A simple and explicit construction of an orthnormal trigonometric
polynomial basis in the spaceC of continuous periodic functions is
presented. It consists simply of periodizing a well-known wavelet on the
real line which is orthonormal and has compactly supported Fourier
transform. Trigonometric polynomials resulting from this approach have
optimal order of growth of their degrees if their indices are powers of 2.
Also, Fourier sums with respect to this polynomial basis are projectors
onto subspaces of trigonometric polynomials of high degree, which implies
almost best approximation properties.