  Usually, complex-valued RKHS are presented as an straightforward application
of the real-valued case. In this paper we prove that this procedure yields a
limited solution for regression. We show that another kernel, here denoted as
pseudo kernel, is needed to learn any function in complex-valued fields.
Accordingly, we derive a novel RKHS to include it, the widely RKHS (WRKHS).
When the pseudo-kernel cancels, WRKHS reduces to complex-valued RKHS of
previous approaches. We address the kernel and pseudo-kernel design, paying
attention to the kernel and the pseudo-kernel being complex-valued. In the
experiments included we report remarkable improvements in simple scenarios
where real a imaginary parts have different similitude relations for given
inputs or cases where real and imaginary parts are correlated. In the context
of these novel results we revisit the problem of non-linear channel
equalization, to show that the WRKHS helps to design more efficient solutions.
