Given all the moments of a probability distribution is it possible to
reconstruct the distribution curve? If so, how is this done in a
general case?

In particular if the first four moments; mean, variance, skew and
kurtosis are equal to m,v,s & k respectively with all other moments
equal to 0, what is the distribution function?

It is theoretically possible to reconstruct the PDF from all the
moments, but it is not possible for all the higher moments to be zero
if the lower ones are non zero.  A distribution with nonzero variance
cannot have a sixth moment of zero.

If you know all the moments, you can determine the pdf by expressing
the characteristic function of the pdf in terms of the raw moments,
and then taking the inverse Fourier transform of the characteristic
function.

The characteristic function, f(t), of a pdf is the Fourier transform
of the pdf, P(x):

  f(t) = Integral from -infinity to +infinity of {P(x)* exp(i*x*t) dx}

Expanding exp(ixt) in a Maclaurin series yields:

  f(t) = Integral from -infinity to +infinity of (SUM from k = 0 to
infinity of [P(x)*(t*x*i)^k/k!) dx}

The raw moments are defined as :

  m_k = Integral from -infinity to +infinity of {P(x)*x^k dx}, k = 0 to infinity

so we can rewrite the characteristic function as:

  f(t) = SUM from k = 0 to infinity of [m_k * (i*t)^k/k!]
The pdf can be recovered from the characteristic function by taking
the inverse Fourier transform of the characteristic function:

  P(x) = 1/(2*pi) * Integral from -infinity to +infinity of {exp(-itx)*f(t) dt}