  Several problems on Fourier series and trigonometric approximation on a
hexagon and a triangle are studied. The results include Abel and Ces\`aro
summability of Fourier series, degree of approximation and best approximation
by trigonometric functions, both direct and inverse theorems. One of the
objective of this study is to demonstrate that Fourier series on spectral sets
enjoy a rich structure that allow an extensive theory for Fourier expansions
and approximation.
