BY COMPLEX SPHERICAL MULTIPLIER OPERATORS

BY COMPLEX SPHERICAL MULTIPLIER OPERATORS

C. P. OLIVEIRA

Received 20 January 2004 and in revised form 9 December 2004

This paper studies, in a partial but concise manner, approximate solutions of equations defined by complex spherical multiplier operators. The approximations are from native spaces embedded in Sobolev-type spaces and derived from the use of positive definite functions to perform spherical interpolation.

1. Introduction

LetΩ2qbe the unit sphere inC^q,q≥2. If the usual inner product betweenzandwinC^q is denoted byz,w, then

Ω2q=

z∈C^q:z,z =1. (1.1)

This paper is concerned with approximate solutions of an equation of the form

T(f)=g, (1.2)

in whichT:L²(Ω2q)→L²(Ω2q) is a spherical multiplier operator. Ideally, the domainDT

ofT should contain complex spherical harmonics up to a certain degree. The term har- monic refers to a function ofqcomplex variables belonging to the kernel of the Laplacian

∆(2q):=4 q j=1

∂²

∂z_j∂z_j, z_∈C^q. (1.3)

A complex spherical harmonic is then the restriction toΩ2qof a harmonic polynomial of bidegree (m,n). A polynomial has bidegree (m,n) when it is homogeneous of degreem with respect toz∈C^qand homogeneous of degreenwith respect toz.

To better explain what a multiplier operator is in our context, we need to introduce some notation. We will write

Ym,n^j :j=1,. . .,d(m,n) (1.4)

Copyright©2005 Hindawi Publishing Corporation Journal of Applied Mathematics 2005:2 (2005) 93–115 DOI:10.1155/JAM.2005.93

to denote a fixed orthonormal basis forᏴm,n(Ω2q), the space of complex spherical har- monics of bidegree (m,n). Orthonormality refers to the usual inner product·,·2 of L²(Ω2q), that is,

f,g2:=

Ω2q

f(z)g(z)dσ_q(z), f,g∈L²Ω2q

, (1.5)

in whichdσqdenotes the Lebesgue measure overΩ2qso normalized that

Ω2q

dσ_q(z) :=ω_q, (1.6)

the surface measure ofΩ2q.

Every function f ofL²(Ω2q) has a Fourier-type expansion in the form

f ∼

m,n∈Z+

d(m,n)

j=1

f_m,n(j)Ym,n^j , f_m,n(j) :=

f,Ym,n^j 2. (1.7) The condensed expansion off is given by

f ∼

m,n∈Z+

sm,n(f), sm,n(f) :=

d(m,n)

j=1

fm,n(j)Ym,n^j . (1.8) A complexspherical multiplier operatorTis then an operator

T:L²Ω2q

−→L²Ω2q

, (1.9)

characterized by the following property: there exists a double-indexed sequence {c_m,n}m,n∈Z+ of complex numbers, thespherical symbol, such thatc_m,n=c_n,m,m,n∈Z+, and

T(f)_m,n(j)=c_m,nf_m,n(j), f ∈L²Ω2q

, j=1,. . .,d(m,n), m,n∈Z+. (1.10) In particular, ifTis a spherical multiplier operator, then

TYm,n^j

=cm,nYm,n^j , j=1,. . .,d(m,n),m,n∈Z+. (1.11) Henceforth, unless stated otherwise, the letterTwill denote a spherical multiplier opera- tor associated with spherical symbol{cm,n}m,n∈Z+.

The approximations in this paper will take place in spaces constructed from a fixed basis functionK:{z∈C:|z| ≤1} →Chaving an expansion in the form (see [9])

K(z)=

m,n∈Z+

a^qm,n(K)P^qm,n⁻²(z), am,n^q (K)≥0,K(1)<∞. (1.12) The functionPm,n^q−2represents thedisk polynomialgiven by

P^qm,n⁻²

re^iθ=r^|m−n|e^i(n−m)θP^(qm∧⁻n^2,|m−n|)(2r²−1), z=re^iθ, (1.13)

wherePm^(q∧⁻n^2,|m−n|)is the Jacobi polynomial of degreem∧nassociated with the pair (q−2,

|m−n|) and so normalized thatPm^(q∧⁻n^2,|m−n|)(1)=1. Details about disk polynomials can be found in [1,4]. For now, we mention theaddition formulafor disk polynomials

Pm,n^q−2 z,w

= ωq

d(m,n)

d(m,n)

j=1

Ym,n^j (z)Ym,n^j (w), z,w∈Ω2q. (1.14) The basis functionKis assumed to be smooth with respect toTin the following sense:

m,n∈Z+

a^qm,n(K)c_m,n²<∞. (1.15) Among other things, such condition allows us to useK to construct spaces where (1.2) makes sense and has solutions. In addition, we have the basis function given by

KT(z) :=

m,n∈Z+

a^qm,n(K)cm,n²Pm,n^q−2(z), a^qm,n(K)cm,n²≥0,KT(1)<∞, (1.16) to be used inSection 3.

We will approximate the solutions f of (1.2) by functions of the form sf=

N j=1

cj

δj◦T^wK·,w

, w∈Ω2q,cj∈C. (1.17) In this equation,

δj(h) :=hwj

, h∈L²Ω2q

, j=1,. . .,N, (1.18) where the pointsw1,. . .,w_N are distinct overΩ2q. The notation (δj◦T^w)K(·,w) means thatδ_j◦Tis acting on the function

w∈Ω2q−→K·,w

. (1.19)

If{δ1,. . .,δN}is linearly independent and f is in an appropriate space, then we will show thats_f is the unique function in the space

spanδj◦T^wK·,w

:j=1,. . .,N (1.20) that is a solution of the interpolation

δ_j◦T(w)=

δ_j◦T(f), j=1,. . .,N. (1.21) Assuming additional smoothness conditions onTand on the pointsw1,. . .,w_N, we will establish error estimates for the approximation of solutions of (1.2) bysf in the norm of spaces of Sobolev-type in which the native spaces are continuously embedded.

We would like to observe that the approach taken here is known (see [3,5,11]). What makes a difference here is that our paper refines and details some ideas presented in other sources, putting them into the more general complex setting. In addition, estimates not considered until now are investigated.

2. Native spaces

In this section, we briefly discuss native spaces associated with sequences and functions.

They are Hilbert spaces where the sequences and functions act as a generalized repro- ducing kernel. For instance, we will see inSection 3that a spherical multiplier operator is surjective when its domain and image are appropriate native spaces inL²(Ω2q). Native spaces in the real setting were considered in [6,7].

Thenative spaceassociated with a complex sequenceβ:= {bm,n}m,n∈Z+is ᏺβ:=

f ∈L²Ω2q

:fβ<∞,sk,l(f)=0, (k,l)∈Aβ

, (2.1)

in which

Aβ:=

(k,l)∈Z²+:bk,l=0, f²_β:= 1

ωq

(m,n)∈Aβ

d(m,n)

bm,nsm,n(f)²₂. (2.2)

It is very easy to see that · βis a norm obtainable from the inner product f,gβ:= 1

ωq

(m,n)∈Aβ

d(m,n) bm,n ^d(m,n)

j=1

fm,n(j)gm,n(j), f,g∈ᏺβ. (2.3)

In addition, we have the following result.

Theorem2.1. The spaceᏺβpossesses these properties:

(i)Ᏼm,n(Ω2q)⊆ᏺβif and only if(m,n)∈A_β; (ii) (ᏺβ,·,·β)is a Hilbert space;

(iii)the spacespan{Ym,n^j :j=1,. . .,d(m,n), (m,n)∈A_β}is dense inᏺβ. Proof. (i) If (m,n)∈A_β, then

s_k,lYm,n^j

=





Ym,n^j , (k,l)=(m,n),

0, (k,l)=(m,n), j=1,. . .,d(m,n). (2.4) Hence,

Ym,n^{j 2}

β= d(m,n)

ω_qb_m,n, s_k,lYm,n^j

=0, (k,l)∈A_β. (2.5) Thus,Ᏼm,n(Ω2q)⊂ᏺβ. The converse is immediate.

(ii) Iff,f_β=0 for some f ∈ᏺβ, thens_m,n(f)=0, (m,n)∈A_β. Sinces_m,n(f)=0, (m,n)∈Aβ, by definition, it follows thatsm,n(f)=0 for allmandn. The linear indepen- dence of the set{Ym,n^j :j=1,. . .,d(m,n),m,n∈Z+}now implies that

fm,n(j)=0, j=1,. . .,d(m,n), m,n∈Z+, (2.6)

that is, f =0. The other properties needed here are standard and require the following two facts: (L²(Ω2q),·,·2) is a Hilbert space and the mapping f ∈L²(Ω2q)→s_m,n(f)∈ L²(Ω2q) is a linear operator.

(iii) It suffices to use (ii) and a known result on total sets in Hilbert spaces (see [4,

page 169]).

Native spaces are continuously imbedded in (Ꮿ(Ω2q), · ∞).

Theorem2.2. If|β|:= {|b_m,n|}m,n∈Z+ is summable, thenᏺβ is a subspace ofᏯ(Ω2q). In addition, there exists a positive constantC0, not depending onf, such that

f_∞≤C0f_β, f ∈ᏺβ. (2.7)

Proof. It suffices to show that the condensed expansion (1.8) of a function f inᏺβ is absolutely convergent. To do that, we use the Cauchy-Schwarz inequality and the addition formula for spherical harmonics to obtain





m,n∈Z+

sm,n(f)(z)





2

=





(m,n)∈Aβ

sm,n(f)(z)





2

=

(m,n)∈Aβ

d(m,n)

j=1

b_m,nω_q^1/2

d(m,n)^1/2 Ym,n^j (z) d(m,n)^1/2

b_m,nω_q^1/2 f_m,n(j)

≤

(m,n)∈Aβ

b_m,nω_q d(m,n)

d(m,n)

j=1

Ym,n^j (z)²

(m,n)∈Aβ

d(m,n) ω_qb_m,n

d(m,n)

j=1

f_m,n(j)²

≤





m,n∈Z+

b_m,n







 1 ωq

(m,n)∈Aβ

d(m,n)

b_m,ns_m,n(f)²₂





=C₀²f²_β, z∈Ω2q,

(2.8)

in whichC₀²=

m,n∈Z+|b_m,n|<∞.

Corollary2.3. If|β|is summable andz∈Ω2q, then the point-evaluation functionalδ_z given byδz(f)=f(z),f ∈ᏺβ, is continuous.

3. Multiplier operators and native spaces

This section contains technical results to be used later and examples of multiplier opera- tors. In particular, we establish a setting where (1.2) has a unique solution.

We begin recalling the concept of spherical convolution. IfKis as in (1.12) and such thata^qm,n(K)=a^qn,m(K), then the spherical convolution operator withKis given by

ΨK(f) :=

Ω2q

Kz,·

f(z)dσ_q(z), f ∈L²Ω2q. (3.1) The addition formula for disk polynomials and the Fubini-Tonellli’s theorem (see [2, page 223]) imply thatΨKis a multiplier operator with spherical symbol

a^qm,n(K)ω_q

d(m,n) , m,n∈Z+. (3.2)

More information about this operator, including a complex version of the Funk-Hecke formula, can be found in [8].

Next, we explore the native space associated with a function K as in (1.12). When the sequenceβin the definition of native space is{a^qm,n(K)}m,n∈Z+, we writeᏺK:=ᏺβ, · K:= · βandA_K:=A_β. An analogous remark applies to basis functionK_T. Theorem3.1. LetTbe a multiplier operator andwa fixed element ofΩ2q. IfKis a function as in (1.12) and obeying (1.15), then functions

z∈Ω2q−→Kz,w

, z∈Ω2q−→T^wKz,w

, z∈Ω2q−→T^zKz,w (3.3) belong toᏺK.

Proof. Writeg(z)=K(z,w),z∈Ω2q. We obtain

gm,n(j)=a^qm,n(K)ωq

d(m,n) Ym,n^j (w), j=1,. . .,d(m,n),m,n∈Z+. (3.4) The addition formula yields

s_m,n(g)²₂=

d(m,n)

j=1

g_m,n(j)²=a^qm,n(K)²ω²_q d(m,n)²

d(m,n)

j=1

Ym,n^j (w)²

=a^qm,n(K)²ωq

d(m,n) , m,n∈Z+.

(3.5)

Hence,

g²K=

(m,n)∈AK

d(m,n) a^qm,n(K)ωq

a^qm,n(K)²ωq

d(m,n) ⁼

(m,n)∈AK

a^qm,n(K)<∞. (3.6)

The proofs in the other two cases are very much alike. We include the details for the functionh(z)=T^zK(z,w),z∈Ω2q. As above

hm,n(j)=a^qm,n(K)ωq

d(m,n) cm,nYm,n^j (w), j=1, 2,. . .,d(m,n),m,n∈Z, (3.7) sm,n(h)²₂=

d(m,n)

j=1

hm,n(j)²=a^qm,n(K)²ωq

d(m,n) cm,n², m,n∈Z. (3.8) Thus,

h²_K=

(m,n)∈AK

d(m,n) a^qm,n(K)ω_q

a^qm,n(K)²ω_q

d(m,n) c_m,n²=

(m,n)∈AK

a^qm,n(K)c_m,n²<∞. (3.9)

Finally, note that if (k,l)∈AK, thensk,l(g)=0 andsk,l(h)=0.

Theorem3.2. LetT andKbe as in the previous theorem. ThenT^zT^w(Kz,w)defines a uniformly convergent series with sumK_T(z,w).

Proof. Forz,w∈Ω2q, the addition formula implies that T^zT^wKz,w

∼

m,n∈Z+

a^qm,n(K)cm,n²Pm,n^q−2 z,w

. (3.10)

Condition (1.15) completes the proof.

Below, we investigate some connections between the native spaces ofK andKT. In particular, we present a setting in whichTbecomes surjective, a condition that guarantees the existence of solutions of the equation T(f)=g. This seems to be forgotten in the real setting.

Theorem3.3. Let T be a multiplier operator andK as described in (1.12). ThenᏺKT⊆ T(ᏺK).

Proof. Letg∈ᏺK_T. The function f ∈L²(Ω2q) with Fourier coefficients given by

fm,n(j)=





 g_m,n(j)

c_m,n , (m,n)_∈A_KT,

0, (m,n)_∈A_K

T,

(3.11)

satisfies

sm,n(f)=





 s_m,n(g)

c_m,n , (m,n)∈A_K

T, 0, (m,n)∈A_K

T.

(3.12)

Consequently,

(m,n)∈AK

d(m,n)

a^qm,n(K)sm,n(f)²₂=

(m,n)∈A_KT

d(m,n)

a^qm,n(K)cm,n²sm,n(g)²₂

=ωqg²_K

T<∞,

(3.13)

that is,fK<∞. Since the definition of f reveals thatsm,n(f)=0, (m,n)∈AK, it is now clear that f ∈ᏺK. Thus,g=T(f). In addition, the Fourier series ofgandT(f) coincide

inL²(Ω2q).

In some situations, the functions in the spaceᏺK need to have some desirable addi- tional smoothness.Theorem 3.4below reveals how one can substitute that space by an- other one composed of smoother functions and not too much different from the original space. It has to do with the compositionΨK◦K(·,w) for somew∈Ω2q.

Theorem3.4. LetKandΨKbe as inTheorem 3.1. Ifw∈Ω2q, then the function z∈Ω2q−→ΨK◦K·,w

(z) (3.14)

is of formK1(·,w), whereK1is representable as in (1.12). In addition,K1obeys (1.15) if Kdoes.

Proof. Using the additional formula and the orthogonality of spherical harmonics, we obtain

ΨK

K·,w (z)=

Ω2q

Kζ,z

Kζ,w dσ_q(ζ)

=

m,n∈Z+

a^qm,n(K)

Ω2q

Kζ,z P^qm,n⁻²

ζ,w dσ_q(ζ)

=

m,n∈Z+

a^qm,n(K)ωq

d(m,n)

d(m,n)

j=1

ΨK

Ym,n^j

(z)Ym,n^j (w)

=

m,n∈Z+

a^qm,n(K)ω_q d(m,n)

2d(m,n)

j=1

Ym,n^j (z)Ym,n^j (w)

=

m,n∈Z+

a^qm,n(K)²ωq

d(m,n) P^qm,n⁻² z,w

:=K1

z,w

, z∈Ω2q.

(3.15)

It follows thatK1has an expansion as in (1.12) because

m,n∈Z+

a^qm,n(K)ω_q

d(m,n) <∞, lim

m,n→∞a^qm,n(K)=0. (3.16) A similar argument resolves the last statement of the theorem.

The functionΨK(K(·,w)) appearing above is frequently called the spherical convo- lution ofKby itself and it is denoted byK∗K. AlthoughA_K∗K=A_K, the spacesᏺK∗K

andᏺKare usually different.

For future use, we mention the series representation ofK_T∗K_T(·,w):

K_T∗K_T·,w

=

m,n∈Z+

a^qm,n(K)²ω_q

d(m,n) c_m,n⁴P^qm,n⁻²

·,w

, w∈Ω2q. (3.17) 4. Native spaces and Sobolev spaces

We will work with estimates in the spacesᏺK and other spaces containing them. Among such spaces are certain Sobolev-type spaces onΩ2qwhich we now define.

Henceforth,−λm,n will denote the sole eigenvalue of the restriction of the Laplace- Beltrami operator to the space Ᏼm,n(Ω2q) while λ will denote the double-indexed se- quence{λ_m,n}m,n∈Z+. Given a real numberµand a real sequenceα:= {α_m,n}m,n∈Z+satis- fying

1 +αm,n≤Cα

1 +λm,n

, m,n∈Z+ (4.1)

for some positive constantCα>0, theSobolev-type spaceassociated withαandµis the space

Ᏼ^µ(α) :=

f ∈L²Ω2q

:fα,µ<∞

, (4.2)

in which

f²_α,µ:=

m,n∈Z+

1 +α_m,n^µs_m,n(f)²₂. (4.3)

Sinceλ_m,n=(m+n)(m+n+ 2q−2), it is not surprising at all that the following equal- ity of Sobolev-type spaces holds. This explains why some authors use the sequenceδin- stead ofλin the above definition.

Theorem4.1. Ifδ:= {m+n}m,n∈Z+, thenᏴ^2µ(δ)=Ᏼ^µ(λ),µ∈R. Proof. Analyzing the graphs of the functions fq: [0,∞)→Rgiven by

f_q(x)=x²+ (2q−2)x+ 1

(1 +x)² , q=2, 3,. . ., (4.4)

we obtain the inequalities

1≤ fq(x)≤q

2. (4.5)

Since fq(m+n)=(1 +λm,n)(1 +m+n)⁻²,m,n∈Z+, it follows that (1 +m+n)²≤

1 +λm,n

≤q

2(1 +m+n)², m,n∈Z+. (4.6) Hence, there is a positive constantK:=K(µ,q) such that

(1 +m+n)^2µ≤

1 +λ_m,n^µ≤K(1 +m+n)^2µ, µ∈R. (4.7)

The assertion of the theorem follows.

Next, we will consider spherical multiplier operators possessing polynomial decay. For such operators, it is possible to visualize a Sobolev-type space setting where (1.2) makes sense and has solutions. This additional assumption on the operators is not as incon- venient as it seems because all important examples considered here and elsewhere have some decay.

A spherical multiplier operatorTispseudodifferential of orderν∈Rif there exist pos- itive constantscandCsuch that

c1 +λm,n

_ν

≤cm,n²≤C1 +λm,n

_ν

, m,n∈Z+. (4.8)

Theorem4.2. LetKbe a function as in (1.12) and obeying (1.15). IfTis pseudo-differential of orderνthenᏺK⊆Ᏼ^ν(λ).

Proof. By using inequality (4.8) and condition (1.12), it is not hard to see that there exists a constantC(ν,q) such that

1 +λ_m,n^ν≤C(ν,q) d(m,n)

a^qm,n(K)ω_q, (m,n)∈A_K. (4.9) Hence,

f²_λ,ν≤C(ν,q)

(m,n)∈AK

d(m,n)

a^qm,n(K)ω_qs_m,n(f)²₂=C(ν,q)f²_K, f ∈ᏺK. (4.10)

The proof is complete.

The reader may verify that the inclusion inTheorem 4.2becomes an equality when a^qm,n(K)>0,m,n∈Z+, andcm,n=0,m,n∈Z+.

Theorem 4.3below establishes a Sobolev-type space setting for solving (1.2). A weaker version of part (ii) was proved in [11]. The setting there was the real one. The proof presented here is simpler.

Theorem4.3. LetT be a pseudodifferential of orderν,αa sequence satisfying (4.1), and µ∈R. The following assertions hold:

(i)ifµ≥ν, there exists a constantC(α,µ,T)such that

T(f)_α,µ−ν≤C(α,µ,T)fλ,µ, f ∈Ᏼ^µ(λ); (4.11) in particular,T(Ᏼ^µ(λ))⊆Ᏼ^µ−ν(α);

(ii)ifµ≥0, there exists a constantc(α,µ,T)such that

fα,µ≤c(α,µ,T)T(f)_λ,µ−ν, f ∈Ᏼ^µ(α). (4.12) Proof. (i) Let f ∈Ᏼ^α(λ). Due to (4.1) and (4.8), we see that

cm,n²

1 +αm,nµ−ν

≤cm,n²C^µα^−ν

1 +λm,nµ−ν

≤CCα^µ−ν

1 +λm,nµ

, m,n∈Z+. (4.13)

Hence,

T(f)²_α,µ−ν=

m,n∈Z+

1 +α_m,n^µ−νc_m,n²s_m,n(f)²₂

≤CCα^µ−ν

m,n∈Z+

1 +λ_m,n^µs_m,n(f)²₂. (4.14)

The constantC(α,µ,T) in the statement (i) satisfiesC(α,µ,T)²=CC^µα^−ν. (ii) It suffices to adapt the proof of (i) using the inequalities

1 +αm,nµ

≤Cα^µ

1 +λm,nµ

≤Cα^µ

c

1 +λm,nµ−νcm,n² (4.15) instead. The constantc(α,µ,T) satisfiescc(α,µ,T)²=Cα^µ. We close the section mentioning without proof a version ofTheorem 4.3which does not involve the conditionµ≥ν. The sequenceαneeds to be replaced withλ. Part (ii) holds even whenνis nonpositive.

Theorem4.4. LetT be a pseudodifferential operator of orderνandµ∈R. The following assertions hold:

(i)there exist a positive constantC(T)such that

T(f)_λ,µ−ν≤C(T)fλ,µ, f ∈Ᏼ^µ(λ); (4.16) in particular,T(Ᏼ^µ(λ))⊆Ᏼ^µ−ν(λ);

(ii)there exists a positive constantc(T)such that

fλ,µ≤c(T)T(f)_λ,µ−ν, f ∈Ᏼ^µ(λ). (4.17) 5. Approximate solutions

In this section, we describe the construction of approximate solutions of the equation T(f)=g. Throughout the section,T will be a multiplicative operator andK will be a function as described in (1.12) and obeying (1.15).

Given a Hilbert space (Ᏼ, [[·,·]]), a closed subspaceV, andu∈Ᏼ, we will denote by p(u,V) the unique element ofVsatisfying [[u−p(u,V),v]]=0,v∈V. Such an element satisfies

[[u−v]]²=[[u−p(u,V)]]²+[[v−p(u,V)]]², v∈V, (5.1) in which|[[·]]|denotes the norm induced by [[·,·]].

IfW= {w1,. . .,wN}is a subset ofΩ2q, we define VW:=spanδj◦T^wK·,w

: 1≤j≤N, V_WT:=spanδ_jK_T·,w

: 1≤j≤N. (5.2)

Due toTheorem 3.2,V_W andV_WT are subspaces ofᏺK andᏺKT, respectively. They are obviously closed.

The interpolatory properties of the spaces defined above are collected in the follow- ing result.

Theorem5.1. LetW= {w1,. . .,w_N}be a subset ofΩ2qsuch that{δ_k◦T:k=1,. . .,N}is linearly independent. The following assertions hold:

(i)if f ∈ᏺK, thens=p(f,V_W)is the unique element ofV_Wsatisfying δj◦T(s)=

δj◦T(f), j=1,. . .,N; (5.3) (ii)ifg∈ᏺKT, thens=p(g,VWT)is the unique element ofVWT satisfying

δj(s)=gwj

, j=1,. . .,N; (5.4)

(iii)ifg∈ᏺKT,f ∈ᏺK, andg=T(f), thenT(p(f,V_W))=p(g,V_WT).

Proof. (i) Let f ∈ᏺK. Due toCorollary 2.3, we know that everyδ_j belongs toᏺ^∗K, the dual ofᏺK.Theorem 3.2guarantees that the interpolation matrix

δ_i◦T^zδ_j◦T^wKz,w

i,j=1,...,N (5.5)

is well defined. We now shows that it is invertible. The quadratic formQdefined by the matrix in (5.5) can be written, with the help of the addition formula for spherical har- monics and ofTheorem 3.3, in the form

Q= N µ=1

N ν=1

c_µc_νδ_µ◦T^zδ_ν◦T^wKz,w

= N µ=1

N ν=1

c_µc_νδ_µ◦T^zδ_ν◦T^w

m,n∈Z+

a^qm,n(K)P^qm,n⁻² z,w

=

m,n∈Z+

a^qm,n(K)ωq

d(m,n)

d(m,n)

j=1

N µ=1

cµ

δµ◦TYm,n^j

2

.

(5.6)

Since the coefficients of this series are nonnegative, the equalityQ=0 occurs if and only if

N µ=1

cµ

δµ◦TYm,n^j

=0, j=1,. . .,d(m,n),m,n∈Z+. (5.7)

However, the fundamentality of{Ym,n^j :j=1,. . .,d(m,n), (m,n)∈A_K}inᏺK (Theorem 2.1) reveals that (5.7) is equivalent to

N µ=1

c_µδ_µ◦T=0, j=1,. . .,d(m,n),m,n∈Z+. (5.8)