... On homogeneous polyharmonic polynomials

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: СОММЕ NT ATIONES MATHEMATICAE X I (1968)

J. Mtjsiałek (Kraków)

On homogeneous polyharmonic polynomials

1. In this paper we give two theorems about the number of linearly independent homogeneous polynomials of degree n in the variables a?!, which are fc-harmonic (n — p > fc) and of the form

(1) P n(X) = P n(xx ... хг/ .

+ ...+ ip = n

We also prove theorems on the expansion of ^-harmonic functions into series of homogeneous A;-harmonic polynomials. Theorem 1 is stated in the paper [2] of K. Kicmeyer, but the proof seems to be incomplete.

2. Lemma 1. The number of the coefficients in polynomial (1) is equal to number of non-negative integer solutions of the equation

h + i% ~f • • lv = n, i.e., it is equal to n + p — 1

P - 1 N ( p , n ) . We omit the simple proof.

Theorem 1 (see [2]). The number of all linearly independent harmonic polynomials of the fojm (1) is equal to

2n + p — 2 In + p — 3 n \ n — 1

ln + р —1\ _ In + p — 3

\ P — 1 ! \ P - ! — M ( p , n ) . Proof. Since APn (X) = 0, where X ~ (xx, ..., xp), we get APn{X) = У 2)(гг + 1 )ai1+2,i2,...,ip +

i x + ...+ ip = n - 2

+ ( Ь +2)(г2 + 1)«г1,(2+2.г3,...,гр+ • * • + (V + 2)(*j) + l)fli1... +

+ 2 (h + 1) (^2+ 1 ) ^ + 112 + 1,13,..„ip + 2 (ix + l ) ( i 3 + l)<*i1 + l,i2,i3 + i,i4,...,ip

“b • • • 4~ 2 ( i p - i ~ b 1 ) {ip~\~ 1 ) йг1,г2,..„ ip _ 2, ip_x + l» ip + i'l — ^

2 8 4 J . M u s i a ł e k

We thus obtain the system of N { p , n — 2) equations

(2) (^3"f" 2) (i3 ⁺ 1) л г1,<2)г3+2,г4,...,гр ~f" i h 2) (г4+ 1) a i 1,i2Ą , i 4i+2,i5 ip ~ f

+ ^{• • •} + ( Ś + 2) (^p+ 1 ) ^ г 1,...,г :р_ 1,гг)+ 2 + 2 ('i3+ 1) (L,+ 1) a i 1,i2 ,ś3 + l,ś 4 + l , i 5, . . . Ji J) H~

+ 2 (г 3 + 1 ) ( ^ 5 + 1 ) а'г1,г2>г3 + Х5г4,г5 + 1,г6,...)г:р + • • • +

+ 2 (г’з + 1) (b?+ 1) tti1,'i2,i3 + aĄ,...Jip+lH“ 1) (%> + 1) яг1,...,гр_1 +1,^+1 = ^ with conditions у > 0 , г4+ ... + iP = п — 2. This is а system of A(^p, n — 2) equations with N { p , n ) nnknows. Ordering lexicographically the indices in the system (2) we obtain a system whose matrix has

rows and \ nJ^P 1 1 columns.

\ P - 1 I

' Let us consider the r x r snbmatrix of the coefficients at the nn­

knows where r = \ ^ „ 4. Its terms on the diagonal are different

\ P - 1 I

from zero, and those below it are equal to zero, hence its determinant is different from zero. Therefore the number of linearly independent solutions of the system (2), i.e., the number of linearly independent har-

2n~\-p — 2 monie polynomials (1) is equal to ---

n 3. ^{Le m m a} 2. Let

I n + p — 3\

P - 1

(3)

p i / y \ p M (p ,n -2(ft-i)) , j r ,

* n-2(k-i)\^-) j ---i jrn-2(k-i) \Л-) be systems of linearly independent homogeneous hay monie polynomials the polynomials of each row being of degree n, n — 2, ..., n — 2(Jc — 1) respec­

tively, then the system of all those polynomials is linearly independent.

Proof. Let

(4) CIP i(X ) + ... + Of«!,'*>(X) +

+ C?JPL*W + ■ • ■ + c f <».*-’ >pM<iv>-2>(X) + . . . +

+ c l P i ^ i)m + - - - + G u {r^ 4k~l))p ^ : T ~ l))(X) = о where Cl are constants, be a linear combination of polynomials (3). Applying to (4) the linear transformation х г — tylt ... j xp — typ, — oo < t < o©?

we get

(5) f [ClPi ( Y) + ... + c**te>n)pM(p,») ( Y ) ] + ... +

+ tn~4k~l) [С&РЦ*-!) ( Г) + .. • + Of(P.*-2(*-1)p^((^-*(*-l)) ( Y ) ] S 0 , where Tc = 1 , 2 , . . . Dividing both sides by f ~ 4 k-') an(j passing to the limit with i -^0 we obtain

c i P i -2,*-i,(T) r i M ((p ,n ~2(k- *)) p^((Pi#-2(fc—1)) / y , -Ł ( » i- 2 - l) к V )

Since the polynomials P^_2(*_i)( T ), ..., Pn-^ik-vf* ^ (T ) are linearly independent we infer that Cl = ... = (j^C>’n- 4 k- 1)) _ q Taking this into account, we now divide (5) by f 1- 2^ - 1) and this leads to (?*_! = ...

_ ęjM(p,n-2(k-2)) _ q

Continuing the procedure step by step we infer that all the coefficients Cl are equal to 0.

Lemma 3. Under the assumptions of Lemma 2, the polynomials

рЦх ) , . . . ,р^ - пЦх ),

e ^ p i _ 2lk. , ( X ) , е^- чр^Кгп' *' -Ц)(Х), where r2 = ^ + ... + , are linearly independent.

Proof. Let

C\Pl{X) + ... + Cl1 (p'n)P f M (X) +

+ r2 [C£Pi _s( X)+ . . . + c f ( X ) ] +. . . +

+ r (‘- 1) { C l P l - w ^ X ) + . . . + C f <”■ (X )]_ 0 >

where C\ are constants. Introducing the spherical coordinates

x x — rcos^j, x 2 = rsinę^cos^, •••? = rsin7?1...sinę5:p_2sin<pi,_1, where <pp-2 ^ ~ , 0 ^ <pp-i ^ 2тг, 0 ^ r < oo, we obtain

rn[C|Pi(<f) - f ... +С?‘{г’’щР?1,№'щ(Ф)]+ ... +

+ ^ [ c i P i . ^ m + . . . + (л»-»(»-1)) р ^ « - 1*,*-1)) (ф)] = о where Ф(<рх, ..., = Ф = (cosqpj, sin ^ c o s^ , .. . , snnjq ... sirкрр_х.

Dividing this equation by rn we obtain (4) and we may apply Lemma 2.

Since the points Ф run through the unit sphere, we obtain C\ = 0.

4. Th eo r em 2. Let the assumptions of Lemma 2 be satisfied. Then polynomials (6) form a maximal system of lc-harmonic linearly independent homogeneous polynomials of degree n in the variables x x, ..., xp and their number is equal to N ( p , n) — N ( p , n — 2k).

Proof. It is well known (see[l]) that each of the polynomials (6) is ^-harmonic, i.e., AkP\(X) = 0. Each of the polynomials (6) is of the form (1), hence, by Lemma 1 has N ( p , n ) coefficients. This polynomial is fe-harmonic; therefore its coefficients satisfy a system of N ( p , n — 2k) equations with N ( p , n ) unknows. Arguing similarly as we did in the proof Theorem 1, we infer that the maximal number of those linearly independent polynomials is equal to N {p, n) — N (p , n — 2Tc).

2 8 6 J . M u s i a ł e k

On the other hand, the number of all polynomials in (6) which are linearly independent is equal to

[ N { p , n ) - N ( p , n - 2 ) ] + . . . + \ N ( p , n - 2 \ l c - l ) ) - N ( p , n - 2 k ]

= N ( p , n ) ~ X ( p , n-~2k), q.e.d.

5. We are ^{п о л у} going to prove two lemmas which enable us to prove a theorem analogous to Theorem 2, concerning the maximal number of linearly independent homogeneous quasi fc-harmonic polynomials of the form (1). By a quasi fc-harmonie polynomial we mean a polynomial satisfying the equation

L kP n{xx, xp) = 0 p

where / / = L L k 1 and Lu = У aik- —-— is an elliptic operator, dxidxk

aik — aki being real numbers.

Let

v

(T) Hi = A.ijX{ (i — 1, . • •, p)

7 = 1

be an linear transformation Y — T ( X ) , Y = (yx, ..., yp) such that X = T~l {Y) transforms the quadratic form F (X ) — aikXiXk into a diagonal form h — 1.

Let D (m) (E) denote the space of real-valued functions on a domain E (E c: B p) which have all partial derivatives of order m on E, and let E x be the image of E inder the transformation T. The transformation T : E E x induces a linear operator T *: В ^ ( Е г) -> D(m\ E ) defined as T* v — u, where x e B ^ i E f } , u{X) = v [ T (X )) for X in E. Since T maps E on to E x, T* is one one. A straightforward computation shows that T* A = LT*, i.e,, Av{Y) = Lv(T(Xj), where Y = T{X). It follows by induction that T* Ak — L kT* for each к (indeed, if it true for a cer­

tain k, then T*Ak+l = T*Ak A = L kT*A = L kLT* = L k+1 A).

As a particular case of the identity T* Ak — L kT* we get the follow­

ing lemma:

Lemma 4. I f v(Y ) is a solution of the equation Aku{Y) — 0, then the function u (X ) = v(T(X)) satisfies the equation L u(X ) = 0.

Now, let B x(X ), R a(X), where s = N ( p , n) — N ( p , n — 2k) be a maximal linearly independent system of ^-harmonic polynomials of degree n, and let Qn{Y) = T*Rn(Y).

Lemma 5. The system Q1, Qs is a maximal system of linearly in­

dependent quasi k-harmonic polynomials of degree n .‘

Proof. If qx, (I > s) were another linearly independent system of quasi й-harmonic polynomials, then T*qx, .. . , T*qx would also be a linearly independent system (indeed, if ]?CiT*qi — 0, then

£ W* = (T*)_1 (Jj ci T *Qi) = 0

and hence cx — ... — сг = 0) and the system R x, . . . , R S would not be maximal.

Combining the foregoing lemmas with Theorem 2 we get the following theorem:

Th e o r e m 3. There are exactly N { p, n) —N ( p, n — 2k) к-quasi har­

monie polynomials of the form (1).

6

.

Le m m a 6. Let the function u(X ) be k-harmonic in a region D c R n (n ^ 2). Then for each ball K ( X 0, R) of centre X 0 and radius R contained in D we have

k - l

(7) m { X 0, R , и) — [QnR n~l)~l S S u ( Y ) d S = JT an>iR 2i A ^ u ( X 0) 9K(Xq,R) i—0

where Qn denotes the area of the n-dimensional sphere in R n, and

1 for i — 0,

^ -- i

[2гИп{п-\-2) ... {n-\-2i-\-2)Yl for 1.

The converse statement is also true. Using this Lemma we shall prove the

Th e o r e m 4. Let un(X) be a sequence of k-harmonic functions in the ball K { X 0, R) uniformly convergent in K { X 0, R) and let the sequence of functions A ^ u n( X ) , 7 = 1, . . . , к — 1, be uniformly convergent in K { X 0, R).

Then the function U (Y) = lim /itw(Y) is k-harmonic in the ball K { X 0, R).

n—>oo

P roof. By Lemma 6

k - l

m(r, X „ «.) = (Й Д* -1)"1 S S un(Y ) d S = У а ,>4Д * А ^ и п( Х 0).

i—1 Passing to the limit as n oo we obtain

m(r, X Q,l\m u n) = {QnR™-1)”1 S S lim wn(Y)di8

k - l

= antiR 21 А {г)Итип( Х 0).

г=1

Since U(Y) = limww(Y), the function U(Y) satisfies (7). By Lemma 6, V is fc-harmonic in D.

2 8 8 J . M u s i a ł e k

The last result enables us to obtain a theorem about the expansion of ft-harmonic functions into a series of homogeneous quasi ^-harmonic polynomials.

Let u (X ) be an analytic solution of the equation L ku ( X ) = 0 in the ball K ( X 0, R), where X 0 = (0, . 0 ) , then it may be expanded in K ( X 0, R) into the Taylor series

00

(8) u( X) = Q0(X) + Q1(X) + ... + Qn(X) + ... = 2 < j t (X)

i = l

where

Qn'yX)

i l + ... + ip = n

= n\{ix\... ip!) 1 du(X) I Ъх\1... (Щ р\х0.

Th e o r e m 5. Let the function u(X) defined by (8) be quasi k-harmonic in the ball K ( X 0, R). Then every of the polynomials Qn{X) is quasi k-harmonic.

Proof. We shall first deel with the of case fc-harmonic functions (i.e., for L = A), by our assumption

A ^ u ( X ) = y j tk>Qi(X) = 0 ;

г=0

therefore A ^Q i(X ) = 0, i = 0 , 1 , 2 , . . . Applying transformations (T) we easily get the general statement.

Co r o l l a r y. Each quasi k-harmonic function in the ball K ( X 0, R ) can be uniformly approximated in this ball by homogeneous quasi k-har­

monic polynomials.

R eferences

[1] M. N ic o le s c o , Les fonctions polyharmoniques, Paris 1936.

[2] H. N ie m e y e r , Lokale und asymptotische EigenscJiaften der Lósungen der Helmholtzschen Schwingungsgleichung, Jahresber. Deutscli. Math. Yerein. 65 (1962),

S. 1-44. \

[3] W. W a lte r , Mittelwertsatze und Hire Verwendung zur Losung von liand- wertaufgagen, Jahresber. Deutsch. Math. Yerein. 59 (1957), S. 93-131.