  The purpose of this investigation is to extend basic equations and
inequalities which hold for functions $f$ in a Bernstein space $B_\sigma^2$ to
larger spaces by adding a remainder term which involves the distance of $f$
from $B_\sigma^2$.
  First we present a modification of the classical modulation space
$M^{2,1}(\mathbb{R})$, the so-called readapted modulation space
$M^{2,1}_\text{a}(\mathbb{R})$. Our approach to the latter space and its role
in functional analysis is novel. In fact, we establish several chains of
inclusion relations between $M^{2,1}_\text{a}(\mathbb{R})$ and the more common
Lipschitz and Sobolev spaces, including Sobolev spaces of fractional order.
  Next we introduce an appropriate metric for describing the distance of a
function belonging to one of the latter spaces from a Bernstein space. It will
be used for estimating remainders and studying rates of convergence.
  In the main part, we present the desired extensions. Our applications include
the classical Whittaker-Kotel'nikov-Shannon sampling formula, the reproducing
kernel formula, the Parseval decomposition formula, Bernstein's inequality for
derivatives, and Nikol'ski\u{\i}'s inequality estimating the $l^p(\mathbb{Z})$
norm in terms of the $L^p(\mathbb{R})$ norm.
