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Abstract. Let $f\in
\Psi(\mathbb R)$ and $I^nf\in N_\Psi(\mathbb R)$, for all $n = 1, 2, ...$.
Then
\[\mathop {\lim
}\limits_{n \to \infty } ||I^n f||_{N_\Psi
()}^{1/n} = \sigma ^{ - 1},\]
where $\sigma := \inf\{|\xi|:
\xi \in \supp f\}, ||.||_N_\Psi(\mathbb R)$ is the norm in the Lorentz
space $N_\Psi (\mathbb R)$, and for $g\in S'(\mathbb R)$, the tempered
generalized function $Ig$ is a primitive of $g$ if $D(Ig) = g$, that is
\[<Ig,\varphi^'>\-<g,\varphi>,
\forall \varphi \in S(\mathbb R)\]
and $S(\mathbb R)$ is the
Schwartz space of rapidly decreasing functions.
In other words, in this paper
we characterize behavior of the sequence of $N_\Psi(\mathbb R) - norms of
primitives of a function by its spectrum (the support of its Fourier
transform).
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