It is well-known that the
Fourier series of a continuous periodic function need not be pointwise convergent.
This fact is a consequence of the unboundedness of the Lebesgue constants, which are
the norms of the partial sum operators. It is equally-known that the Fourier series of
a continuous function is uniformly (C,1)-summable to the value of the function.
Thus, the question naturally arises as to which summability matrices are effective in
the limitation of Fourier series of continuous functions. In this paper we
consider a very general class of matrices, the (f,dn) means, and show that
their Lebesgue constants are unbounded. An interesting corollary is that the
Fourier series of a continuous periodic function need not be everywhere almost
convergent.
