A Quaternionic Bernstein Theorem

We prove a four-dimensional version of a Bernstein’s theorem, with complex polynomials being replaced by quaternionic polynomials. Moreover, using an Almansi-type decomposition of polynomials, we formulate the quaternionic Bernstein’s inequality in terms of four-dimensional zonal harmonics and Gegenbauer polynomials.


Introduction
In 1930, S. Bernstein [3] proved the following result: Theorem (A).Let p(z) and q(z) be two complex polynomials with degree of p(z) not exceeding that of q(z).If q(z) has all its zeros in {|z| ≤ This note deals with a four dimensional version of such classic results, with complex polynomials being replaced by quaternionic polynomials.The extension of Bernstein's inequality to the quaternionic setting has already appeared in [5].The proof given there is based on a quaternionic version of the Gauss-Lucas Theorem.Unfortunately, this last result is valid only for a small class of quaternionic polynomials, as it has been recently showed in [8], where another version of the Gauss-Lucas Theorem, valid for every polynomial, has been proved.Recently, a different proof of the quaternionic Bernstein's inequality has been given in [13], using the Fejér kernel and avoiding the use of the Gauss-Lucas Theorem.
We refer the reader to [6] and [9] for definitions and properties concerning the algebra H of quaternions and many aspects of the theory of quaternionic slice regular functions, a class of functions which includes polynomials and convergent power series.The ring H[X] of quaternionic polynomials is defined by fixing the position of the coefficients with respect to the indeterminate X (e.g. on the right) and by imposing commutativity of X with the coefficients when two polynomials are multiplied together (see e.g.[11, §16]).Given two polynomials P, Q ∈ H[X], let P • Q denote the product obtained in this way.A direct computation (see [11, §16.3]) shows that if P (x) = 0, then In particular, if P has real coefficients, then (P • Q)(x) = P (x)Q(x).In this setting, a (left) root or zero of a polynomial where H * := H \ {0}.In particular, for any imaginary unit I ∈ H, S I = S is the 2-sphere of all imaginary units in H.It it is well-known (see e.g.[6, §3.3]) that if P ≡ 0, the zero set V (P ) consists of isolated points or isolated 2-spheres of the form (2).
We show that the quaternionic version of Theorem (A) holds true after imposing a necessary assumption on the second polynomial.We require that Q ∈ H[X] has every coefficients belonging to a fixed subalgebra of H.This restricted version of the Bernstein Theorem is however sufficient to deduce the quaternionic Bernstein's inequality, i.e. the analog of Theorem (B).In Section 3, we restate the inequality in terms of four-dimensional zonal harmonics and Gegenbauer polynomials.To obtain this form, we use results of [12] to obtain an Almansi type decomposition of a quaternionic polynomial.

Bernstein Theorem and inequality
Let I ∈ S and let C I ⊂ H be the real subalgebra generated by I, i.e. the complex plane generated by 1 and I.If C I contains every coefficient of P ∈ H[X], then we say that P is a C I -polynomial.Every C I -polynomial P is one-slice-preserving, i.e.P (C I ) ⊆ C I .If this property holds for two imaginary units I, J, with I = ±J, then it holds for every unit and P is called slice-preserving.This happens exactly when all the coefficients of P are real.
Let Theorem 1.Let P, Q ∈ H[X] be two quaternionic polynomials with degree of P not exceeding that of Q. Assume that there exists polynomials and then they can be identified with elements of In view of Rouché's Theorem for polynomials in Proof.Let M = P and apply the previous theorem to P (X) and Q(X) = M X d .Since Q is slice-preserving, the thesis of Theorem 1 holds for every I ∈ S.
The inequality of Corollary 2 is best possible with equality holding if and only if P is a multiple of the power X d .Proof.We can assume that P (X) is not constant.Let b = P ′ (y) −1 and set Q(X) This means that the C I -polynomial Q I , considered as an element of C I [X], satisfies the equality in the classic Bernstein's inequality.The same inequality implies that This implies that π I (a d ) = c, π I (a k ) = 0 for each k = 1, . . ., d − 1 and Q can be written as Q This inequality forces Q to be the zero polynomial and then We now show that in Theorem 1, the assumption on Q to be one-slice-preserving is necessary.

Bernstein inequality and zonal harmonics
Since the restriction of a complex variable power z m to the unit circumference is equal to cos(mθ) + i sin(mθ), the classic Bernstein inequality for complex polynomials can be restated in terms of trigonometric polynomials.In this section we show that a similar interpretation is possible in four dimensions, by means of an Almansi type decomposition of quaternionic polynomials and its relation with zonal harmonics in R 4 .Quaternionic polynomials, as any slice regular function, are biharmonic with respect to the standard Laplacian of R 4 [12, Theorem 6.3].In view of Almansi's Theorem (see e.g.[1, Proposition 1.3]), the four real components of such polynomials have a decomposition in terms of a pair of harmonic functions.The results of [12] can be applied to obtain a refined decomposition of the polynomial in terms of the quaternionic variable.
In the following we will consider polynomials in the four real variables x 0 , x 1 , x 2 , x 3 of the form A(x) = d k=0 Z k (x)a k , with quaternionic coefficients a k ∈ H.They will be called zonal harmonic polynomials with pole 1.All these polynomials have an axial symmetry with respect to the real axis: for every orthogonal transformation T of H ≃ R 4 fixing 1, it holds A • T = A.
Proposition 5 (Almansi type decomposition).Let P ∈ H[X] be a quaternionic polynomial of degree d.There exist two zonal harmonic polynomials A, B with pole 1, of degrees d and d − 1 respectively, such that The restrictions of A and B to the unit sphere S 3 are spherical harmonics depending only on x 0 = Re(x).
Proof.Let P (X) = d k=0 X k c k .Formula (7) follows immediately from (6) setting The restriction of Z k (x) to the unit sphere S 3 is equal to the Gegenbauer (or Chebyshev of the second kind) polynomial C k (x 0 ), where x 0 = Re(x) (see [12, Corollary 6.7(e)]).This property implies immediately the last statement.Remark 6.The zonal harmonics A and B of the previous decomposition can be obtained from P through differentiation.Since P (x)−P (x) = A(x)−xB(x)−A(x)+xB(x) = 2 Im(x)B(x), the function B is the spherical derivative of P , defined (see [7]) on H\R as P ′ s (x) = (2 Im(x)) −1 (P (x)− P (x)).In [12] it was proved that the spherical derivative of a slice regular function, in particular of a quaternionic polynomial, is indeed the result of a differential operation.Given the Cauchy-Riemann-Fueter operator Defining A and B by formulas (8) and using results from [12], it can be easily seen that the Almansi type decomposition f (x) = A(x) − xB(x) holds true for every slice regular function f , with A and B harmonic and axially symmetric w.r.t. the real axis.Observe that B = f ′ s is the spherical derivative of f and A = f • s + x 0 f ′ s , where f • s (x) = 1 2 (f (x) + f (x)) is the spherical value of f (see [7]).
Remark 10.Some of the results presented in this note can be generalized to the general setting of real alternative *-algebras, where polynomials can be defined and share many of the properties valid on the quaternions (see [7]).The polynomials of Proposition 4 can be defined every time the algebra contains an Hamiltonian triple i, j, k, i.e. when the algebra contains a subalgebra isomorphic to H (see [4, §8.1]).This is true e.g. for the algebra of octonions and for the Clifford algebras with signature (0, n), with n ≥ 2. In all such algebras we can repeat the previous proofs and get the analog of Theorem 1, as well as of the Bernstein inequality (see also [13] for this last result).

d k=1 X k− 1
ka k be the derivative of P .For every I ∈ S, let π I : H → H be the orthogonal projection onto C I and π ⊥ I = id − π I .Let P I (X) := d k=1 X k a k,I be the C I -polynomial with coefficients a k,I := π I (a k ).We denote by B = {x ∈ H | |x| < 1} the unit ball in H and by S 3 = {x ∈ H | |x| = 1} the unit sphere.
is a quaternionic polynomial of degree d, and |P ′ (y)| = d P at a point y ∈ S 3 , then P (X) = X d a, with a ∈ H, |a| = P .
e. max x∈CI ∩S 3 |Q I |CI (x)| = 1/d.Therefore the restriction of Q I to C I coincides with the function x d c, with c ∈ C I , |c| = 1/d: