus, is the polyanalytic version of the S-functional calculus.…”