Vol. 100, No. 2, 1982

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On the transformation of Fourier coefficients of certain classes of functions

Kenneth F. Andersen

Vol. 100 (1982), No. 2, 243–248
Abstract

Suppose f(x) L1(0) and let a = {aν} (b = {bν}) denote the Fourier cosine (sine) coefficients of f extended to (π,π) as an even (odd) function, that is

      ∫                  ∫
2- π              2-  π
a0 = π 0  f(x)dx, aν = π  0 f(x)cosνx dx,
2∫ π                               ν = 1,2,⋅⋅⋅
bν = π-   f(x)sinνx dx.
0

The sequence transformations T and Tare defined by

                      ν              ∞
(T a) = a ,  (T a) = 1-∑  a , (T ′a) = ∑  (a ∕j),  ν = 1,2,⋅⋅⋅ .
0   0      ν   ν j=1 j       ν  j=ν  j

The purpose of this note is to characterize those rearrangement invariant function spaces Lσ(0) which are left invariant by the operators T and Tacting on Fourier coefficients of functions in these spaces. Our results include and improve some results of Hardy, Bellman and Alshynbaeva.

Mathematical Subject Classification 2000
Primary: 42A16
Secondary: 46E30
Milestones
Received: 21 May 1980
Published: 1 June 1982
Authors
Kenneth F. Andersen