Polynomial Approximation in Quaternionic Bloch and Besov Spaces
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Advances in Applied Clifford Algebras
Polynomial Approximation in Quaternionic Bloch and Besov Spaces Sorin G. Gal∗
and Irene Sabadini
Abstract. In this paper we continue our study on the density of the set of quaternionic polynomials in function spaces of slice regular functions on the unit ball by considering the case of the Bloch and Besov spaces of the first and of the second kind. Among the results we prove, we show some constructive methods based on the Taylor expansion and on the convolution polynomials. We also provide quantitative estimates in terms of higher order moduli of smoothness and of the best approximation quantity. As a byproduct, we obtain two new results for complex Bloch and Besov spaces. Mathematics Subject Classification. Primary 30G35, Secondary 30E10. Keywords. Slice regular functions, Bloch and Besov space of the first and second kind, Approximating polynomials, Taylor expansion, Convolution polynomials, Quantitative estimates, Moduli of smoothness, Best approximation.
1. Introduction and Preliminary Results We work in the algebra of quaternions H which is a noncommutative algebra generated by the imaginary units i, j, k satisfying i2 = j 2 = k 2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j. A quaternion q is of the form q = x0 + x1 i + x2 j + x3 k, where xi ∈ R, i = 0, 1, 2, 3. The norm |q| of a quaternion is given by |q| = x20 + x21 + x22 + x23 , while the real part of q is x0 and the vector part (also called imaginary part) of q is x1 i + x2 j + x3 k. By the symbol Br , r > 0, we mean the open ball in H of radius r, namely Br = {q = x0 + ix1 + jx2 + kx3 , such that x20 + x21 + x22 + x23 < r2 }. This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29–August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen. ∗ Corresponding
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S. G. Gal, I. Sabadini
Adv. Appl. Clifford Algebras
The symbol S denotes the sphere of unit purely imaginary quaternion, i.e. S = {q = ix1 + jx2 + kx3 , such that x21 + x22 + x23 = 1}. Any I ∈ S is such that I 2 = −1 and for any fixed I ∈ S we can consider the complex plane CI = {x + Iy; | x, y ∈ R}. Note that CI , H= I∈S
in fact, any non real quaternion q is uniquely associated to the element Iq = (ix1 + jx2 + kx3 )/|ix1 + jx2 + kx3 | ∈ S and q belongs to the complex plane CIq . It is immediate that the real axis belongs to CI for every I ∈ S. In the sequel, we introduce convolution operators of a quaternion variable, and to this end we need a suitable exponential function of quaternion variable. For any I ∈ S, we consider eIt = cos(t) + I sin(t),
t ∈ R,
see [14] and we note that the Euler’s formula holds : (cos(t) + I sin(t))k = cos(kt) + I sin(kt), and therefore we have (eIt )k = eIkt . Let us recall, see [14], that for any q ∈ H\R if we set r = |q|, there exists a ∈ (0, π) such that cos(a) = xr1 and a is unique, moreover there exists a unique Iq ∈ S, such that x3 x4 x2 q = reIq a , with Iq = iy + jv + ks, y = ,v= ,s= . r sin(a) r sin(a) r sin(a
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