aa r X i v : . [ m a t h . C V ] O c t Bohr’s Theorem for Monogenic PowerSeries
K. G¨urlebeck, J. Morais ∗ October 6, 2007
Abstract
The main goal of this paper is to generalize Bohr’s phenomenon fromcomplex one-dimensional analysis to higher dimensions in the frameworkof Quaternionic Analysis.
MSC 2000 : 30G35
Keywords : spherical monogenics, homogeneous monogenic polynomials, Bohr’sTheorem.
In 1914 Bohr discovered that there exists a radius r ∈ (0 ,
1) such that if a powerseries of a holomorphic function converges in the unit disk and its sum has amodulus less than 1, then for | z | < r the sum of the absolute values of its termsis again less than 1. This radius does not depend on the function. Theorem 1.1 (Bohr, 1914)
Let f be a bounded analytic function in the openunit disk, with Taylor expansion f ( z ) = ∞ X n =0 a n z n convergent in the unit diskand with modulus less than . Then ∞ X n =0 | a n | r n < for ≤ r < . This inequality known as Bohr’s inequality is true for 0 ≤ r < and the constant cannot be improved.Originally, this theorem was proved for 0 ≤ r < but soon improved to thesharp result. In Bohr’s paper [20] his own proof was published as well as a proofby Wiener based on function theory methods. Later, S. Sidon gave a differentproof (see [22]).Recently, several papers were published, generalizing Bohr’s theorem to func-tions of n complex variables (see [8], [21], [1]). Using the standard multi-indexnotations α := ( α , α , ..., α n ) ∈ N n with | α | = α + ... + α n , z := ( z , ..., z n ), z i ∈ C , z α := z α z α . . . z α n n , it is shown in [21] that if a power series P α c α z α has a modulus less than 1 in the unit polydisc { ( z , ..., z n ) : max ≤ j ≤ n | z j | < } ,then the sum of the moduli of the terms is less than 1 in the polydisc of radius √ n . ∗ Institute of Mathematics / Physics, Bauhaus-University, Weimar, Germany.
1n [15], the result shows the possibility to obtain a Bohr type theorem formonogenic functions in the ball in the Euclidean space R with the additionallycondition f (0) = 0. It is shown that for r < . | f ( x ) | < B (0).Having in mind the analogy to the one-dimensional complex function theorywe want to know if the result can be proved for a ball in the Euclidean spaceand not for a polydisc. It is not the goal here to find a sharp estimate for themost general class of functions. Let { e , e , e , e } be an orthonormal basis of the Euclidean vector space R .The vector e is the scalar unit while the generalized imaginary units e , e , e satisfy the following multiplication rules e i e j + e j e i = − δ i,j e , i, j = 1 , , e e i = e i e = e i , i = 0 , , , . This non-commutative product generates the algebra of real quaternions de-noted by H . The real vector space R will be embedded in H by identifying a := ( a , a , a , a ) ∈ R with the element a = a e + a e + a e + a e ∈ H , where a i ( i = 0 , , ,
3) are real numbers. Remark that e = (1 , , , T is themultiplicative unit element of H and by identifying e with 1, it will thereforeneglected in the following notation.The real number Sc a := a is called the scalar part of a and Vec a := a e + a e + a e is the vector part of a . Analogously to the complex case,the conjugate of a is the quaternion a := a − a e − a e − a e . The norm of a is given by | a | = (cid:0) a + a + a + a (cid:1) / and coincides with the correspondingEuclidean norm of a , as a vector in R . Considering the subset A := span R { , e , e } of H , the real vector space R can be embedded in A by the identification ofeach element x = ( x , x , x ) ∈ R with the reduced quaternion x = x + x e + x e ∈ A . As a consequence, we will often use the same symbol x to represent a point in R as well as to represent the corresponding reduced quaternion. Note that theset A is only a real vector space but not a sub-algebra of H .Let us consider an open set Ω ⊂ R with a piecewise smooth boundary. An H -valued function is a mapping f : Ω −→ H such that f ( x ) = f ( x ) + f ( x ) e + f ( x ) e + f ( x ) e , where the coordinates f i are real-valued functions defined in Ω. For continuouslyreal-differentiable functions f : Ω −→ H , the operator D = ∂ x + e ∂ x + e ∂ x (1)2s called the generalized Cauchy-Riemann operator. We define the conjugategeneralized Cauchy-Riemann operator by D = ∂ x − e ∂ x − e ∂ x . (2)A function f : Ω ⊂ R −→ H is called left (resp. right ) monogenic in Ω if Df = 0 in Ω (resp ., f D = 0 in Ω) . From now on we only use left monogenic functions. For simplicity, we willcall them monogenic. The generalized Cauchy-Riemann operator (1) and itsconjugate (2) factorize the Laplace operator in R . In fact, it holds∆ = DD = DD and implies that any monogenic function is also a harmonic function.From now on, we will consider the following notations: B := B (0) is the unitball in R centered at the origin, S = ∂B its boundary and dσ is the Lebesguemeasure on S . In what follows, we will denote by L ( S ; X ; F ) (resp. L ( B ; X ; F ))the F -linear Hilbert space of square integrable functions on S (resp. B ) withvalues in X ( X = R or A or H ), where F = H or R . For any f, g ∈ L ( S ; A ; R )the real-valued inner product is given by h f, g i L ( S ) = Z S Sc ( f g ) dσ. (3)Each homogeneous harmonic polynomial P n of order n can be written inspherical coordinates as P n ( x ) = r n P n ( ω ) , ω ∈ S, (4)its restriction, P n ( ω ), to the boundary of the unit ball is called sphericalharmonic of degree n . From (4), it is clear that a homogeneous polynomialis determined by its restriction to S . Denoting by H n ( S ) the space of real-valued spherical harmonics of degree n in S , it is well-known (see [3] and [16])that dim H n ( S ) = 2 n + 1 . It is also known (see [3] and [16]) that if n = m , the spaces H n ( S ) and H m ( S )are orthogonal in L ( S ; R ; R ).Homogeneous monogenic polynomial of degree n will be denoted in generalby H n . In an analogously way to the spherical harmonics, the restriction of H n to the boundary of the unit ball is called spherical monogenic of degree n . Wedenote by M n ( H ; F ) the subspace of L ( B ; H ; F ) ∩ ker D ( B ) of all homogeneousmonogenic polynomials of degree n . Sudbery proved in [17] that the dimensionof M n ( H ; H ) is n + 1. In [5], it is proved that the dimension of M n ( H ; R ) is4 n + 4.Consider, for each n ∈ N , a basis { H νn : ν = 1 , ..., dim M n ( H ; F ) } of M n ( H ; F ), F = H or F = R . Taking into account that the coordinates of H νn are harmonic, for arbitrary n, k = 0 , , ... , we have h H νn , H µk i L ( B ; H ; F ) = δ n,k n + k + 3 h H νn , H µk i L ( S ; H ; F ) . (5)3 Homogeneous Monogenic Polynomials
Based on the Fueter variables z = x − e x and z = x − e x , several systemsof homogeneous monogenic polynomials are constructed and used for differentpurposes (see, e.g., [4, 9, 7, 10, 12, 17]). Following [12], being γ = ( γ , γ ) amulti-index with γ + γ = n , the generalized powers (or also Fueter polynomials)of degree n are defined by z γ × z γ = z × z × · · · × z | {z } γ times × z × z × · · · × z | {z } γ times = 1 n ! X π ( i ,...,i n ) z i · · · z i n , where the sum is taken over all permutations π ( i , . . . , i n ) of (1 , · · · , | {z } γ , , · · · , | {z } γ ).The general form of the Taylor series of a monogenic function f : Ω ⊂ R −→ H in the neighborhood of the origin (see, e.g., [4, 12]) is given by f = ∞ X n =0 X | γ | = n ( z γ × z γ ) c γ , (6)where c γ = γ ! γ ! ∂ γ x ∂ γ x f ( x ) (cid:12)(cid:12)(cid:12) x =0 ∈ H are the Taylor coefficients.In order to prove a collection of inequalities related to Bohr’s inequality, weneed also the Fourier expansion of monogenic functions.In ([5] and [6]) R -linear and H -linear complete orthonormal systems of H -valued homogeneous monogenic polynomials in the unit ball of R are con-structed. The main idea of these constructions is based on the factorization ofthe Laplace operator. We take a system of real-valued homogeneous harmonicpolynomials and apply the D operator to get systems of H -valued homogeneousmonogenic polynomials. To be precise, we introduce the spherical coordinates, x = r cos θ, x = r sin θ cos ϕ, x = r sin θ sin ϕ, where 0 < r < ∞ , 0 < θ ≤ π , 0 < ϕ ≤ π . Each point x = ( x , x , x ) ∈ R \{ } admits a unique representation x = r w , where for each i = 0 , , w i = x i r and | w | = 1. Now, we apply for each n ∈ N , the operator D to the homogeneousharmonic polynomials, { r n +1 U n +1 , r n +1 U mn +1 , r n +1 V mn +1 , m = 1 , ..., n + 1 } n ∈ N (7)formed by the extensions in the ball of the spherical harmonics (considered, e.g.,in [18]), U n +1 ( θ, ϕ ) = P n +1 (cos θ ) U mn +1 ( θ, ϕ ) = P mn +1 (cos θ ) cos mϕ (8) V mn +1 ( θ, ϕ ) = P mn +1 (cos θ ) sin mϕ, m = 1 , ..., n + 1 . P n +1 stands for the Legendre polynomial of degree n + 1, given by P n +1 ( t ) = [ n +12 ] X k =0 a n +1 ,k t n +1 − k P ( t ) = 1 , t ∈ ( − , , with a n +1 ,k = ( − k n +1 (2 n + 2 − k )! k !( n + 1 − k )!( n + 1 − k )! , where [ s ] denotes the integer part of s ∈ R . Also, we stipulate this sum to bezero whenever the upper index is less then the lower one.The functions P mn +1 are called the associated Legendre functions, P mn +1 ( t ) := (1 − t ) m/ d m dt m P n +1 ( t ) , m = 1 , ..., n + 1 . (9)We remark that if m = 0, the corresponding associated Legendre function P n +1 coincides with the Legendre polynomial P n +1 .Notice that the Legendre polynomials together with the associated Legendrefunctions satisfy several recurrence formulae. We point out only some of them,which will be used several times in the next section. Following [2], Legendrepolynomials and associated Legendre functions are solutions of a second orderdifferential equation, called Legendre differential equation , given by(1 − t )( P mn +1 ( t )) ′′ − t ( P mn +1 ( t )) ′ + (cid:18) ( n + 1)( n + 2) − m − t (cid:19) P mn +1 ( t ) = 0 ,m = 0 , ..., n + 1. They also satisfy the recurrence formula(1 − t )( P mn +1 ( t )) ′ = ( n + m + 1) P mn ( t ) − ( n + 1) tP mn +1 ( t ) , (10) m = 0 , ..., n + 1. An additional and useful identity is given by P mm ( t ) = (2 m − − t ) m/ , (11) m = 1 , ..., n + 1.These functions are mutually orthogonal in L ([ − , Z − P mn +1 ( t ) P mk +1 ( t ) dt = 0 , n = k and their norms are Z − ( P mn +1 ( t )) dt = 22 n + 3 ( n + 1 + m )!( n + 1 − m )! , m = 0 , ..., n + 1 . For a detailed study of Legendre polynomials and associated Legendre func-tions we refer, for example, [2] and [18].5estricting the functions of the set (7) to the sphere, we obtain the sphericalmonogenics X n := (cid:18) D (cid:19) ( r n +1 U n +1 ) (cid:12)(cid:12)(cid:12)(cid:12) r =1 X mn := (cid:18) D (cid:19) ( r n +1 U mn +1 ) (cid:12)(cid:12)(cid:12)(cid:12) r =1 (12) Y mn := (cid:18) D (cid:19) ( r n +1 V mn +1 ) (cid:12)(cid:12)(cid:12)(cid:12) r =1 , m = 1 , ..., n + 1 . For each n ∈ N , taking the monogenic extensions of the spherical monogenicsinto the ball, we obtain the set of homogeneous monogenic polynomials { r n X n , r n X mn , r n Y mn : m = 1 , ..., n + 1 } . (13)We need norm estimates of our functions in terms of its Taylor and Fourierexpansion are needed. In this way, we begin now to write the homogeneousmonogenic polynomials in Cartesian coordinates. In parts, these results werealready obtained in [13] and [14], without proof. Lemma 3.1
The homogeneous monogenic polynomials r n X ln ( l = 0 , , ..., n + 1) in terms of Cartesian coordinates can be written as: r n X ln ( x ) = [ r n X ln ( x )] + [ r n X ln ( x )] e + [ r n X ln ( x )] e , where [ r n X ln ( x )] = [ n − l ] X k =0 β n +1 ,l,k ( n + 1 − k − l ) x n − k − l r k [ l ] X j =0 ( − j (cid:18) l j (cid:19) x l − j x j + [ n +1 − l ] X k =1 β n +1 ,l,k (2 k ) x n +2 − k − l r k −
1) [ l ] X j =0 ( − j (cid:18) l j (cid:19) x l − j x j [ r n X ln ( x )] = [ n +1 − l ] X k =1 β n +1 ,l,k (2 k ) x n +1 − k − l r k −
1) [ l ] X j =0 ( − j +1 (cid:18) l j (cid:19) x l − j +11 x j + [ n +1 − l ] X k =0 β n +1 ,l,k x n +1 − k − l r k [ l − ] X j =0 ( − j +1 (cid:18) l j (cid:19) ( l − j ) x l − j − x j [ r n X ln ( x )] = [ n +1 − l ] X k =1 β n +1 ,l,k (2 k ) x n +1 − k − l r k −
1) [ l ] X j =0 ( − j +1 (cid:18) l j (cid:19) x l − j x j +12 + [ n +1 − l ] X k =0 β n +1 ,l,k x n +1 − k − l r k [ l ] X j =1 ( − j +1 (cid:18) l j (cid:19) (2 j ) x l − j x j − , being β n +1 ,l,k = ( − k n +2 (cid:18) n + 2 − kn + 1 − k (cid:19)(cid:18) n + 1 − kk (cid:19) ( n + 1 − k ) l − and ( n + 1 − k ) l − stands for the Pochhammer symbol. roof. Let us consider the spherical monogenics given by (12), explicitly de-scribed in (8). By the definition of the Legendre polynomials we have P (1) n +1 ( t ) = ddt [ n +12 ] X k =0 a n +1 ,k t n +1 − k = [ ( n +1) − ] X k =0 a n +1 ,k ( n + 1 − k ) t ( n +1 − k ) − . Now, derivating recursively in order to t ( l −
1) times, ∂ lt P n +1 ( t ) = P ( l ) n +1 ( t )= [ ( n +1) − l ] X k =0 a n +1 ,k ( n + 1 − k )( n + 1 − k − · · · ( n + 1 − k − ( l − t ( n +1 − k ) − l . By simplicity, we set β n +1 ,l,k = 2( n + 1 − k )( n + 1 − k − · · · ( n + 1 − k − ( l − , so that, finally we get for (9) the expression P ln +1 (cos θ ) = [ n +1 − l ] X k =0 β n +1 ,l,k ( sinθ ) l (cos θ ) n +1 − k − l . In order to express the set { X ln : l = 0 , , ..., n + 1 } in cartesian coordinates, weconsider the coordinate’s relation:cos θ = x r cos ϕ = x p x + x sin θ = p x + x r sin ϕ = x p x + x . Now, using cos( mϕ ) = [ l ] X j =0 ( − j (cid:18) l j (cid:19) (cos ϕ ) l − j (sin ϕ ) j and substituting in (12) we obtain r n +1 U ln +1 ( x ) = 2 [ n +1 − l ] X k =0 [ l ] X j =0 β n +1 ,l,k x n +1 − k − l r k ( − j (cid:18) l j (cid:19) x l − j x j . Applying the hypercomplex derivative ( D ) to this expression carries our poly-nomials in Cartesian coordinates, respectively.Similar results holds for r n Y mn m = 1 , ..., n + 1). Let us consider now thefollowing function: Definition 3.1
Let i, j ∈ N . The function g i,j is given by g i,j = (cid:26) , if i and j have the same parity0 , if i and j have different parity . roposition 3.1 The Taylor coefficients of the homogeneous monogenic poly-nomials r n X ln ( l = 0 , , ..., n + 1) are given by [ a lα ] = g l,n g α ,l g α , β n +1 ,l, n − l [ l ] X j =0 ( − j (cid:18) l j (cid:19) (cid:18) n − l α − l + j (cid:19) [ a lα ] = g l − ,n g l − ,α g α , [ n − l +12 ] X p =1 β n +1 ,l,p (2 p ) (cid:18) p − l − n − + p (cid:19) [ l ] X j =0 ( − j +1 (cid:18) l j (cid:19) (cid:18) n − l − α − l − + j (cid:19) + [ n − l +12 ] X p =0 β n +1 ,l,p (cid:18) p l − n − + p (cid:19) [ l − ] X j =0 ( − j +1 (cid:18) l j (cid:19) ( l − j ) (cid:18) n − l +12 α − l +12 + j (cid:19) [ a lα ] = g l − ,n g l,α g α , [ n − l +12 ] X p =1 β n +1 ,l,p (2 p ) (cid:18) p − l − n − + p (cid:19) [ l ] X j =0 ( − j +1 (cid:18) l j (cid:19) (cid:18) n − l − α − l + j (cid:19) + [ n − l +12 ] X p =0 β n +1 ,l,p (cid:18) p l − n − + p (cid:19) [ l ] X j =1 ( − j +1 (cid:18) l j (cid:19) (2 j ) (cid:18) n − l +12 α − l + j (cid:19) Proof.
The proof follows directly from Lemma 3.1 by applying the partialderivatives ∂ α x ∂ α x .Again, we obtain analogous formulae for the Taylor coefficients of r n Y mn ( m =1 , ..., n + 1). Corollary 3.1
Let γ = ( γ , γ ) be a multi-index with | γ | = n . The Taylorcoefficients a γ , a mγ and b mγ of the homogeneous monogenic polynomials r n X n , r n X mn and r n Y mn satisfy the following inequalities: | a γ | ≤ γ ! ( n + 1)! r π ( n + 1)2 n + 3 | a mγ | ≤ γ ! ( n + 1)! s π ( n + 1)( n + 1 + m )!2(2 n + 3)( n + 1 − m )! | b mγ | ≤ γ ! ( n + 1)! s π ( n + 1)( n + 1 + m )!2(2 n + 3)( n + 1 − m )! , m = 1 , ..., n + 1 . Proof.
Let B r ( x ) ⊂ R be a ball of radius r centered at x . From [11] we knowthe Cauchy integral formula for the ball B ( x ), f ( x ) = 14 π Z S x − y | x − y | n ( y ) f ( y ) dS y , (14)8here n stands for the outward pointing normal unit vector to S at y . Forsimplicity we just present the proof for the homogeneous monogenic polynomi-als r n X mn ( m = 1 , ..., n + 1). Applying the Cauchy integral formula to thesepolynomials in the ball B and taking partial derivatives with respect to x and x , we get a m, ∗ γ = 1 γ ! ∂ γ x ∂ γ x X mn ( x ) (cid:12)(cid:12) x =0 = 1 γ ! 14 π Z S ∂ γ x ∂ γ x x − y | x − y | (cid:12)(cid:12)(cid:12)(cid:12) x =0 n ( y ) X mn ( y ) dS y taking the modulus and applying the Schwarz inequality we finally obtain | a m, ∗ γ | ≤ γ ! ( n + 1)! s π n + 1) ( n + 1 + m )!( n + 1 − m )! , where a m, ∗ γ denotes the Taylor coefficients associated to the functions X mn . Theprevious inequality is based on [5] where the following relation is proved k X mn k L ( S ) = k Y mn k L ( S ) = s π n + 1) ( n + 1 + m )!( n + 1 − m )! , m = 1 , ..., n + 1and on the paper [26] where it was obtained that (cid:13)(cid:13)(cid:13)(cid:13) ∂ γ x ∂ γ x x − y | x − y | (cid:12)(cid:12)(cid:12)(cid:12) x =0 (cid:13)(cid:13)(cid:13)(cid:13) L ( S ) ≤ ( n + 1)! | y | n +2 . Using the relation (5) we get the Taylor coefficients associated to the homoge-neous monogenic polynomials r n X mn . The case m = 0 is proved analogously.Besides norm estimates we also need pointwise estimates of our basis poly-nomials. Proposition 3.2
For n ∈ N the homogeneous monogenic polynomials satisfythe following inequalities: | r n X n ( x ) | ≤ r n ( n + 1)2 n r π ( n + 1)2 n + 3 | r n X mn ( x ) | ≤ r n ( n + 1)2 n s π n + 1)(2 n + 3) ( n + 1 + m )!( n + 1 − m )! | r n Y mn ( x ) | ≤ r n ( n + 1)2 n s π n + 1)(2 n + 3) ( n + 1 + m )!( n + 1 − m )! , m = 1 , ..., n + 1 . Proof.
Again, we prove only the case of the polynomials r n X mn ( m = 1 , ..., n +1), the proof for r n Y mn being similar. We write these polynomials as a Taylorexpansion (6) r n X mn ( x ) = X | γ | = n ( z γ × z γ ) a mγ . Consequently, modulus of r n X mn leads to | r n X mn ( x ) | ≤ r n ( n + 1)! s π n + 1)(2 n + 3) ( n + 1 + m )!( n + 1 − m )! 2 n n ! . | z γ × z γ | ≤ r n for every multi-index γ = ( γ , γ ) with | γ | = n .For future use in this paper we will need estimates for the real part of thespherical monogenics described in (12). Theorem 3.1
Given a fixed n ∈ N , the spherical harmonics (cid:8) Sc { X n } , Sc { X mn } , Sc { Y mn } : m = 1 , ..., n (cid:9) are orthogonal to each other with respect to the inner product (3). Proposition 3.3
Given a fixed n ∈ N , the moduli of the spherical harmonics Sc ( X n ) , Sc ( X mn ) and Sc ( Y mn ) satisfy the following inequalities | Sc { X ln }| ≤
12 ( n + 1 + l )! n ! , l = 0 , ..., n | Sc { Y mn }| ≤
12 ( n + 1 + m )! n ! , m = 1 , ..., n. Proof.
According to the results from [5], the real parts of the spherical mono-genics are given by Sc { X n } = A ,n ( θ ) Sc { X mn } = A m,n ( θ ) cos( mϕ ) Sc { Y mn } = A m,n ( θ ) sin( mϕ ) , where A l,n ( θ ) = 12 (cid:18) sin θ ddt (cid:2) P ln +1 ( t ) (cid:3) t = cosθ + ( n + 1) cosθP ln +1 ( cosθ ) (cid:19) , l = 0 , ..., n. For simplicity sake we only present the proof for the spherical harmonics Sc ( X mn )( m = 1 , ..., n + 1). Making the change of variable t = cosθ and using therecurrence formula (10), it follows that Sc { X mn } = 12 ( n + 1 + m ) P mn ( t ) . Applying the modulus in the previous expression and using the inequality provedin [19] | P mn ( t ) | ≤ ( n + m )! n ! , for − ≤ t ≤ n ≥ m , we finally obtain the estimate | Sc { X mn }| ≤
12 ( n + 1 + m )! n ! . Some of the basis polynomials described in (13) play a special role. Applyingresults from ([5], Proposition 3 . .
3) we get:10 roposition 3.4
For n ∈ N , the spherical monogenics X n +1 n and Y n +1 n aregiven by X n +1 n = − C n +1 ,n cosnϕ e + C n +1 ,n sinnϕ e (15) Y n +1 n = − C n +1 ,n sinnϕ e − C n +1 ,n cosnϕ e where C n +1 ,n = n + 12 1 sinθ P n +1 n +1 ( cosθ ) , and their monogenic extensions into the ball belong to kerD ( B ) ∩ kerD ( B ) . Remark 3.1
The spherical monogenics X n +1 n and Y n +1 n are monogenic con-stants, i.e, monogenic functions which depend only on x and x . Moreover,they play the role of constants with respect to the hypercomplex differentiation ( D ) . Proposition 3.5
Given a fixed n ∈ N , the spherical harmonics Sc ( X n +1 n e ) and Sc ( Y n +1 n e ) are orthogonal to each other with respect to the inner product(3) and their moduli satisfy the following inequalities | Sc { X n +1 n e }| ≤
12 ( n + 1)(2 n + 1)!2 n n ! | Sc { Y n +1 n e }| ≤
12 ( n + 1)(2 n + 1)!2 n n ! . Proof.
Again, we present the proof for the spherical harmonics Sc { X n +1 n e } ,the one for Sc { Y n +1 n e } being similar. According to (15), the real part of thespherical harmonic X n +1 n e is given by Sc { X n +1 n e } = C n +1 ,n cos nϕ. Making the change of variable t = cosθ and applying the modulus in the previousexpression, we get | Sc { X n +1 n e }| = n + 12 (cid:12)(cid:12)(cid:12)(cid:12) √ − t P n +1 n +1 ( t ) (cid:12)(cid:12)(cid:12)(cid:12) , and due to the recurrence formula (11) we finally obtain | Sc { X n +1 n e }| = n + 12 (cid:12)(cid:12)(cid:12)(cid:12) √ − t (2 n + 1)!!(1 − t ) n +12 (cid:12)(cid:12)(cid:12)(cid:12) ≤
12 ( n + 1)(2 n + 1)!! . Proposition 3.6
Given a fixed n ∈ N , the norms of the spherical harmonics Sc ( X n ) , Sc ( X mn ) and Sc ( Y mn ) are given by k Sc ( X n ) k L ( S ) = ( n + 1) r π n + 1 and k Sc ( X mn ) k L ( S ) = k Sc ( Y mn ) k L ( S ) = s π n + 1 + m )(2 n + 1) ( n + 1 + m )!( n − m )! , m = 1 , ..., n. roposition 3.7 Given a fixed n ∈ N , the spherical harmonics Sc ( X n +1 n e ) and Sc ( Y n +1 n e ) are orthogonal to each other with respect to the inner product(3) and their norms are given by k Sc ( X n +1 n e ) k L ( S ) = k Sc ( Y n +1 n e ) k L ( S ) = 12 p π ( n + 1)(2 n + 2)! . We will denote by X , ∗ n , ... the normalized basis functions in L ( S ; H ; H ). Theorem 4.1 (see [5]) Let M n ( R ; A ) be the space of A -valued homogeneousmonogenic polynomials of degree n in R . For each n , the set of n + 3 homo-geneous monogenic polynomials (cid:8) √ n + 3 r n X , ∗ n , √ n + 3 r n X m, ∗ n , √ n + 3 r n Y m, ∗ n , m = 1 , ..., n + 1 (cid:9) (16) forms an orthonormal basis in M n ( R ; A ) . In [15], a first version of a quaternionic Bohr’s theorem was considered,therein we restricted ourselves to the case of functions with f (0) = 0 and weobtained an estimate in terms of a radius of r = 0 . | f ( x ) | < B ,estimating a value for the radius. Theorem 4.2
Let f be a square integrable A -valued monogenic function with | f ( x ) | < in B , Sc { f } be positive and let ∞ X n =0 √ n + 3 r n ( X , ∗ n α n + n +1 X m =1 [ X m, ∗ n α mn + Y m, ∗ n β mn ] ) be its Fourier expansion. Then ∞ X n =0 √ n + 3 r n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( X , ∗ n α n + n +1 X m =1 [ X m, ∗ n α mn + Y m, ∗ n β mn ] )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < holds in the ball of radius r , with ≤ r < . .Proof. According to Theorem 4.1, a monogenic L -function f : Ω ⊂ R −→ A can be written as Fourier series f = ∞ X n =0 √ n + 3 r n ( X , ∗ n α n + n +1 X m =1 [ X m, ∗ n α mn + Y m, ∗ n β mn ] ) , where α n , α mn and β mn ( m = 1 , ..., n + 1) are the associated Fourier coefficients.Let us denote by Sc { f } the real part of f . Then, Sc { f } = f + f ∞ X n =0 √ n + 3 r n ( Sc { X , ∗ n } α n + n X m =1 [ Sc { X m, ∗ n } α mn + Sc { Y m, ∗ n } β mn ] ) . f in the following way f = √ α X , ∗ + √ α X , ∗ + √ β Y , ∗ + ∞ X n =1 √ n + 3 r n ( X , ∗ n α n + n X m =1 [ X m, ∗ n α mn + Y m, ∗ n nβ mn ] ) + ∞ X n =1 √ n + 3 r n (cid:2) X n +1 , ∗ n α n +1 n + Y n +1 , ∗ n β n +1 n (cid:3) . Based in this spliting, we introduce f = 12 r π α + ∞ X n =1 √ n + 3 r n ( X , ∗ n α n + n X m =1 [ X m, ∗ n α mn + Y m, ∗ n β mn ] ) f = − r π α e − r π β e + ∞ X n =1 √ n + 3 r n (cid:2) X n +1 , ∗ n α n +1 n + Y n +1 , ∗ n β n +1 n (cid:3) , so that f = f + f . Then, we have f (0) = f (0) + f (0)where f (0) = 12 r π α f (0) = − r π α e − r π β e . Let us assume that there exists 0 < δ < | f | < δ and | f | < − δ .In this way, the modulus of f is preserved. We start now to study the function f . The main idea is to compare each Fourier coefficient with the coefficient α .In fact, multiplying both sides of the expression Sc { δ − f } = δ − Sc { f } (17)by each real part of the homogeneous monogenic polynomials described in (13)and integrating over the sphere, we get these relations. For simplicity we justpresent the idea applied to the coefficients of X , ∗ n , i.e, α n . Multiplying bothsides of the expression (17) by Sc { X k } and integrating, we obtain −√ k + 3 α k = Z S Sc { δ − f } Sc { X k } dσ with 0 < δ <
1. Now, applying the modulus we obtain finally | α k |√ k + 3 ≤ √ π | Sc { X k }|k Sc { X k }k L ( S ) δ − r π α ! . (18)In an analogous way, we can state the following results: | α pk |√ k + 3 ≤ √ π | Sc { X pk }|k Sc { X pk }k L ( S ) δ − r π α ! . | β pk |√ k + 3 ≤ √ π | Sc { Y pk }|k Sc { Y pk }k L ( S ) δ − r π α ! , p = 1 , ..., k. | Sc { X k }|k Sc { X k }k L ( S ) ≤ π (2 k + 1) k + 1 | Sc { X pk }|k Sc { X pk }k L ( S ) = | Sc { Y pk }|k Sc { Y pk }k L ( S ) ≤ π (2 k + 1)( k − p )!( k + 1 + p ) k ! , p = 1 , ..., k. Finally, the previous expressions can be rewritten | α k |√ k + 3 ≤ √ π (2 k + 1) k + 1 δ − r π α ! | α pk |√ k + 3 ≤ √ π (2 k + 1)( k − p )!( k + 1 + p ) k ! δ − r π α ! | β pk |√ k + 3 ≤ √ π (2 k + 1)( k − p )!( k + 1 + p ) k ! δ − r π α ! . Consequently, we can state the following inequalities: | X , ∗ k || α k |√ k + 3 ≤ √ π (2 r ) k (2 k + 1) δ − r π α ! k X p =1 | X p, ∗ k || α pk |√ k + 3 ≤ √ π (2 r ) k (2 k + 1) δ − r π α ! k X p =1 | Y p, ∗ k || β pk |√ k + 3 ≤ √ π (2 r ) k (2 k + 1) δ − r π α ! . Now, using the previous inequalities we end with | f | ≤ r π α + ∞ X n =1 √ n + 3 r n " | X , ∗ n || α n | + n X m =1 ( | X m, ∗ n || α mn | + | Y m, ∗ n || β mn | ) ≤ r π α + 5 √ π δ − r π α ! ∞ X n =1 (2 r ) n (2 n + 1) . Thus, we have that | f | ≤ δ = ⇒ √ π ∞ X n =1 (2 r ) n (2 n + 1) ≤ , and, the last series is convergent for r < .
05. In the same way, we can studythe function f . Let f = √ α r X , ∗ + √ β r Y , ∗ + ∞ X n =1 √ n + 3 r n (cid:2) X n +1 , ∗ n α n +1 n + Y n +1 , ∗ n β n +1 n (cid:3) . f in the right side by e we get˜ f := f e = √ α ( r X , ∗ e ) + √ β ( r Y , ∗ e )+ ∞ X n =1 √ n + 3 r n (cid:2) ( X n +1 , ∗ n e ) α n +1 n + ( Y n +1 , ∗ n e ) β n +1 n (cid:3) . We want to apply the same idea previously used for f . Taking in considerationthat f is an A -valued function, we obtain an estimate for the coefficient α . Ina similar way, we obtain an estimate for β if we multiply f at right by e .This leads to the inequalities | α k +1 k |√ k + 3 ≤ r π | Sc { X k +1 k e }|k Sc { X k +1 k e }k L ( S ) (1 − δ ) − r π α ! | β k +1 k |√ k + 3 ≤ r π | Sc { Y k +1 k e }|k Sc { Y k +1 k e }k L ( S ) (1 − δ ) − r π α ! . being | Sc { X k +1 k e }|k Sc { X k +1 k e }k L ( S ) = | Sc { Y k +1 k e }|k Sc { Y k +1 k e }k L ( S ) ≤ π n ( n + 1)! . Consequently, we have proved: | X k +1 , ∗ k e || α k +1 k |√ k + 3 ≤ √ π r k k ! (1 − δ ) − r π α ! | Y k +1 , ∗ k e || β k +1 k |√ k + 3 ≤ √ π r k k ! (1 − δ ) − r π α ! . With the previous inequalities we get | ˜ f | = | f | ≤ r π α + 4 √ π (1 − δ ) − r π α ! ∞ X n =1 r n n ! . Finally, we end with | f | ≤ − δ = ⇒ √ π ∞ X n =1 r n n ! ≤ , and, the last series is convergent for r < .
56. Finally, ∞ X n =0 √ n + 3 r n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( X , ∗ n α n + n +1 X m =1 [ X m, ∗ n α mn + Y m, ∗ n β mn ] )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ≤ r < .
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