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A sequence of complex exponentials      { e i λ  n t } \{ {e^{i{\lambda _n}t}}\}       is said to be a Riesz basis for      L 2 ( −  π  , π  ) {L^2}( - \pi ,\;\pi )       if each function in the space has a unique representation      f = Σ  c n e i λ  n t f = \Sigma {c_n}{e^{i{\lambda _n}t}}      , with      A Σ  | c n | 2 ⩽  | | f | | 2 ⩽  B Σ  | c n | 2 A\Sigma |{c_n}{|^2} \leqslant ||f|{|^2} \leqslant B\Sigma |{c_n}{|^2}      . It is known, for example, that if      | λ  n −  n | ⩽  L &gt; 1 / 4 ( −  ∞  &gt; n &gt; ∞  ) |{\lambda _n} - n| \leqslant L &gt; 1/4( - \infty &gt; n &gt; \infty )      , then      { e i λ  n t } \{ {e^{i{\lambda _n}t}}\}       is a Riesz basis. In this note we show that not only the orthonormal basis      { e i n t } \{ {e^{int}}\}      , but any Riesz basis of complex exponentials can be suitably perturbed.

On perturbing bases of complex exponentials in 𝐿² (-𝜋,𝜋)