Let $\{g(\cdot-k),k\in\mathbb Z\}$ be a Riesz basis, and let $\varphi\in L^2(\mathbb R)$ be a function defined by its Fourier transform $$\hat{\varphi}(\xi)=\frac{\hat{g}(\xi)}{\Gamma(\xi)},$$ where $$\Gamma(\xi)=\left(\sum_k |\hat{g}(\xi+2k\pi)|^2\right)^{1/2}$$ Then $\{\varphi(\cdot-k),\ k\in\mathbb Z\}$ is an orthonormal system.

The proof of this result uses Parseval's identity and the fact that the Fourier transform of $\varphi(x-k)$ is $e^{-ik\xi}\hat{\varphi}(\xi)$. Hence, after some steps, one arrives to: $$\frac{1}{2\pi}\int_0^{2\pi}e^{-i(k-l)\xi}d\xi=\delta_{kl}$$ A complete proof is located - for instance - in:

Härdle, Wolfgang, et al. Wavelets, approximation, and statistical applications. Vol. 129. Springer Science & Business Media, 2012.

Let us consider a "perturbed system" $\{g(\cdot-\lambda_k),\ \lambda_k\in\mathbb R, \ k\in\mathbb Z\}$ that is a Riesz basis for $|\lambda_k-k|\leq L<1$. Is it possible to derive an orthonormal system by this Riesz basis?

References and answers are welcome.