  Fourier series multiscale method, a concise and efficient analytical approach
for multiscale computation, will be developed out of this series of papers. The
second paper is concerned with simultaneous approximation to functions and
their (partial) derivatives. On the basis of sufficient conditions of 2r (r is
a positive integer) times term-by-term differentiation of Fourier series, a
one-dimensional or two-dimensional function with general boundary conditions is
decomposed into the linear combination of a corner function that describes the
discontinuities at corners of the domain (only for the two-dimensional
function), boundary functions that describe the discontinuities on boundaries
of the domain and an internal function that describes the smoothness within the
domain. It leads to the methodology of simultaneous approximation of functions
and their (partial) derivatives with composite Fourier series. And
specifically, for the algebraical polynomial based composite Fourier series
method, the reproducing property of complete algebraical polynomials is
theoretically analyzed and the approximation accuracy is validated with
numerical examples. This study generalizes the Fourier series method with
supplementary terms to simultaneous approximation of functions and their
(partial) derivatives up to 2rth order.
