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Is the set

$$\left\{e^{-\frac{\sqrt 5}{12} t} e^{i\left(n+\frac{1}{6}\right) t}: n=1,2,3,...\right\}$$

a Riesz basis in $L^2[-\pi,\pi]$?

Any suggestions please?

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Riesz basis is the image of an orthonormal basis under an invertible linear operator of $L^2$ onto itself. An example of an invertible linear operator is multiplication by a function $\varphi$ such that $\varphi\in L^\infty$ and $\varphi^{-1}\in L^\infty$. Looking at the family of the functions you have, it is natural to take $\varphi(t) = \exp(-\sqrt{5} t/12) \exp(it/6)$. Thus, the given set is the image of $E=\{e^{int}:n=1,2,3\dots\}$ under an invertible linear operator.

However, $E$ is not a basis of $L^2$; it is orthogonal to any $e^{int }$ with integer $n\le 0$. Therefore, your set is not a basis either; it does not even span $L^2$.

If you extend the set by allowing arbitrary $n\in \mathbb Z$, then the above argument shows that you have a Riesz basis.

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