Nonsaturated Estimates of the Kotelnikov Formula Error

The error of approximation by the Kotelnikov sums

$$ {U}_Tf(x)=\sum_{j\in \mathbb{Z}}f\left(\frac{j}{T}\right)\operatorname{sinc}\left( Tx-j\right).T>0,\operatorname{sinc}\ z=\frac{\sin \pi z}{\pi z} $$

is estimated. Let f ∈ A, i.e.,$f(x)=\underset{\mathbb{R}}{\int }g(y){e}^{ixy} dy,g\in {L}_1\left(\mathbb{R}\right),$ and let ${\left\Vert f\right\Vert}_{\textbf{A}}=\underset{\mathbb{R}}{\int}\left|g\right|$ be the Wiener norm of f. Then the sharp inequality

$$ {\left\Vert f-{U}_Tf\right\Vert}_{\textbf{A}}\le 2{A}_{T\pi}{(f)}_{\textbf{A}} $$

holds, where A_σ(f)_A is the best approximation of f in the Wiener norm by entire functions of exponential type not exceeding 𝜎. Several non-saturated uniform estimates are also established.