Wiener Algebra for the Quaternions

Abstract

Abstract. We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-Lévy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators.

Quaternionic Wiener Algebras, Factorization and Applications

“…We begin with the scalar-valued case. The following definition is analogous to the one for the discrete Wiener algebra (see [1]).…”

“…In [1] it was shown that the discrete and continuous Wiener algebras are real Banach algebras. Along the same lines, we have: is countable, so we may enumerate it as (w m ) m∈N .…”

“…In [1], a map ω was introduced as a way of viewing the quaternionic Wiener algebras as complex matrix-valued Wiener algebras. Using the same symbol, we analogously define a map ω = ω i,j depending on the choice of an imaginary unit i ∈ S and of a j ∈ S orthogonal to i, that is, satisfying ij = −ji.…”

“…The scalar-valued case was also extended in [1] to the quaternionic setting. The initial motivation behind this paper was the desire to extend the connection between the Wiener algebras and certain functional equations (of three types: difference, convolution and more generally integro-difference) from the complex setting to the quaternionic one.…”

Quaternionic Hankel operators and approximation by slice regular functions

“…For instance, such a series does converge if the sequence of coefficients {a n } belongs to ℓ 2 (Z, H). Power series with coefficients in ℓ 1 (Z, H) are considered in [4]. Let L s (∂B) denote the space of measurable slice functions and, for 1 ≤ p < +∞, let L p (∂B) denote the space of (equivalence classes of) functions f : ∂B → H such that f p p := ∂B |f | p dΣ < ∞.…”