By Brian Randell

ALGOL 60 implementation: the interpretation and use of ALGOL 60 courses on a computer

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Ak X. I f w e define k fn(x) then COROLLARY. 6. z x Aj j=l = into - flx)Il < l / n Let X. p f o r every x E X. be a f i n i t e B o r e l m e a s u r e on a c o m p a c t L e t f f n ) b e a s e q u e n c e o f continuous mappings F w h i c h c o n v e r g e s u n i f o r m l y on X t o a mapping f. i s c o n t i n u o u s and For each A E z. So f a r w e have only c o n s i d e r e d Bochner i n t e g r a t i o n w i t h re- s p e c t t o f i n i t e p o s i t i v e measures, b u t the extension t o real measures o r t o complex measures may p r o c e e d e x a c t l y as i n the s c a l a r case.

A, E = U A and each A m i s c l o s e d i n E . S i n c e E i s a m= 1 m B a i r e s p a c e , some A m has nonempty i n t e r i o r . 8 t h a t P k i s i d e n t i c a l l y z e r o on E f o r e v e r y k > m . The proof i s now complete. Then 24 MUJ ICA Theorems 3 . 6 and 3 . 7 reduce t h e s t u d y of p o l y n o m i a l s t o the case where t h e domain s p a c e i s one d i m e n s i o n a l . Next w e p r e s e n t r e s u l t s t h a t r e d u c e t h e s t u d y o f p o l y n o m i a l s t o t h e case where t h e r a n g e s p a c e i s one d i m e n s i o n a l .

For each faft) = f f a + t) Pmfa(t) = Pmf(a + t ) i s a v e c t o r space iso- fa a;F). E. two Given functions J€(u) and fg E rn P r n ( f g )( x ) x = pm-j f ( x ) P j g ( x ) j=O for a l l m E liVo and x - a) E U. D. Let m=O P,(x b e a power series from w i t h r a d i u s o f convergence R > 0 and w i t h each Let f : B(a;R) F be d e f i n e d by E co x m=O Pm(x - a) f o r each x E F, Pm cmtinuous. -+ f(x) = into B(a;R). MUJ I CA 40 Show t h a t f i s holomorphic on t h e b a l l B ( a ; R / e I .

### ALGOL 60 Implementation: The Translation and Use of ALGOL 60 Programs on a Computer by Brian Randell

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