Abstract: For any fixed ε > 0, an explicit construction of anorthonormal trigonometric polynomial basis {T_k}_k=1^∞ in L²[0,1) with deg T_k ≤ 0.5(1 +ε)k is presented. Thus weimprove the results obtained by D. Offin and K. Oskolkov in[4] and by Al. A. Privalov in[6]. and practically solve the open problemasked in[4],[8] and[9]. Moreover, as in[4], Fourier sums with respectto this polynomial basis are projectors onto subspaces of trigonometricpolynomials of high degree, which implies almost best approximation-properties.

Key words: Trigonometric polynomial bases, wavelets, best approximation