Vol. 16, No. 3, 1966

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ISSN: 0030-8730
A transplantation theorem for ultraspherical coefficients

Richard Allen Askey and Stephen Wainger

Vol. 16 (1966), No. 3, 393–405
Abstract

Let f(𝜃) be integrable on (0) and define

     ∫ π                      ∫ π
an =   f (𝜃)cosn𝜃 d𝜃, bn = n1∕2   f (𝜃)Pn(cos𝜃)(sin𝜃)1∕2d𝜃
0                        0

where Pn(x) is the Legendre polynomial of degree n. Then

    ∞               ∞
c ≦ ∑  |a |p(n+ 1)α∕∑  |b |p(n+ 1)α ≦ C
n=0 n          n=0  n
(1)

for 1 < p < , 1 < α < p 1, where C and c depend on p and α but not on f. From this we obtain a form of the Marcinkiewicz multiplier theorem for Legendre coefficients. Also an analogue of the Hardy-Littlewood theorem on Fourier coefficients of monotone coefficients is obtained. In fact, any norm theorem for Fourier functions can be transplanted by (1) to a corresponding theorem for Legendre coefficients.

Actually, the main theorem of this paper deals with ultraspherical coefficients and (1) is just a typical special case, which is stated as above for simplicity.

Mathematical Subject Classification
Primary: 42.15
Secondary: 33.00
Milestones
Received: 5 October 1964
Published: 1 March 1966
Authors
Richard Allen Askey
Stephen Wainger