Taken from Austin, Anthony P., and Lloyd N. Trefethen. "Trigonometric interpolation and quadrature in perturbed points." SIAM Journal on Numerical Analysis 55.5 (2017): 2113-2122.:
Kadec's theorem is an answer to a question of sampling theory that originates with Paley and Wiener. The exponentials $\left\{\exp \left(i \lambda_k x\right)\right\},-\infty<k<\infty$, form an orthonormal basis for $L^2[-\pi, \pi]$ if $\lambda_k=k$ for each $k$. Thus, the sampling theorist would say that one can recover a function $f \in L^2[-\pi, \pi]$ from its inner products with the functions $\left\{\exp \left(i \lambda_k x\right)\right\}$. Now suppose these wave numbers are perturbed so that $\left|\tilde{\lambda}_k-k\right| \leq \alpha$ for some fixed $\alpha$. Can one still recover the signal? Specifically, does the family $\left\{\exp \left(i \tilde{\lambda}_k x\right)\right\}$ form a Riesz basis for $L^2[-\pi, \pi]$, that is, a basis that is related to the original one by a bounded transformation with a bounded inverse? Paley and Wiener showed that this is always the case for $\alpha<1 / \pi^2$, and Levinson showed that it is not always the case for $\alpha \geq 1 / 4$. Kadec's theorem shows that Levinson's construction was sharp: for any $\alpha<1 / 4$, the family $\left.\exp \left(i \tilde{\lambda}_k x\right)\right\}$ forms a Riesz basis.
From this passage, it appears that the counterexample to Kadec's theorem comes from Levinson's book Gap and Density Theorems.
Another important reference, instead, states:
The first result on the nonharmonic Fourier series is due to Paley and N. Wiener: if $\lambda_n \in \mathbb{R}, n \in \mathbb{Z}$, and $$ \sup _{n \in \mathbb{Z}}\left|\lambda_n-n\right| \leqslant \delta $$ with $\delta<\pi^{-2}$ (that is, $\left\{e^{i \lambda_n t}\right\}$ is close to the standart orthogonal basis $\left.\left\{e^{i n t}\right\}\right)$, then $\left\{e^{i \lambda_n t}\right\}$ is a Riesz basis in $L^2(0,2 \pi)$. Later, M. Kadec [21] showed that the same is true for any $\delta<1 / 4$, whereas, by a result of A. Ingham, the system $\left\{e^{i \lambda_n t}\right\}$ may fail to be a basis if $\delta=1 / 4$.
The reference in this case is Baranov, Anton. "Stability of the bases and frames reproducing kernels in model spaces." Annales de l'institut Fourier. Vol. 55. No. 7. 2005.
Unfortunately, no work by Ingham is cited in this article, although I suspect it may be the following one: Ingham, Albert Edward. "Some trigonometrical inequalities with applications to the theory of series." Mathematische Zeitschrift 41.1 (1936): 367-379. Unfortunately I can't find any evidence in it that allows me to say that "the system $\left\{e^{i \lambda_n t}\right\}$ may fail to be a basis if $\delta=1 / 4$".
In the book 'An Introduction to Nonharmonic Fourier Series' by Young, the counterexample is provided without citing the source, although it appears similar to the one given by Levinson.
Does anyone know to whom the counterexample to Kadec's 1/4 theorem is attributed? Levinson or Ingham? Or have both of them provided one? If yes, in which of Ingham's works was this counterexample introduced?
