Approximation by polynomials in Bergman spaces of slice regular functions in the unit ball

Abstract

In this paper, we show that the set of quaternionic polynomials is dense in the Bergman spaces of slice regular functions in the unit ball, both of the first and of the second kind. Several proofs are presented, including constructive methods based on the Taylor expansion and on the convolution polynomials. In the last case, quantitative estimates in terms of higher‐order moduli of smoothness and of best approximation quantity are obtained.

Polynomial Approximation in Slice Regular Fock Spaces

“…According to [3], in the framework of slice regular functions, can be considered two kinds of (weighted) Bergman spaces. In the recent paper [12], the properties of density for quaternionic polynomials in these kinds of spaces were obtained. Let us mention here that in the complex case, convolution polynomials were used to obtain constructive approximation results in complex Bergman spaces, see [8].…”

“…To this end, we note that since the restriction of f r is entire on C I 0 , it is continuous, which evidently implies that pointwise we have lim r→1 − f (rq) = f (q), for all q ∈ C I 0 . Now, for f ∈ F p α,I 0 (H), by setting rq = w and taking into account that as in the proof of Theorem 2.1 in [12] (see also [2]), we have dm I 0 (q) = 1 r 2 dm I 0 (w), we obtain…”

“…We will reason similar to the complex case in the proof of Proposition 2.9, p. 38 in [20], taking into account that f : H → H can be written componentwise as 4 and that applying the Lemma 3.17, p. 66 in [15] to f is equivalent to apply it to each real-valued function of four real variables f k (x 1 , x 2 , x 3 , x 4 ), k = 1, 2, 3, 4. By using the componentwise form, since f is entire slice regular it follows that it is continuous on H and it is immediate that lim r→1 −1 f (rq) = f (q), for all q ∈ H. Now, for f ∈ F p α (H), changing the variable rq = w and taking into account that as in the proof of Theorem 2.1 in [12], we have dm(q) = 1 r 4 dm(w), we obtain…”