An analogue of a theorem of Titchmarsh for Walsh-Fourier transformations

B. I. Golubov

Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 707-725

Résumé

Let $\hat f_c$ be the Fourier cosine transform of $f$. Then, as proved for functions of class $L^p(\mathbb R_+)$ in Titchmarsh's book 'Introduction to the theory of Fourier integrals' (1937), $$ \mathscr H(\hat f_c)=\widehat {\mathscr B(f)}_c, \qquad \mathscr B(\hat f_c)=\widehat {\mathscr H(f)}_c, $$ for the Hardy operator $$ \mathscr H(f)(x)=\int _x^{+\infty }\frac {f(y)}y\,dy, \qquad x>0, $$ and the Hardy-Littlewood operator $$ \mathscr B(f)(x)=\frac 1x\int _0^xf(y)\,dy, \qquad x>0. $$ In the present paper similar equalities are proved for functions of class $L^p(\mathbb R_+)$, $1$, and the Walsh-Fourier transformation.