What follow s ar e notes covering lectures given at a Regional Conferenc e a t Fayette -
ville, Arkansas in Jun e 1971 , sponsored b y CBMS and th e NSF . M y aim in thes e lecture s
was to introduc e som e o f th e more recen t development s o n functio n algebras , presuming n o
detailed knowledg e o n th e listener's part. O f cours e the n i t wa s necessary t o develo p som e
of th e foundations , an d som e olde r result s are interspersed amon g th e newe r ones . Som e
areas are also slighted considerably : fo r example , we shall have nothing t o sa y abou t analyt
ic structure i n the spectrum ; and becaus e o f th e technica l aspect , we will really cove r onl y
one resul t i n rationa l approximatio n i n th e plan e (an are a where considerabl e recen t progres
has been made via a combination o f th e classica l and abstrac t techniques) , and wil l wholly
ignore higher dimensions .
Here, briefly, i s what w e attempt t o cover : (1) some basic notions, results and
examples o f uniform algebras , (2) interpolation , (3 ) orthogona l measures , (4) rationa l
approximation, an d (5 ) recen t characterization s o f C(X).
I am greatly indebte d t o various friends fo r kindl y providin g preprints an d th e lates t
and simples t version s of proofs , particularly A . Bernard, B. Cole, A. M. Davie, T. W. Gameli
J. Garnett, S . J. Sidney , and N. T. Varopoulos. Finally , thanks ar e du e th e Departmen t o f
Mathematics o f th e Universit y o f Arkansas , and especiall y Professor Alla n Cochran, withou t
whose untiring effort s th e conferenc e coul d no t hav e take n place .
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