Móricz Ferenc honlapja

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Cím:On the $L\sp 1$-convergence of Fourier transforms.
Szerző:Dang Vu Giang; M\'oricz, Ferenc
Forrás:J. Aust. Math. Soc., Ser. A 60, No.3, 405-420 (1996).
Nyelv:English
Absztrakt:We study cosine and sine Fourier transforms defined by $F(t):= (2/\pi) \int^\infty_0 f(x) \cos tx dx$ and $\widetilde F(t):= (2\pi) \int^\infty_0 f(x) \sin tx dx$, where $f$ is $L^1$-integrable over $[0, \infty)$. We also assume that $F$ and $\widetilde F$ are locally absolutely continuous over $[0, \infty)$. In particular, this is the case if both $f(x)$ and $xf(x)$ are $L^1$-integrable over $[0, \infty)$. Motivated by the inversion formulas, we consider the partial integrals $s_\nu(f, x):= \int^\nu_0 F(t)\cos xt dt$ and $\widetilde s_\nu(f, x):= \int^\nu_0 \widetilde F(t)\sin xt dt$ and the modified partial integrals $u_\nu(f, x):= s_\nu(f, x)- F(\nu)(\sin \nu x)/x$ and $\widetilde u_\nu(f, x):= \widetilde s_\nu(f, x)+ \widetilde F(\nu)(\cos \nu x)/x$, where $\nu> 0$. We give necessary and sufficient conditions for the $L^1[0, \infty)$-convergence of $u_\nu(f)$ and $\widetilde u_\nu(f)$ as well as for the $L^1[0, X]$-convergence of $s_\nu(f)$ and $\widetilde s_\nu(f)$ to $f$ as $\nu\to \infty$, where $0< X< \infty$ is fixed. On the other hand, in certain cases we conclude that $s_\nu(f)$ and $\widetilde s_\nu(f)$ cannot belong to $L^1[0, \infty)$. Consequently, it makes no sense to speak of their $L^1[0, \infty)$-convergence as $\nu\to \infty$.\par As an intermediate tool, we use the Ces\`aro means of Fourier transforms. Then we prove Tauberian type results and apply Sidon type inequalities in order to obtain Tauberian conditions of Hardy-Karamata kind.\par We extend these results to the complex Fourier transform defined by $$G(t):= {1 \over (2\pi)} \int^\infty_{-\infty} g(t) e^{-ixt} dt,$$ where $g$ is $L^1$-integrable over $(- \infty, \infty)$. 
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