The negative coefficients convey phase information. Ludwik's example
uses a square pulse of height 7 between x = 2 and 5. Because this
function is neither odd nor even, a Fourier transform will
necessarily involve sinusoidal functions of myriad phases. In fact,
the function he describes cannot be expanded in terms of cosines
alone because the result of doing so would necessarily be an even
function. The set of all cosine functions do not span the infinite
dimensional space of all real functions of one variable. He would
need to add sine functions as well.
Having found the expansion in terms of coefficients
A(k)=Integral of f(x)*cos(kx)*dx
and
B(k)=Integral of f(x)*sin(kx)*dx
he could then combine the A(k) and B(k) to find an amplitude C(k) and
a phase F(k) for each value of k and then expand the function as
Integral of C(k)*sin(kx+F(k))*dk
Alternatively, he could simply shift his origin so that the function
is a square pulse between x = -1.5 and +1.5 in which case B(k) = 0
and the cosine functions would suffice with phases restricted to 0
and pi. The contributions with phase = pi would be indicated by the
negative coefficients.
It's easy to see that all values of |k| up to |k| = pi/(halfwidth of
the pulse) will be positive so that the "width" of the central
maximum of the transform A(k) is inversely proportional to the pulse
width.