ROCZNIKI POLSKIEGO TOWARZYSTWA MA TEMATYCZNEGO Séria I: PR ACE MATEMATYCZNE XXVI (1986)
T eresa M arkiewicz and L ucyna R empulska (Poznan)
On the application o f the Fou rier-Legendre series for the Laplace equation
1. Let r, s, t be the spherical coordinates of points, Q = {(r, s, t): 0 ^ r
< 1 ; 0 ^ t < 2n] and Q = {(r, s, t): 0 ^ r ^ 1 ; 0 ^ s ^ л; 0 < t
< 2 n\.
Denote by X m(Q) (m is a non-negative integer) the class of all real functions defined in Q, having the values independent of t and having the partial derivatives of the order ^ m continuous in Q. Analogously will be interpreted the symbol X m(Q).
Let A be the Laplace operator, i.e.,
_ d2 2 d 1 d_ 1
^ dr2 + r dr \ 2 ds2 + r 2 tan s ds + r2 sin 2 s dt2 ’
and A”и = A (A "~1 u)
for n = 2, 3, ... and u eX ^ iQ ) (A 1 == A).
In this paper we shall give a solution of the Dirichlet problem for the equation Anu(r, s, t) = 0 in Q. We shall construct the function и such that u<=Xn(Q) (n $5 1), An+l u{r, s) = 0 in Q, u (l,s) = /(s) and (dpu/drp)r= j = 0 for se< 0 , 7 i ) and p = 1 , ..., n, where / is a fixed function continuous in
< 0 , я).
The solution of Dirichlet’s problem for the equation Au{r, s) = 0 in Q was given in [3], p. 256. This problem for the Laplace equation of the order n in the unit disc was investigated in [ 1 ].
First we shall give some properties of the operator An.
Using the mathematical induction, we can prove
A” — r — Anu(r, s) + 2nAnu(r, s); ô
v r /
+ 2 (« — l)zl" 11 6u(r, s) + r2 Au(r, s) + 4r-r~
\ dr
and
for (r, s, t)eQ.
Lemma 1 and linearity of the operator An yield
L emma 2. Suppose that ue X 2n + 2(Q), n ^ 1. I f Anu{r, s) — 0 in Q, then the function v,
satisfies the equation An+1 v(r, s) = 0 in Q.
It is easy to verify the successive auxiliary result, too.
L emma 3. Suppose that u e X n+2(Q), w ^ l . I f (dpu/drp)r=1 = 0 for s e (0 ,n } and p = \ ,...,n , then v defined by ( 1 ) satisfies the condition ( dpv/ôrp)r=l = 0 for se <0, 7 t> and p = 1, . . n+ 1.
2. Let L2< —1, 1) be the class of all real functions Lebesgue-integrable with 2-nd power in <—1, 1). Let Lip a, 0 < a ^ 1, be the set of functions / e L 2< — 1 , 1 ) such that \/{x)—f(y)\ < M\x—y\* for — 1 , 1 >
(M = const > 0 ; see [4], p. 33).
The symbol E(f ] { f) will denote the best approximation of /eL2< —1,1) by algebraic polynomials of the order n (L5], p. 41).
( 1) v(r, s) = u(r, s)T
Let
oc
( 2 ) Z Cn(f) Pn(x) (X € < -1 , 1 »
be the Fourier-Legendre series of / e L 2 < - l , 1), i.e.,
,3) = and
i 2/7 ■*}“ 1 Ç
cnU ) = — j ~ f W P„(x)dx for n = 0 , 1 , ...
- 1
([3], p. 64, 74).
It is known ([4], p. 26, 153, 386) that if / has the derivative f ip)e U p ot (p is an integer ^ 1 ; a e ( 0 , 1 » , then
(4) max \ f ( x ) - f ^ c k( f ) P k(x )\ ^ M l n~p~a + 112
•X6 < - 1 , 1 > f c = 0
and
( f ( f i x ) - Z ck{ f ) P k(x )f dx)1'2 = £(„2)( / K M 2n~p~a
- 1 fc= о
(M j, M 2 = const >0). Arguing as in [2], p. 391, we obtain
L emma 4. I f f has the derivative / (p)e L ip a (p ^ 1, ae(0, 1 », then
00
k = 1
3. Let as in [1]:
(5) D °(rk) = rk; D"(r*) = ° п~1 (гк) + Г ~ ^ ~ for к = 0 , 1 , ..., n = 1 , 2 ,... and re < 0 , 1 ).
Using the mathematical induction, we can prove L emma 5. I f к = 0, 1, ... and n = 1,2,..., then
" d 9
(6) Z)"(r*) = r *+ Z Щ г’ (rE <0’
where Wq are some algebraic polynomials with coefficients depending on n only and such that
dp
dr* WAn n)
r = 1
0 if РФ q,
( - l ) p if p = q
for p = 0 , 1 , ..., n and q = 1 , ..., n (see [ 1 ]).
(7) \Dn(rk — rk+ l)| ^ M 3 (n)(l — r)n+ 1 (k + l ) nrk (к, n = 0 , 1 , re < 0 , 1 )) with M 3 (n) = const > 0 .
4. Let / e L 2< — 1, 1). Then the function
Hn( r , x j ) = £ m ^ ) c M ) p t (x)
k=
0
with Pk, ck{ f ) and Dn given by (2), (3) and (5), exists for re<0, 1) and
xe < - l , 1 >. 00
By the Abel transformation and the equality £ Dn(rk — rk+i) = 1 for
k = о
r e (0, 1 ) and n = 0, 1 ,
( 8 ) H .(r, x \ f)—f (x) = £ D” (r * - r * +,)( £ c ,(/ )P ,(x )-/(.*))
k=0
0
(re < 0 , 1 ), xe < — 1 , 1 » for
Formula ( 8 ) and inequalities (4) and (7) lead to
T heorem 1. I f f has the derivative f {p) ( p is a integer > 1) continuous in <— 1, 1) and f {p)e Lip a, ae(0, 1), then
max I/(x) — Hn(r, x ;/ ))<
-1
( l - r )"+1
( l - r )"+1 |log(l — r)l for re < 0 , 1 ) (Af| = M 4 (n, p, a) = const > 0 ).
if p + oc- 1 > n,
if p + a —I = «
5. In this part we shall consider functions defined in <0, тс). In this case the symbol Lip a, ae(0, 1>, will be defined analogously as in Section 2.
T heorem 2. Suppose that f has the derivative f (2n+1), with an integer n
^ 0, continuous in <0, к} and f (2n+ 1)e Lip a, ae(0, 1). Then the function Un( f ) defined by formula
(9) U„(ry s ;f ) = £ D "(r*)dk(/) Pk(cos s), k = 0
where Pk, D" are defined by (3), (5) and
Я
dk i f ) = 2 I/ (s) (cos s) sin s ^s’
о
satisfies the conditions:
1° Un( f ) e X n(Q),
2° An+1Un( r , s ; f ) = Ofor (r, s, t)eQ, 3° Un( 1, s ;f ) = f(s ) for se <0, те),
4° = 0 f or SE <°> пУ and P = U ■■■, n.
Proof. It is known ([3], p. 256) that the function U0( f ) has properties 1°—3° if — eLip a, ae(0, 1). df
ds
The inequality
|Pk(x ) K 1 ( x e < - l , 1>; к = 0, 1, ...)
([3], p. 69) and the Bernstein inequality ([5], p. 232) imply the estimation
&1
dsq pk (cos s) < ( * + W ( sg < 0 , tu >; k, q = 0 , 1 , ...).
Hence, by Lemma 4, L 0(/)e X 00^ ) and U0( f ) e X 2n(Q).
By ( 6 ) and (9),
m d9
(10) Um(r , s ; f ) = U0( r , s ; f ) + £ Wq(r; v$— U0{r, s\f) for (r, s, t)eQ and 1 ^ m ^ n.
Consequently,
(11) V m( f ) e X » { Q ) and Um( f ) e X 2f- m(Q) for 0 ^ m ^ n.
Clearly, by (5) and (9),
у d
(12) Um(r, s ;f ) = t/w_ ! (r, * ; / ) + - “ Vm - 1 (Г, 5 ;/)
for (r, s, O g Q and 1 ^ m ^ n.
Applying the properties of U0(f ), (11)-(12) and Lemma 2, we obtain 2°
for (/„(/).
By Lemma 5, (9) and (4),
Un( 1, s;/) = £ dk(/ )Pk(cos s) = /(s) lc= 0
for s g <0, tc >. Hence, we have 3°.
Formulae (12) and (11) imply the equality L i (r, s;/)^ = 0 for
sg <0, tt >. Now, by (11)—(12), we can apply Lemma 3. For U „{f) we obtain 4°. Thus the proof is completed.
1 — Prace matematyezne 26.1
[1 ] Z. D o p ie r a la , L. R e m p u ls k a , On the summability o f series by harmonie methods, Comment. Math. 23 (1983), 11 25.
[2 ] S. K a c z m a r z , H. S te in h a u s , Théorie der Orthogonalreihen (in Russian), Moscow 1958.
[3 ] N . N. L e b ie d ie v , Special functions and their application (in Russian), Moscow 1963.
[4 ] P. K. S u ie t in , Classical orthogonal polynomials (in Russian), Moscow 1979.
[5 ] A. F. T i m a n , The theory o f approximation o f real functions (in Russian), Moscow 1960.