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

In this paper, by using the convolution method, we obtain quantitative results in terms of various moduli of smoothness for approximation of polyanalytic functions by polyanalytic polynomials in the complex unit disc. Then, by introducing the polyanalytic Gauss–Weierstrass operators of a complex variable, we prove that they form a contraction semigroup on the space of polyanalytic functions deﬁned on the compact unit disk. The quantitative approximation results in terms of moduli of smoothness are then extended to the case of slice p -polyanalytic functions on the quaternionic unit ball. Moreover, we show that also in the quaternionic case the Gauss–Weierstrass operators of a quaternionic variable form a contraction semigroup on the space of polyanalytic functions deﬁned on the compact unit ball.


Introduction and Preliminaries
Given a natural number p, a complex-valued function f of a complex variable is called a p-analytic or polyanalytic of order p, in an open set where ∂ p is the p-power of the Cauchy-Riemann operator, i.e. ∂ = ∂/∂z. It can be proved that f necessarily has the representation where f 0 , . . . , f p−1 are analytic (holomorphic) in G. If all f 0 , . . . , f p−1 are polynomials, then f is called p-analytic polynomial and the degree deg( f ) (with respect to z) of a p-analytic polynomial f , is defined as max{deg( f j ); j = 0, . . . , p − 1}. For simplicity, everywhere in the paper we assume that the degree of f is considered with respect to z. The concept of a polyanalytic function was introduced in 1908 by Kolossov, see [34][35][36][37], to study elasticity problems. This stream of research was later on continued by his student Muskhelishvili, see the book [42].
It is also worth mentioning the early paper by Pompeiu [47] and, one decade later, the work of Burgatti, see [12], and in the thirties Teodorescu's doctoral dissertation, see [48]. However, a systematic study of polyanalytic functions was done by the Russian school under the supervision of Balk, see his book [10].
Although the representation (1) suggests that the building blocks of polyanalytic functions are holomorphic functions, the class of polyanalytic functions presents deep differences from the class of holomorphic functions, see [10] for more information.
For functions p-analytic in G and continuous in G, the available results on approximation using p-analytic polynomials are of qualitative type.
Thus, the first goal of the present paper is to obtain, in Sect. 2, quantitative uniform approximation results in terms of various moduli of smoothness in the particular case when G = D-the open unit disk in C. Section 3 introduces the polyanalytic Gauss-Weierstrass complex operators, for which one proves that they form a contraction semigroup on the space of polyanalytic complex functions in the unit disk. We then move to the quaternionic case, and in Sect. 3 we consider the particular case when G = B is the open unit ball and we obtain quantitative results, similar to those ones in the complex case, in uniform approximation by slice quaternionic polyanalytic polynomials. Finally, Sect. 5 deals with similar properties for the polyanalytic Gauss-Weierstrass quaternionic operators. 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].

Approximation by Polyanalytic Polynomials
For p ∈ N and D the open unit disk in C, let us denote by H p (D) the space of all p-analytic functions in D and continuous in D, endowed with the uniform norm · . Definition 2.1 Let K n (v) be an even trigonometric polynomial of degree d n ∈ N, with K n (v) ≥ 0, for all v ∈ [0, 2π ] and n ∈ N.
For i = √ −1, f ∈ H p (D) and n ∈ N, let us define the convolution operator where By the formula in (1), it is immediate that L n ( f )(z) can be written in the form Let us set The first main result is the following.
In addition, if there exists a constant M > 0 (independent of n) and α n → +∞, such that That is, since lim n→∞ α n = +∞, it follows that L n ( f ) → f uniformly on D.
Proof Taking z = re i x ∈ D, 0 ≤ r ≤ 1, we can write is a trigonometric polynomial of degree d n , we have the representation where for each j = 0, . . . , p − 1, we can write with c ( j) l ∈ C. Inspired by formula (4), we compute Now, by integrating as in formula (4) and taking into account that 2π 0 e iv (λ+k) it easily follows that for each fixed j ∈ {0, . . . , p − 1}, 2π 0 S 1 dv reduces to a finite sum of powers of z l (for those l, q ≥ 0 with l + q = j) and 2π 0 S 2 dv reduces to a finite sum of powers of z l (for those l, q ≥ 0 with l − q = j). Moreover, it is clear that the maximum for l is obtained in the sum S 2 and it is given from the formula l − q = j, i.e. l = q + j, therefore is attained for for j = p − 1.
In other words, this means that L n ( f )(z) is a p-analytic polynomial of degree d n + p − 1 and this proves the first part of the theorem.
To prove the second part of the theorem, we use formula (2) and we have and the proof is complete.

Remark 2.3
By taking in Theorem 2.2 as K n (v) other approximate units, we will get various other approximation results. For example, if we choose as K n (v) the so-called Jackson's kernel, then we deduce the following result on the p-analytic polynomials L n ( f ) given by (2): where n = [n/2] + 1, n ∈ N and therefore (see [22]), then, L n ( f )(z), n ∈ N are p-analytic polynomials of degree n + p − 1, which satisfy the quantitative estimate Proof In this case, since by relation (5) in Lorentz [38, p. 57], the case r = 2, we have it follows that we can choose α n = n, for all n ∈ N, which by the estimate (5) leads to But by [38, pp. 55-56] K n (v) is a trigonometric polynomial of degree n, which by Theorem 2.2 implies that the degree of L n ( f )(z) is n + p − 1, proving the corollary.

Remark 2.5
The step 1/n inside the modulus of continuity in Corollary 2.4 can be put in accordance with the degree n + p − 1 of the p-analytic polynomials . Indeed, it is good enough to choose C p > p, which will imply that Let us define higher moduli of smoothness of f ∈ H p (D) by where q ∈ N, q ≥ 2 and The error estimate in the approximation of f by L n ( f )(z) as in Corollary 2.4 can be expressed in terms of ω 2 ( f ; δ) ∂D , as follows.
Writing z = re i x , we now easily get where for the last inequality we have applied the relations in Lorentz [38, p. 56].
Here we also have applied the property ∂D . Also, notice that for ω 2 we used here a definition equivalent to (6) (in fact it is obtained from (6) by the simple substitution x + h := y) The theorem is proved.
More generally, let us attach to f ∈ H p (D) the Jackson-type convolution operator given by the formula where r is the smallest integer for which r ≥ (q + 3)/2, q ∈ N and [38, p. 57] K n,r is a trigonometric polynomial of degree n. Since f ∈ H p (D), by using formula (1), we immediately obtain Reasoning as in the proof of Theorem 2.2, we have that each I n,q ( f )(z) is a p-analytic polynomial.
, then the p-analytic polynomials I n,q ( f )(z) are of degree n + p − 1 and give the error estimate Proof As in [38, pp. 57-58] by taking into account the formula (7) and denoting z = re i x , we get The theorem is proved.

Remark 2.8
Reasoning as in Remark 2.5, the estimate in Theorem 2.7 can be replaced by one of the form

Remark 2.9
For fixed order p and degree m, let us denote by P p,m the class of all p-analytic polynomials of degree ≤ m and for f ∈ H p (D) let us denote by the best approximation of f by p-analytic polynomials of degree ≤ m, where · D denotes the uniform norm. Concerning this quantity E p,m ( f ), there exist two interesting open problems: one is to find the degree of E p,m ( f ) for p-analytic functions in various subclasses of H p (D), and the second one is, for given f , p and m, to prove the existence of P * p,m ∈ P p,m with E p,m ( f ) = f − P * p,m D and even to construct with C > 1 a constant independent of m (and possibly also independent of f ).
In the first case, it is known for example that for Gevrey polyanalytic classes of functions f of order p, the degree of E p,m ( f ) was obtained in [51].

Polyanalytic Gauss-Weierstrass Complex Operators
In this section we deal with the approximation properties of the convolution based on the classical Gauss-Weierstrass kernel given by K t (u) = e −u 2 /(2t) , u ∈ R, t > 0, by introducing the polyanalytic Gauss-Weierstrass complex operator and showing that the family of these operators has all the properties of a semigroup on the space of polyanalytic functions of a given order. More exactly, if f ∈ H p (D), then the p-analytic Gauss-Weierstrass complex operator is defined by We have: (ii) The following estimate holds: where C > 0 is a constant independent of t, z and f . (iii) The following estimate holds: where C s > 0 is a constant depending on f , independent of z and t and V s is any neighborhood of s. (iv) The operator W t is a contraction, that is, is given by the formula Proof (i) We obtain where for the last equality we used, for example, the formula in Theorem 2.2.1 (i) in the book [25, p. 27] (see also [26,Thm. 2
It remains to prove that W t ( f )(z) is continuous on all of D. In this sense, let z 0 ∈ D and consider a sequence z n ∈ D, n ∈ N, with z n → z 0 as n → ∞.
We get Therefore, passing to the limit with n → ∞, since f is continuous on D it follows from the continuity of W t ( f )(z) for z ∈ D.
(ii) We obtain Let us set √ t = a, √ s = b. By the mean value theorem, there is a value c ∈ (a, b), such that which combined with the fact that immediately implies the desired inequality for W t .
(iv) Since If z ∈ D, z = re iϕ , 0 < r < 1, then by (i), we can write It is easy to see that , for all t, s > 0. If z is on the boundary of D, then we may take a sequence (z n ) n∈N of points in D with lim n→∞ z n = z and we apply the continuity property from the above point (i). Also, denoting W t ( f )(z) by T (t)( f ), it is easy to see that the property lim t 0 T (t)( f ) = f , the continuity of T (·) and its contraction property follow from (ii), (iii) and (iv), respectively. Therefore, all these facts show that (W t , t ≥ 0) is a (C 0 )-contraction semigroup of linear operators on the space H p (D). Furthermore, since from (i) the above series representation for W t ( f )(z) is uniformly convergent in any compact disk included in D, it can be differentiated term by term, with respect to t and ϕ. Then, we easily get that Also, from the same series representation, it is easy to see that 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.

Approximation by Slice Quaternionic Polyanalytic Polynomials
The analogue of polyanalytic functions in the slice quaternionic setting have been introduced in [7-9] and subsequent papers.
To explain our results we need to introduce the necessary definitions and notation. The skew field of quaternions is defined to be where the imaginary units satisfy the relations In H the conjugate and the norm of q are defined respectively by The set contains all the imaginary units, namely all the elements q such that q 2 = −1. Any quaternion q ∈ H \ R can be written in a unique way as q = x + I y for some real numbers x and y > 0, and imaginary unit I ∈ S, in fact For every I ∈ S, we set C I = R + RI which is isomorphic to the complex plane C. It is immediate that H = I ∈S C I . In this work, we are interested in the specific case of functions defined on the unit ball B = {q ∈ H; |q| < 1} and in this case slice p-polyanalytic functions are of the form where f j (q) = +∞ l=0 q l c where the series is convergent in B, i.e., f j (q) is a slice regular function. In particular, f j (q) can be a polynomial and if f j (q) is a polynomial for all j = 0, . . . , p − 1 we say that f is a slice p-polyanalytic polynomial whose degree deg( f ) is defined as the maximum degree of the f j 's. We refer the reader to [16,32] for more information on this class of functions and to [30] for a summary of the approximation results in this framework.
To introduce the corresponding convolution operators of a quaternion variable, we need a suitable exponential function of a quaternion variable. For any I ∈ S, we choose the following well-known definition for the exponential: e I t = cos(t) + I sin(t), t ∈ R, see [33]. The Euler's formula holds: (cos(t) + I sin(t)) k = cos(kt) + I sin(kt), and therefore we can write [e I t ] k = e I kt .
For any q ∈ H \ R, let r := q ; then, see [33], there exists a unique a ∈ (0, π) such that cos(a) := x 1 /r and a unique I q ∈ S, such that q = re I q a , with I q = iy + jv + ks, y = x 2 r sin(a) , v = x 3 r sin(a) , s = x 4 r sin(a) . Now, if q ∈ R, then we choose a = 0, if q > 0 and a = π if q < 0, and as I q we choose an arbitrary fixed I ∈ S. So that if q ∈ R \ {0}, then again we can write q = q (cos(a) + I sin(a)) (but with a non unique I ). The above is called the trigonometric form of the quaternion number q = 0. For q = 0 we do not have a trigonometric form for q (exactly as in the complex case). Analogously to the case of a complex variable, we can introduce the following convolution operator of a quaternionic variable. Also, let K n,r (v) be an even, classical, positive-valued, trigonometric polynomial of degree d n,r ∈ N, with K n,r (v) ≥ 0, for all v ∈ [0, 2π ] and n, r ∈ N.
For f ∈ S P p (B), q ∈ H \ R and n ∈ N, let us define the convolution operator where In this section, we will use the trigonometric kernels .
According to Lorentz [38, p. 55] they are even and positive trigonometric polynomials of degree r (n − 1), which can be written in the form with A r ,s ∈ R, for all r ∈ N, r ≥ 2 and s = 1, . . . , r (n − 1). Firstly, we prove the following: Proof Since q, q, e I q v and e −I q v are on the slice C I q determined by I q , they commute. Therefore it is immediate that L n,r ( f )(q) can be written in the form From formula (15), we need to calculate 2π 0 e −I q jv · f j (ze I q v ) · K n (v)dv.
Again from the fact that I q , e −I q jv , q l and e I q lv commute, we get Now, by integrating the sum S 1 with respect to v from 0 to 2π , it easily follows that the only terms which are different from zero are the terms for which l = j − 1, with the maximum value l = p − 1 + nr − r , while integrating S 2 we get that all its terms are equal to zero. Consequently, formula (15) shows that L n,r ( f )(q) is a slice ( p − 1)-polyanalytic polynomial of degree p − 1 + nr − 1. Denoting we are now in position to prove the first main result of this section. Theorem 4.2 For f ∈ S P p (B), q ∈ H \ R and n ∈ N, let us define the convolution operator Then L n,2 ( f )(q), n ∈ N are slice p-polyanalytic polynomials of degree n + p − 1, which satisfy the quantitative estimate where M > 0 is a constant independent of q, f and n.
Proof Taking q = re I q x ∈ B, we can write It follows that and using calculations similar to those in the proof of the second part of Theorem 2.2 (by replacing α n , i, z by n, I q , q respectively) we obtain the required estimate.

Remark 4.3
Reasoning as in Remark 2.5, the step 1/n inside the modulus of continuity in Theorem 4.2 can be put in accordance with the degree n + p − 1 of the slice ppolyanalytic polynomials L n,2 ( f )(q), since there exists C p > 0 (depending only on p) such that ω 1 ( f ; 1/n) B ≤ C p ω 1 ( f ; 1/(n + p − 1)) B , for all n ∈ N and all f ∈ S P p (B). Indeed, it is good enough to choose C p > p, which will imply that Now, if we define higher moduli of smoothness of f ∈ S P p (B) by where m ∈ N, m ≥ 2 and then the error estimate in the approximation of f by L n,2 ( f )(q) as in Theorem 4.2, can be expressed in terms of ω 2 ( f ; δ) ∂B , as follows.

Theorem 4.4 For f ∈ S P p (B)
, the slice p-polyanalytic polynomials L n,2 ( f )(q) defined in (16) give the estimate where C > 0 is an absolute constant.
Proof Indeed, we can write Then, writing q = re I q x , and reasoning as in the proof of Theorem 2.6 (where i, z must be replaced by I q , q respectively) we now easily get the assertion.
More generally, for f ∈ S P p (B), let us attach the generalized Jackson-type convolution operator given by the formula where r is the smallest integer for which r ≥ (m + 3)/2, m ∈ N, and , n = n r + 1, with λ n ,r determined by π −π K n,r (v)dv = 1. According to [38, p. 57] K n,r is an even trigonometric polynomial of degree n.
We now set K n,r (v) = n s=0 A r ,s cos(s · v) and we consider f ∈ S P p (B). Using reasonings and calculations similar to those ones in the proof of Lemma 4.1, by formula (12) we immediately obtain (ii) The following estimate holds: where C > 0 is a constant independent of t, q and f . (iii) The following estimate holds: where C s > 0 is a constant depending on f , independent of q and t and V s is any neighborhood of s. (iv) The operator W t is a contraction, that is, for all t > 0, f ∈ SP p (B).
(v) (W t , t ≥ 0) is a (C 0 )-contraction semigroup of linear operators on the space S P p (B) and the unique solution u(t, q) ∈ S P p (B), for each fixed t, of the Cauchy problem ∂v ∂t (t, q) = 1 2 ∂ 2 v ∂ϕ 2 (t, q), (t, q) ∈ (0, +∞) × B, q = re I q ϕ , q = 0, (21) v(0, q) = f (q), q ∈ B, f ∈ S P p (B), is given by the formula Proof (i) Since q j , q l , e I q lu , e −I q ju are on the same slice, they commute and therefore we obtain