  The variance-gamma (VG) distributions form a four parameter family that
includes as special and limiting cases the normal, gamma and Laplace
distributions. Some of the numerous applications include financial modelling
and approximation on Wiener space. Recently, Stein's method has been extended
to the VG distribution. However, technical difficulties have meant that bounds
for distributional approximations have only been given for smooth test
functions (typically requiring at least two derivatives for the test function).
In this paper, which deals with symmetric variance-gamma (SVG) distributions,
and a companion paper \cite{gaunt vgii}, which deals with the whole family of
VG distributions, we address this issue. In this paper, we obtain new bounds
for the derivatives of the solution of the SVG Stein equation, which allow for
approximations to be made in the Kolmogorov and Wasserstein metrics, and also
introduce a distributional transformation that is natural in the context of SVG
approximation. We apply this theory to obtain Wasserstein or Kolmogorov error
bounds for SVG approximation in four settings: comparison of VG and SVG
distributions, SVG approximation of functionals of isonormal Gaussian
processes, SVG approximation of a statistic for binary sequence comparison, and
Laplace approximation of a random sum of independent mean zero random
variables.
