Archiv der Mathematik Holomorphic approximation of L2-functions on the unit sphere in R^3

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1 Manuscript Number: Archiv der Mathematik Holomorphic approximation of L-functions on the unit sphere in R^ --Manuscript Draft-- Full Title: Article Type: Corresponding Author: Holomorphic approximation of L-functions on the unit sphere in R^ Original Article Nele De Schepper, Ph.D. UGent Ghent, BELGIUM Corresponding Author Secondary Information: Corresponding Author's Institution: UGent Corresponding Author's Secondary Institution: First Author: Nele De Schepper, Ph.D. First Author Secondary Information: Order of Authors: Nele De Schepper, Ph.D. Tao Qian, Professor Frank Sommen, Professor Jinxun Wang Order of Authors Secondary Information: Abstract: In this paper we construct an embedding of holomorphic functions in two complex variables into the unit ball in R^. This leads to a closed subspace of the L-functions on the unit sphere spanned by quaternionic polynomials for which we construct orthonormal bases and study the related convergence properties. Powered by Editorial Manager and ProduXion Manager from Aries Systems Corporation

2 Manuscript Click here to download Manuscript: DeQSW - Holomorphic approximation on the unit sphere - Arch Click here to view linked References Holomorphic approximation of L -functions on the unit sphere in R Nele De Schepper, Tao Qian, Frank Sommen and Jinxun Wang Abstract. In this paper we construct an embedding of holomorphic functions in two complex variables into the unit ball in R. This leads to a closed subspace of the L -functions on the unit sphere spanned by quaternionic polynomials for which we construct orthonormal bases and study the related convergence properties. Mathematics Subject Classification (00. C. Keywords. Quaternionic analysis, orthogonal polynomials, holomorphic signals, holomorphic polynomials.. Introduction The most widely used approximation for L -functions on the unit sphere in R m is the approximation by spherical harmonics. In concreto a spherical harmonic of degree k is a homogeneous polynomial S k (x which is harmonic, i.e. x S k (x 0, x m j x j being the Laplacian, and every f L (S m admits an orthogonal decomposition in spherical harmonics of the form f(ω k0 S k(ω, ω x x Sm. Clifford analysis forms a refinement of harmonic analysis. It starts with the construction of a Clifford algebra with generators e,..., e m and relations e j, e j e k e k e j, j k and leads to the Dirac operator x m j e j xj for which x x ; solutions of x f 0 are called monogenic functions. Spherical monogenics of degree k are then defined as homogeneous polynomials P k (x which are monogenic: x P k (x 0 and the series k0 P k(ω L (S m form the closed subspace of monogenic signals ML (S m. For any scalar function f L (S m there exists a monogenic signal k0 P k(ω g for which [g] 0 f, [. ] 0 denoting the scalar part. The monogenic function theory has a lot of interesting properties; we refer to the extended literature containing the books [,,, ] etc. One disadvantage is however the fact that the product of monogenic functions is no longer monogenic. This leads us to the idea of constructing an

3 Nele De Schepper, Tao Qian, Frank Sommen and Jinxun Wang embedding of holomorphic functions of several complex variables into the unit ball in Euclidean space. In R, it works as follows: consider Clifford generators e, e with defining relations e e and e e e e. Then the holomorphic correspondence is the map from the holomorphic functions in two complex variables g(z, z p,q0 zp zq C p,q into the corresponding series in R : g(z, z (x 0 + x p (x e e x q C p,q p,q0 with x x e + x e. This embedding leads to a closed subspace of L (S called the space of holomorphic signals on S. The first task in this paper (see section is to construct the Gram- Schmidt orthonormal basis that corresponds to the holomorphic polynomials (x 0 + x p (x e e x q. In section we also study convergence properties for these holomorphic bases and for q fixed the series converges for all x 0 + x with x 0 + x < (see Proposition.. In case both p and q are variable, we are still able to prove convergence for x 0 + x < (see Proposition... Preliminaries We will work in the algebra H of quaternions. Let e, e be two imaginary units of H, satisfying the multiplication rules e e and e e e e. The conjugation in H is determined by e e and e e. For any x x 0 + x e + x e R we also write x x 0 + x, where x x e + x e. Let S be the unit sphere in R. The space L (S consists of all functions defined on S, taking values in H, and being square integrable on S with respect to the surface area element ds. The inner product on L (S is defined by f, g : f(ξg(ξds, f, g L (S, π ξ S which leads to an induced norm given by f : f, f, f L (S. It is known that the set of polynomials is dense in L (S. Moreover, we observe that L (S span{x a 0x b (x e e x c : a, b, c N}. Since x 0 x + x, x x x, and on the unit sphere x x, we have L (S span{x p (x e e x q : p Z, q N}. In this paper we will restrict ourselves to half of the above generating set, we consider namely the set of holomorphic polynomials HP {x p (x e e x q : p, q N}. The closed formula for the orthonormalization of HP will be given. These kind of holomorphic polynomials are in fact closely related to the spherical monogenics. To see this, let us first recall some definitions.

4 Holomorphic approximation of L -functions on the unit sphere in R A quaternion-valued function f, defined in an open set Ω R, is called left monogenic in Ω if it is in Ω a null solution of D, i.e. Df 0, where the differential operator D : x0 + x ( / x 0 + e ( / x + e ( / x is the so-called Cauchy Riemann operator. A left monogenic polynomial of degree k is called a left inner spherical monogenic of degree k. The collection of all such monogenic polynomials is denoted by M k. A useful tool to construct a monogenic function from a given smooth function R Ω x f(x H is the Cauchy Kowalevski (CK extension (see [, ], given by ( x CK(f(x e x0 x 0 n f x n f(x. n! Since for any p, q N we arrive at CK(x p (x e e x q x (x p (x e e x q x p (x e e x q, p C q p,j xp j x j 0 (x e e x q j0 (A p,q (x 0, x + xb p,q (x 0, x (x e e x q, where C q p,j, A p,q and B p,q are real-valued, and x A p,q (x 0, x +xb p,q (x 0, x is an axial monogenic function ([, 0] when q 0. From Proposition. (see the next section we conclude that B k {CK(x p (x e e x q : p, q N, p + q k} is an orthogonal basis of M k with respect to the inner product on L (S. This construction should be compared with the results in [, ]. The study of the projection operator from L (S to spanhp and the related approximation problems will be our next research objectives.. Orthonormalization of HP For any p, q N, let α p,q (x x p (x e e x q (x 0 + x p (x e e x q. The aim of this section is to orthogonalize this sequence. For m N and γ >, we have (see [], p.,., formula : π cos (mθ(sin θ γ dθ γ π cos ( mπ Γ(γ + 0 Γ( m + γ Γ( + m + γ (.. The above result leads to the following proposition. Proposition.. For p, p, q, q N: α p,q, α p,q R, and when q q, or p p is odd, we have α p,q, α p,q 0.

5 Nele De Schepper, Tao Qian, Frank Sommen and Jinxun Wang Proof. Assume p p. Using spherical coordinates, set x 0 cos θ, x sin θ cos β and x sin θ sin β with 0 θ π, 0 β < π, we get ds sin θdθdβ, and α p,q (xα p,q (x (x + e e x q (x 0 x p (x 0 + x p (x e e x q (sin θ q+q (cos (q β + e e sin (q β ( cos ((p p θ x x sin ((p p θ (cos (q β e e sin (q β (sin θ q+q cos ((p p θ (cos ((q q β + e e sin ((q q β (sin θ q+q sin ((p p θ (cos (q β + e e sin (q β x x (cos (q β e e sin (q β (sin θ q+q cos ((p p θ (cos ((q q β + e e sin ((q q β (sin θ q+q sin ((p p θ (e cos ((q + q + β + e sin ((q + q + β. We thus obtain α p,q, α p,q π δ q,q cos ((p p θ(sin θ q+ dθ 0 q πγ(q + cos( p p π Γ( p p + q Γ( + p p + q δ q,q, (. where in the last line we have used (.. So, the orthogonalization of HP {α p,q : p, q N} is equivalent to the separate orthogonalization of {α s,q : s N} and {α s+,q : s N} for each fixed q N. Since by (. it follows that α s,q, α s,q and ( s s q πγ(q + Γ( s + s + q Γ( + s s + q α s +,q, α s+,q, (. α s,q q πγ(q + (Γ( + α s+,q, q we just need to consider the orthogonalization of {α s,q : s N} with q being fixed (see also Remark.. For convenience, we now change notations. Let α s,q (x α s+ (x, s N. Then according to the Gram Schmidt orthogonalization process, the sequence {α n } n can be orthogonalized by setting β : α, n β n : α n i α n, β i β i, β i β i, n. Thus {B n } : { βn β n } becomes an orthonormal polynomial system. (.

6 Holomorphic approximation of L -functions on the unit sphere in R Remark.. The orthogonalization of {α s+,q : s N} with q fixed, is then given by {xβ n }, since xf, xg f, g as xx for x S. By (. and straightforward calculations, one obtains β α, β q πγ(q + (Γ( + q, β α + q + q + α, β q πγ(q + (Γ( +, q etc. In fact, we have the following result. α, β β, β q + q +, α, β + (q β, β (q + Theorem.. Let {β n } n be defined through (., then for any n N, we have n ( n Γ( β n + k + qγ( k + n + q k Γ( + qγ( + n + q α k, and k β n (n! q πγ(q + n + (Γ( + n +. q Moreover, for any i, j N with i > j, it holds that α i, β j β j, β j ( i j ( i i j Γ( + qγ( + j + q Γ( + i + qγ( i + j + q. To prove these results, we need the following lemmas. Lemma.. For all non-negative integers s < j, we have j ( j ( k k s 0. k k0 This lemma has a close connection with the Stirling numbers of the second kind (see e.g. [], and is well-known (see for e.g. [], p., 0., formula. Lemma.. For any positive integer j, we have j ( j ( k Γ( k +k+qγ( k+j+q Γ( i + j + q Γ( + i + q Γ( i + k + q Γ( + i k + q k ( j Γ(q + j + Γ( + qγ( + q (i (i (i j +. (. Γ(q + Proof. We observe that Γ( i + j + q Γ( + i + q Γ( i + k + q Γ( ( + ( i k + q i + (j + q i + (j + q ( ( + (i + q + (i + q ( i + k + q ( + i k + q

7 Nele De Schepper, Tao Qian, Frank Sommen and Jinxun Wang is a polynomial in i of degree (j k + (k j, and so is the case of the right hand side of (.. Thus, it suffices to show that: (a The left hand side of (. has roots i,,..., j. (b The coefficients of i j in both sides are equal, namely, j k ( j k Γ( + k + qγ( k + j + q Γ(q + j + Γ( + qγ( + q. Γ(q + When i is a positive integer and less than j, we can see that Γ( + k + qγ( k + j + q Γ( i + k + qγ( + i k + q ( + (k + q( + (k + q ( i + k + q ( k + (j + q( k + (j + q ( + i k + q (. is a polynomial in k of degree (i + (j i j. Hence (a follows immediately by Lemma.. Now we prove (. by induction. The case j is clear. Suppose that (. is true for some certain j, then for the next integer j +, we get j+ ( j Γ( k + k + qγ( k + (j + + q k j+ k j k + [( j + k ( ] j Γ( k + k + qγ( k + (j + + q ( j ( k k + j + qγ( + k + qγ( k + j + q j k ( + j + q ( j ( k + k + qγ( + k + qγ( k + j + q j k ( j Γ( k + k + qγ( k + j + q ( + j + q Γ(q + j + Γ( + qγ( + q Γ(q + Γ(q + (j + + Γ( + qγ( + q, Γ(q + where in the second last line we have used the induction hypothesis. Proof of Theorem.. For n,,..., let n ( n Γ( β n + k + qγ( k + n + q k Γ( + qγ( + n + q α k, k

8 Holomorphic approximation of L -functions on the unit sphere in R where α k x k (x e e x q as before. Then from (. and (., we obtain α i, β j j ( j Γ( + k + qγ( k + j + q ( i k π q Γ(q + k Γ( + qγ( + j + q Γ( i + k + qγ( + i k + q k ( i j π q Γ( + qγ(q + j + (i (i (i j + Γ( + i + qγ( + j + qγ( i + j + q, (. which implies that α i, β j 0 for i < j. Since β i is a linear combination of α,..., α i, we conclude that β i, β j 0 for i < j, and hence it is true for all i j, since β i, β j β j, β i. We also note that β j β j, β j α j, β j (j! q πγ(q + j + (Γ( + j + q, where we have used (.. Therefore, for i j, we have α i, β j β j, β j ( i j ( i j Moreover, it is clear that n β i α n α n, β i β i β i i Consequently, n i n β n α n Γ( + qγ( + j + q Γ( + i + qγ( i + j + q. n α n, β i β i, β i β α n, β i i β n + β i i, β i β i. i α n, β i β i, β i β i, which means that {β n } n is exactly the outcome of the Gram Schmidt orthogonalization process of (... Pointwise convergence of the series For any positive integer n and q N, we let n ( n Γ( β n,q (x + k + qγ( k + n + q k Γ( + qγ( + n + q α k, k then from the previous section we know that ( {β n,q (x} {xβ n,q (x} q0 n consists an orthogonal system on the unit sphere S in R. We have the following result.

9 Nele De Schepper, Tao Qian, Frank Sommen and Jinxun Wang Proposition.. Suppose {λ n } l (i.e. n λ n <, then for each q N, β n,q (x λ n β n,q n is convergent in the open unit ball {x : x < }. Proof. Changing the order of summation, we obtain β n,q (x λ n β n n,q n ( n q+ Γ( λ + k + qγ( n k + n + q k n k πγ( + q (n!γ(q + n + α k Γ( C + k + q ( Γ( k + n + q (n! q (k! (n k! Γ(q + n + λ n α k, k nk where C q q+ πγ( +q. Observe that Γ( k + n + q (n! (n k! Γ(q + n + λ n nk Γ( + n + q (n + k! n! Γ(q + n + k + λ n+k Γ( + n + q n! n! Γ(q + n + λ n+k, (n+k! Γ(q+n+k+ since is decreasing in k. Using Cauchy Schwarz inequality, we find Γ( k + n + q (n! (n k! Γ(q + n + λ n nk (Γ( + n + q λ n n!γ(q + n + n m (Γ( lim + n + q λ n m n!γ(q + n + n (Γ( lim + m + q m ( + q λ n m!γ(q + m + λ n + q, n where we have made use of the following identity m (Γ( + n + q n!γ(q + n + (Γ( + m + q ( + q m!γ(q + m +, n

10 Holomorphic approximation of L -functions on the unit sphere in R which can be proved by induction on m, and lim m (Γ( +m+q m!γ(q+m+. Moreover, we note that Γ( + k + q (k! k q+ so the series is always convergent in x <. (k, Considering also the summation over q, we obtain the following convergence result. Proposition.. Let C q0 n λ n,q <, then β n,q (x λ n,q β n,q q0 n is convergent when x <. Proof. Again changing the order of summation, we get β n,q (x λ n,q β q0 n n,q ( ( n q+ Γ( + k + qγ( k + n + q k q0 πγ( + q (n!γ(q + n + λ n,q α k. k q0 nk So, similar to the above proof, we find β n,q (x λ n,q β q0 n n,q ( q+ Γ( C + k + q q0 k πγ( + q(k! + q x k x e e x q C ( Γ( + k + q π Γ( + q x k ( x e e x q. (. (k! k Applying Maclaurin series, we have that ( x (q+ Γ(q + + i x i Γ(q + Γ(q + + k x k i! Γ(q + (k!. i0 k Hence, (. becomes C ( x C q ( x e e x q π π ( x q0 which converges if x < and q0 ( q x x, x x <, hence if x <. Acknowledgment This work was supported by Macao FDCT 0/00/A and research grant of the University of Macau No. UL0/0-Y/MAT/QT0/FST.

11 0 Nele De Schepper, Tao Qian, Frank Sommen and Jinxun Wang References [] S. Bock, K. Gürlebeck, R. Lávička, V. Souček, Gelfand Tsetlin bases for spherical monogenics in dimension. Rev. Mat. Iberoam., (0,. [] F. Brackx, R. Delanghe and F. Sommen, Clifford analysis. Research Notes in Mathematics,, Pitman (Advanced Publishing Program, Boston, MA,. [] A. K. Common, F. Sommen, Axial monogenic functions from holomorphic functions. J. Math. Anal. Appl., (, 0. [] R. Delanghe, F. Sommen and V. Souček, Clifford algebra and spinor-valued functions. Mathematics and its Applications,, Kluwer Academic Publishers Group, Dordrecht,. [] J. Gilbert and M. Murray, Clifford algebras and Dirac operators in harmonic analysis. Cambridge University Press, Cambridge,. [] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products. Academic Press, New York - London - Toronto - Sydney - San Francisco, 0. [] K. Gürlebeck and W. Sprössig, Quaternionic and Clifford calculus for physicists and engineers. Mathematical Methods in Practice, Wiley, Chichester,. [] C. Jordan, Calculus of finite differences. Third edition. Chelsea Publishing Company, New York,. [] F. Sommen, A product and an exponential function in hypercomplex function theory. Applicable Analysis, (,. [0] F. Sommen, Special functions in Clifford analysis and axial symmetry. J. Math. Anal. Appl., 0 (, 0. Nele De Schepper Ghent University - Department of Mathematical Analysis Galglaan, B-000 Gent, Belgium nds@cage.ugent.be Tao Qian University of Macau - Faculty of Science and Technology - Department of Mathematics Taipa, Macao fsttq@umac.mo Frank Sommen Ghent University - Department of Mathematical Analysis Galglaan, B-000 Gent, Belgium fs@cage.ugent.be Jinxun Wang University of Macau - Faculty of Science and Technology - Department of Mathematics Taipa, Macao wjxpyh@gmail.com