Proper Definitions and Demonstration of Dirichlet Conditions

Fourier theory is the backbone of the area of Signal Processing (SP) and
Communication Engineering. However, Fourier series (FS) or Fourier
transform (FT) do not exist for some signals that fail to fulfill a
predefined set of Dirichlet conditions (DCs). We note a subtle gap in
the explanation of these conditions as available in the popular signal
processing literature. They lack a certain degree of explanation
essential for the proper understanding of the same. For example,

the original second Dirichlet condition is the requirement of bounded
variations over one time period for the convergence of Fourier Series,
where there can be at most infinite but countable number of maxima and
minima, and at most infinite but countable number of discontinuities of
finite magnitude. However, a large body of the literature replaces this
statement with the requirements of finite number of maxima and minima
over one time period, and finite number of discontinuities. The latter
incorrectly disqualifies some signals from having valid FS
representation. Similar problem holds in the description of DCs for the
Fourier transform. Likewise, while it is easy to relate the first DC
with the finite value of FS or FT coefficients, it is hard to relate the
second and third DCs as specified in the signal processing literature
with the Fourier representation as to how the failure to satisfy these
conditions disqualifies those signals from having valid FS or FT
representation.