Abstract. First, we study cosine and sine Fourier transforms. We prove that if $f$ is an absolutely continuous function over $[0,\infty )$ such that $\lim_{t\to\infty }f(t)=0$ and for some $p>1$ we have $$\int_0^\infty\left ({1\over u}\int_u^{2u}|f'(t)|^pdt\right )^{1/p}du < \infty,$$ then $(*) \hat{f}_c(x):=\int_0^{\to\infty }f(t)\cos xt dt$ converges for any $x>0, \hat{f}_c$ is Lebesgue integrable over $[0,\infty )$, and the inversion formula holds. Under the above conditions, $(**) \hat{f}_s(x):=\int_0^{\to\infty }f(t)\sin xt dt$ converges for any $x>0$, $\hat{f}_s$ is $L$-integrable over $[0,\infty )$ if and only if $f(t)/t$ is $L$-integrable over $[0,\infty )$, in which case the inversion formula holds. Our basic tools are two Sidon type inequalities which we elaborate also in this paper. Second, we extend these results to complex Fourier transforms over $(-\infty,\infty )$. Third, we deduce sufficient conditions for the $L$-integrability of trigonometric series over $[-\pi,\pi ].$ In this way, we obtain new proofs of the theorems by G.A. Fomin, J. Fournier and W. Self, which are known presently as the most general ones. Fourth, as a by-product, we obtain sufficient conditions under which a function is the complex Fourier transform of an $L$-integrable function over $(-\infty,\infty ).$

AMS Subject Classification (1991): 42A38, 26A46

Keyword(s): cosine and sine Fourier transforms, complex Fourier transform, absolutely continuous function, cosine and sine series, complex trigonometric series, Lebesgue integrability, Fourier series, Hausdorff-Young inequality, inequality of Titchmarsh, Sidon type inequalities, inversion formula

Received September 1, 1994. (Registered under 5636/2009.)