On asymptotics of Fourier transform for functions of certain classes (Q1313681)

The author proves asymptotic formulas for the Fourier transform of functions belonging to certain classes, which are more general than those known, e.g., in the case of convex functions. These results can be considered to be the analogues of the known tests for integrability of trigonometric series found by Boas, Fomin, Sidon, Teljakovskij, etc. The main results can be summarized as follows. Assume that a function \(f\) is locally absolutely continuous on \(R_ +=[0,\infty)\), \(\lim_{x \to \infty} f(x)=0\), \[ \hat f_ c(y)= \int^ \infty_ 0 f(x) \cos xy dx,\quad \hat f_ s(y)=\int^ \infty_ 0 f(x) \sin xy dx. \] If \((f')_ s \in H^ 1(R)\) where \((f')_ s\) is the odd continuation of \(f'\) from \(R_ +\) to \(R\) (Theorem 2) or if \((u^{-1} \int^ \infty_ u | f'(x) |^ p dx)^{1/p} \in L^ 1(R_ +)\) for some \(p>1\) (Theorem 3), then \(\hat f_ c(y) \in L^ 1(R_ +)\) and \(\hat f_ s(y)-y^{-1} f(\pi/2y) \in L^ 1(R_ +)\). Parts of the results are extended to multiple Fourier transforms and series.