Vol. 21, No. 3, 1967

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ISSN: 0030-8730
A transplantation theorem for Jacobi coefficients

Richard Allen Askey

Vol. 21 (1967), No. 3, 393–404
Abstract

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

         ∫
α,β    α,β  π      (α,β)        𝜃 α+(1∕2)    𝜃 β+(1∕2)
an  = tn  0  f(𝜃)Pn   (cos𝜃)(sin2)      (cos2)     d𝜃

where Pn(α,β)(x) is the Jacobi polynomial of degree n, order (α,β) and

[tαn,β]2 = (2n-+-α+-β-+-1)Γ (n-+-1)Γ (n+-α-+-β +-1).
Γ (n + α + 1)Γ (n + β + 1)

Then if α,β,γ,δ 12 we have

∑∞                   ∞∑
|a(γm,δ)|p(n + 1)σ ≦ A    |a(nα,β)|p(n + 1)σ
n=0                  n=0

for 1 < p < ,1 < σ < p 1 whenever the right hand side is finite.

From this result any norm inequality for Fourier coefficients can be transplanted to give a corresponding norm inequality for Fourier-Jacobi coefficients.

Mathematical Subject Classification
Primary: 42.15
Secondary: 33.00
Milestones
Received: 1 April 1966
Published: 1 June 1967
Authors
Richard Allen Askey