 
Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICALSOCIETY
Volume 207, 1975
CONVERGENCEAND DIVERGENCEOF SERIES CONJUGATE
TO A CONVERGENTMULTIPLEFOURIER SERIES
BY
J. MARSHALL ASH(1) AND LAWRENCE GLUCK
ABSTRACT. In this note we consider to what extent the classical
theorems of Plessnerand Kuttner comparing the set of convergence of a trig
onometric series with that of the conjugate trigonometric series can be gen
eralized to higher dimensions. We show that if a function belongs to LP, p > 1,
of the 2torus, then the convergence ( unrestricted rectangular convergence) of
the Fourier series on a set implies its three conjugate functions converge almost
everywhere on that set. That this theorem approaches the best possible may be
seen from two examples which show that the dimension may not be increased
to 3, nor the required power of integrability be decreased to 1.
We also construct a continuous function having a boundedly divergent
Fourier series of power series type and an a.e. circularly convergent double
Fourier series whose yconjugate diverges circularly a.e.
Our LP result depends on a theorem of L. Gogoladze (our proof is
