Let G be finite Jordan domain bounded a Dini smoth curve ? in the complex
plane C. We investigate the approximation properties of the partial sums of
the Fourier series and prove direct theorem for approximation by polynomials
in the subspace of Morrey spaces associated with grand Lebesgue spaces.
Also, approximation properties of the Faber-Laurent rational series
expansions in spaces Lp),? (?) are studied. Direct theorems of approximation
theory in grand Morrey-Smirnov classes, defined in domains with a Dini-
smooth boundary, are proved.