0. Introduction
0.1 Many multiplier theorems of Fourier analysis have analogs for ultra-
spherical expansions. But what was a single theorem in the Fourier
setting becomes an entire family of theorems in this more general set-
ting. The problem solved in this paper is that of organizing the
children of the Fourier theorems, and many new theorems besides, into a
coherent theory. The most critical step in this organization is identi-
fying a family of Banach spaces which include the sequences described in
the classical multiplier theorems as special cases. Once this family is
found, the next step is to develop the methods of interpolation neces-
sary to show that this family forms a scale of spaces-in the sense that
if two spaces in the family act as multipliers on L , then all spaces
"between" these two spaces act as multipliers on L .
Neither the family of Banach spaces nor the methods of interpola-
tion mentioned above utilize facts about ultraspherical expansions or
are restricted in application to ultraspherical expansions. This
material, which deals with the properties of Banach spaces of functions
on the real line, equivalent norms, Sobolev modules, techniques of
interpolation, etc. has been gathered together in the first six sections
to make these results more accessible to those who would like to apply
them to some other setting -- and in fact, we have included three sec-
tions of applications of these techniques to other settings. For the
Received by the editors October 18, 197^ and, in revised form,
May 27, 1976.
Both authors were partially supported by AFOSR grant No. 71-2047,
University of Missouri Faculty Research Fellowships, and summer research
fellowships at the University of New Hampshire, 1973.
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