Equivalence of the trigonometric system and its perturbations in Lp(−π,π)

Abstract

Let B be one of the spaces L^p(−π,π), 1 ≤ p < ∞, p ≠ 2, and C[−π,π]. Sufficient conditions under which the “perturbed” trigonometric system \({e^{i{{\left( {n + {\alpha _n}} \right)}^t}}}\), n ∈ Z, is equivalent in B to the trigonometric system e^int, n ∈ Z, are found. Under an additional requirement on (α_n), a necessary condition is obtained. One of the results is as follows. If (α_n) ∈ l^s, where 1/s = 1/p - 1/2, then the equivalence specified above takes place, and the exponent s is exact; the space C corresponds to p = ∞. The proofs are based on the application of Fourier multipliers.