Geodesic
An analogue of a theorem of Titchmarsh for Walsh-Fourier transformations
Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 707-725 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{SM_1998_189_5_a3,
     author = {B. I. Golubov},
     title = {An analogue of a theorem of {Titchmarsh} for {Walsh-Fourier} transformations},
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     pages = {707--725},
     year = {1998},
     volume = {189},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_5_a3/}
}
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B. I. Golubov. An analogue of a theorem of Titchmarsh for Walsh-Fourier transformations. Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 707-725. http://geodesic.mathdoc.fr/item/SM_1998_189_5_a3/

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