Estimates for spherical harmonics

Citation for published version (APA):van Berkel, C. A. M., & Eijndhoven, van, S. J. L. (1989). Estimates for spherical harmonics. (RANA : reports onapplied and numerical analysis; Vol. 8920). Eindhoven: Technische Universiteit Eindhoven.

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Eindhoven University of Technology Department of Mathematics and Computing Science

RANA 89-20

September 1989

ESTIMATES FOR

SPHERICAL HARMONICS

by C.A.M. van Berkel

and

S.J.L. van Eijndhoven

Repons on Applied and Numerical Analysis

Department of Mathematics and Computing Science

Eindhoven University of Technology

P.O. Box 513

5600 MB Eindhoven

The Netherlands

Summary

ESTIMATES FOR SPHERICAL HARMONICS by

C.A.M. van Berkel and

S.J.L. van Eijndhoven

In this paper we present Loo- and Lrestimates for spherical harmonics in which partial differentiation operators and multiplication operators are involved. Our results are com­pared with classical results of Stein, Seeley and Calderon and Zygmund.

A.M.S. Classifications: 26DlO, 33A45. Key Words: spherical harmonics, homogeneous polynomials, reproducing kernels.

-2-

1. Introduction

The present discussion starts with the space IRq. q ~ 3, provided with the Euclidean inner

product X· Y = Xl Y 1 + ... + Xq Yq. By cfl-l we denote the surface measure on the unit sphere Sq-l in IRq, normalized by the relation

dx = r q - l dr dcfl- l (co) (i)

where dx = dxl ... dxq • So we have

coq := J dcfl-1 = 21tq12 r(q 12). S,-1

(ii)

Consider the Hilbert space LZ(Sq-l) of square integrable functions on Sq-I with the

natural inner product

(f.g>L2(Sf-I ):= J f(~) g(~) dcfl-l(~) ,f.g E LZ(Sq-I).

5q- 1

We introduce the following spaces of polynomials in q variables

pq : the linear space of all polynomials

(iii)

HZt : the linear space of all m-homogeneous hannonic polynomials (iv)

yZt : the linear space of all restrictions of polynomials in HZt to Sq-l .

All spaces can be endowed with the Lz(sq-l)-inner product. The vector spaces HZt and

yZt are finite dimensional each with dimension d1,. given by

dZt = (2m+q-2)(m+q-3)! . (q-2)! m!

And there is the orthogonal decomposition

... Lz(Sq-l):::: $ YZt·

m=O

In pq we introduce another inner product,

(PloPZ)pq :=Pl("V)P2(X) Ix=o, Pl,PZ E pq

where V represents the gradient vector [ a; 1 ••••• a:,],

(v)

(vi)

(vii)

For harmonic homogeneous polynomials this pq-inner product is related to the LZ(Sq-l)­

inner product as follows. Let hI and h2 be harmonic homogeneous polynomials of degree

m and n respectively. Then

- 3 -

(viii)

(for a proof we refer to [Ma, Lemma 2.14 and Lemma 2.25] or [Be, Chapter 1 D· The Gegenbauer polynomials C~, m E iNo, A> -I, introduced in [MOS, p. 218 ff.] obey

the orthogonality relations

Now for mE iNo and q~ 3 we set

P'/n = m! (q - 3)! C7j2-1. (q+m-3)!

Then P'/n(l) = 1 and according to (ix)

I 00 f P'/n(t) P%(t) (l-t2 )t(q-3) dt = q ~lIm. -I ooq_1 d'/n

It is a well-known result, cf. [Mil, Lemma 7, p.14]. that for all I E Y'/n and S E Sq-l

dq

I(s) = ~ J P'/n(s' 00)/(00) doll-I (00). ooq sq-I

(ix)

(x)

(xi)

(xii)

dq

So (S. 00) f-7 ..-!'!... P'/n(s' 00) is the reproducing kernel of the Hilbert space Y'/n and for any ooq

orthonormal basis {e'ln,j : 1:::;; j:::;; d'ln} of Y'ln we have

(xiii)

Using the reproducing kernel property of the P'In we obtain for all h E H'In and; E Sq-l

[dq ]\i [dqj\i

I he;) I :::;; 00: P'/n(s- S) IIhIlL2(SQ-I) = ~ IIhIlL2(Sq-l) , (xiv)

In this paper we present various Loo- and Lz-estimates for spherical harmonics. We conclude the introduction with some notations concerning multi-indices and partial derivatives. By 10k we denote the multi-index with components eJ =~kj, k,j E {l, ... , q},

and 10 = (1, ... , 1). Furthermore, for all k E IN 0, X E IR q, and for all multi-indices

a, ~ E IN6

I al = al + ... + aq

a+k~= (al+k~), ... ,aq+k~q)

as: ~ :~ al s: ~1>"" aqs: ~q

(a)p = (al)!lJ· ... • (aq)p,

-4-

here (. ) j denotes the Pochhammer symbol, which is defined by

(b),'= r(b+j) . 012 J' r(b) ,J = , , , ....

Utilizing the inequalities

[ n+nm~kk-ll «::; [n +nm] _ _ ,n;:::k;:::O ,m:?:.I;:::O,

it follows that for all multi-indices a, ~ E !Nt, with ~;::: £,

(~)aS: (1~I-q+1)lal'

(xv)

(xvi)

(xvii)

(xviii)

By akfwe denote the partial derivative of fwith respect to the k-th variable. As usual, for

all multi-indices a E !Nt, we write

Clearlyaa maps H~ onto H~-Ial' where H% is taken to be the trivial space for n < O.

2. Loo- and Lrestimates for spherical harmonics

We start with integrating monomials oof'> over the unit sphere Sq-I.

Lemma 1. Let ~ E JN6 be a multi-index.

(i) If~jisoddforsomejE {l, ... ,q},then J oof'>da'l-l(oo)=O. sq-I

(ii) For all multi-indices a E !N'!, satisfying as: ~, we have

(a+1. £)p.-a J OOlP da'1-1(00) = z J 002a da'l-I (00).

S,-I (Ial q)Ip.-a1 S,-I

Proof

(xix)

Consider the integral J ooP da'l-l(oo). Substituting OO=tel + ",,1-t2 'fl, with -1 s: tS: 1 and sq-I

T\ E span {ez, ... , eq } we derive

-5

1 J ro?> d<fl-I(ro)= J t?>'(l_t2)t(q+IJ3H,-3) dt J T1~ ••. T1!q d<fl-2 (T1) = Sq-l -I Sq-2

{

o , if~1 is odd = r(l(Pl+l»r(l(q+I~I-~l-l»

2 ret (~+ I PI» /2 T1~ ... T1~q daq-2

(T1) , if ~1 is even.

Now induction arguments yield the first part of the lemma. FurthelIDore, with the aid of the above fOlIDula and the relation z rez) = rez + 1) we obtain

~1 +l J Ol2@+£') d<fl-1 (ol) = ; J ol2P d<fl-1 (ro). sql 1 PI + 2 q sql

And from this fOlIDula we deduce for all m E IN 0 and k E {I, ... , q},

(Pk+t )m J Ol2<J3+me') daq-1(Ol) = 1 J Ol2f} d<fl-1(ro). sq-l (IPI +2 q )m sq-l

U sing these integral relations the second part of the lemma follows by induction. I]

Taking a. = 0 in the preceding lemma, we get explicit expressions for integrals of mono­mials:

Corollary 2. For all multi-indices P E lNg,

27tq12 (l E)?> J Ol2?> d<fl-1 (ro) = 1 : . sq-l r(2 q) (2 q)I?>1

I]

Theorem 3.

Let p be an m-homogeneous polynomial and let a E IN6 be a multi-index. Then

(m+l)lal IIx o. pllt2(sQ-l):::; (}2 IIpllt2(sQ-l).

m+2 q)lal

Proof.

The proof is by induction to the length of a E INg. Obviously equality holds for a. = O.

Suppose the estimates are valid for all multi-indices a E INg with length I a 1 ::;; n for some n 2 O. Let P E INg be a multi-index with length 1131 = n + 1. Then there exists j E {l, ... , q} such that Pj 2 1. Since p is an m-homogeneous polynomial, x 2$-ei) Ip 12 is a 2(m +n)-homogeneous polynomiaL Therefore, x2(J3-ei) Ip 12 is a linear combination of monomialsxY with Iyl =2(m+n):

-6-

x 2(]Hi) Ip(x)1 2 = L c(Y)xY• Iyl =2(m+n)

With the aid of Lemma 1 we estimate

J ro2P lp(ro)1 2 dcfl-1(ro)= L C(y) J roy+-2r!dcfl-1(ro)= S9-1 Iyl =2(m+n) So-I

YI + 1 J == L c(y) roy dcfl-J (ro) S; Iyl =2(m+n) 2(m +n) + q S.-I

S; +1 J ro2(]Hi) Ip(ro)1 2 dcfl-I (ro). 2(m +n) + q S.-I

SO using the induction hypothesis we obtain

(m+i)n+l J ro2P Ip(ro)12 dcfl-1(ro)S; J Ip(ro)1 2 dcfl-1(ro) S,-1 (m +i q)n+l S.-I

which completes the proof.

Note that equality in the above theorem holds if (l = m Ek and p(x) = (Xk)m.

n

Next we pay attention to estimates for partial derivatives of spherical harmonics. But we gather some auxiliary results first.

Lemma 4. Let p be a homogeneous polynomial of degree m. Then for each k e {I, ... , q}

J (akP)(ro)dcfl-1(ro)==(m+q-l) J rokP(ro)dcfl-1(ro). S'I-I S.-I

Proof.

We may as well assume that k == 1. Let 'II denote a compactly supported differentiable function on /R+. Using the homogeneity of a1 p we have

J 'I'(r)rq- 1 dr J (a1P)(ro)dcfl-1(ro)= o S.-I

== f {J Ixl-m+I 'V(lxl)(a1P)(x)dxd dx2 ... dxq . /Rq-l IR

Applying partial integration the latter integral equals

J {f Ixl-m- 1 [(m-l)'II(lxl)-lxl ¥(lxl)]XIP(x)dxddx2 '" dxq = IRq-I IR

= J [(m-l)'V(r)-r¥(r)]rq-1dr J rolP(ro)dcfl-1(ro) o ~~

and the result follows.

- 7 -

Corollary S. Let f and g be homogeneous polynomials of degree m and n. Then for each k E {I, ... , q}

Proof.

Putting p = f· Ii in Lemma 4, we are done. o

Note that the above relation is a special case of the integral formula established in [Ma, Theorem 1.16]. However, our proof is based on simpler arguments.

Lemma 6.

Let hE Ht and let k E {I, ... , q}. Then we have

ndkhlll,(sq-I) = (2m +q - 2) [em ++ q) IIXk hlll;2(sQ-I) - + IIhlll;2(s4-1)].

Proof.

Let f = hand g = dA; h in the previous corollary. Then we derive, writing (. " ) instead of (. " k,(S4 1)

ndA;hIlL2(S4-1) (i3th, dkh)=(dA;h, dA;h)+(h, dth)=

= (2m+q-2) (xA;h, dA;h) =

= (2m +q - 2). + [(dA;(XA; h), h) + (Xk h, dkh) - (h,h)] =

= (2m +q -2) [em +t q) (xt h, h) - + (h,h)]

where in the last step Corollary 5 is used again.

As a consequence of Lemma 6 and Theorem 3 we get immediately

Corollary 7.

Leth E Ht and letk E {I, ... , q}. Then

ndkhIl1,(S4-'):S; m(2m +q - 2) Ilhlll,(S4-1).

(]

(]

We recall that for each multi-index a E IN'!" the operator da maps Ht onto Ht-Ial. Now we arrive at the main theorem of this paper covering Loo(Sq-l) - and L 2(Sq-l) - estimates for partial derivatives of spherical harmonics.

- 8 -

Theorem 8.

(i) Let 0. E INZ. Then for all h E HZt

lIaahlit SO-I :s;; m!(2m+q-2)!! IIhllt2(so-I).

2( ) (m-lo.l)! (2m+q-2-210.1)!!

(ii) Let 0. E INZ. Then for all h E HZt and ~ E Sq-l

[ dZt-lal m! (2m+q-2)!! ]Ih

l(aah)(~)I:S;; (i)q (m-lo.l)! (2m+q-2-210.1)!! IIhIlL2(SQ-I).

Proof.

The first part of the theorem follows from Corollary 7 and a simple induction argument. The second part is a direct consequence of the inequality (xiv) presented in the introduc­tion and the first part of this theorem. 11

We now give another, very elegant proof of Theorem 8(i) wltt!rt! the correspondence (viii) between the pq -inner product and the L 2(Sq-l )-inner product is utilized.

Second proof of Theorem 8(i).

Let 0. E INZ with 10.1 :s;; m. Since hE HZt, h can be written as

hex) = L b(~) xj} 1j}I=m

and so

(aa h) (x) = L (~-o.+£)a b(~)xlMx. 1j}I=m

From relation (viii) it follows that

n-q/2 2m-

1 r(m +t q) (h,hk2(sO-I) = (h,h)pq = L ~! I b(~) 12. 1j}I=m

The polynomial aa h belongs to HZt-lal and so, applying (viii) once more and using (xviii) thereafter, we estimate

n-ql2 2m-lal-l r(m - 10.1 +t q) (aa h, aa hk2(so-l) =

= (aa h, aah)pq = L ~! (~-a+£)a Ib(~)12:s;; 1j}I=m

:s;; (m-Ial +l)lal L ~! Ib(~)12= 1j}I=m

m! -q12 2m- 1 r( 1) (h h) = ·n m+-q, L (SO-I) , (m - 10. I)! 2 2

yielding the wanted result. n

-9-

Using Stirling's fonnula we obtain from Theorem 8(ii) the following asymptotic fonnula.

Corollary 9. For each multi-index a E IN6 there exists a constant Cq•1al > 0 such that for all hE H;k and ~ E Sq-l

[]

Although the result of Corollary 9 is known (cf. [CZ, fonnula (4)] or [Se, Theorem 4b]) our proofs are more concise and based on elementary techniques.

3. Comparison with corresponding results.

Stein has proved the following pointwise estimate, cf. [St, p.276].

For any multi-index a E IN6 there exists a constant Cq,a such that for all hE H;k and for all ~ E Sq-l

(xx)

From Corollary 9 we see that instead of mtq+1al , in Stein's estimate, we can take

mtq-l+lal. This improvement by a factor m has consequences for the continuity and com­

pactness of the differential operators aa on certain function spaces of hannonic functions (cf. [Li2, Theorem 7]). Liu and Martens proved in [LM, Theorem 4] the following result.

Let k E {l, ... , q}, let~ E Sq-l and let hE H;k. Then

I (d. h)(l;) I ~ m [ ~ ] " IlhIlL,~ .. ). (xxi)

We shall show that this estimate is best possible. According to the estimate of Theorem 8(ii), we get

I (ak h) (~) I ::; [ 2m +q -4] ~ . m [ d;k] 'h IIhIlL2(Sq-l).

m+q-3 ffiq (xxii)

So the estimate (xxi), settled by Liu and Martens, is about a factor 12 sharper than ours. In their proof they explore the Gegenbauer polynomials. Using an orthonormal basis {4,1 : 1::; j::; d'ln} in H'In they write

- 10-

~ h(~) = L (h, e7n,jk1(SQ-l) e7n,j(~).

j=l

Hence, by the Cauchy-Schwarz inequality,

[

tf! ]Ih 1 Cdlc h) (~) 1 ~ IIhIlL2(S9 I) ~ 1 (die e'ln,J) (~) 12

J=1

It turns out that

from which the result can be obtained.

Take k = I and ~ = el = (1,0, ... ,0). We define ho E H7n by

Then

~----ho = L (a l e7n,j) (el) e7n,j'

j=l

I (a, ho) (e,)1 =m [ roo] "llhOIIL,(s'-»'

(xxiii)

(xxiv)

(xxv)

(xxvi)

(xxvii)

So their estimate is best possible. This answers a question raised in LLM]. We note that in [LM] no estimates for I (d<X h) (~) I with I al > 1 are given. In fact, their techniques are not appropriate for deriving these estimates. Finally we present another method, based on the reproducing kernel of Y7n-l, to obtain supremum norm estimates for the partial deriva­tives dlch. Let h E H7n and let k E {I, ... , q}. Using the reproducing kernel of Y7n-l, the orthogonality relations of spherical harmonics, Corollary 5 and the Cauchy-Schwarz ine­quality we get for all ~ E Sq-l,

(xxviii)

If we take ~ = elc and substitute ill = telc + ;}1-t2 11 in the integral, a tedious calculation of the integral yields

[ dq ]14

I (dlc h) (ek) I ~ B7n 00: IIhIl L2(S'-') ' with (xxix)

- 11 -

B1n = [1 (m +q -2) (m +q-3) (m+t q-3) +(m -1) (m-2) (m+t q-1) ]lh . 2(2m+q-4) (m+t q - 1) (m+tq-3)

• [ (2m+ q -2)(2m+q -4)m] liz (xxx) m+q-3

Since B'ln > m, an exact calculation of the integral on the right hand side of (xxviii) will

yield an inequality which is less accurate than the inequality (xxi) of Liu and Martens.

Hence equality in (xxviii) will never be attained. This leads to the following observation. In his thesis [Lil, p.104], Liu proves that the function ro 1-7 rokh(ro), defined on the

sphere Sq-I , extends to a hannonic polynomial in the space H'ln-l 6) H'ln+l' i.e. there exist

polynomials Pm-l E H'ln-l and Pm+l E H'ln+l such that

Pm-l (ro) + Pm+l (ro) = rok hero) • ro E sq-l. (xxxi)

Since equality in (xxviii) will not occur it follows that neither Pm-l nor Pm+l is identically equal to zero.

- 12-

References

[Be] Berkel, C.A.M. van, Gel'fand-Shilov's function spaces of type S for several independent variables. EUT -Report, to appear.

fCZ] Calderon, A.P. and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math., 79 (1957) 901-921.

lLi1] Liu. G.-Z., Evolution equations and scales of Banach spaces, thesis, June '89, Eindhoven.

[Li2] Liu, G.-Z., Hilbert spaces of harmonic functions in which differentiation opera­tors are continuous or compact, RANA 89-09, June 1989.

lLM] Liu, G.-Z. and F.lL. Martens, Improvement of an estimate of spherical harmon­ics, RANA 89-10, June 1989.

[Ma] Martens, F.lL., Function spaces of harmonic and analytic functions in infinitely many variables, EUT-Report, 87-Wsk-02.

[MOS] Magnus, W., F. Oberhettinger and R.P. Soni, Formulas and theorems for mathematical physics, Springer, Berlin, 1966.

[Mii [ Miiller, C., Spherical harmonics, LNM 17 , Springer-Verlag, Berlin-Heidelberg, New-York, 1966.

[SeJ Seeley, R.T., Spherical harmonics, Amer. Math. Monthly, 73 (1966) 115-121.

[St] Stein, E.M., Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, New Yersey, 1970.