Geodesic
An analogue of a theorem of Titchmarsh for Walsh-Fourier transformations
Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 707-725 
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/
@article{SM_1998_189_5_a3,
     author = {B. I. Golubov},
     title = {An analogue of a theorem of {Titchmarsh} for {Walsh-Fourier} transformations},
     journal = {Sbornik. Mathematics},
     pages = {707--725},
     year = {1998},
     volume = {189},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_5_a3/}
}
TY  - JOUR
AU  - B. I. Golubov
TI  - An analogue of a theorem of Titchmarsh for Walsh-Fourier transformations
JO  - Sbornik. Mathematics
PY  - 1998
SP  - 707
EP  - 725
VL  - 189
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/SM_1998_189_5_a3/
LA  - en
ID  - SM_1998_189_5_a3
ER  - 
%0 Journal Article
%A B. I. Golubov
%T An analogue of a theorem of Titchmarsh for Walsh-Fourier transformations
%J Sbornik. Mathematics
%D 1998
%P 707-725
%V 189
%N 5
%U http://geodesic.mathdoc.fr/item/SM_1998_189_5_a3/
%G en
%F SM_1998_189_5_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Zigmund A., Trigonometricheskie ryady, T. 2, Mir, M., 1965 | MR

[2] Krein S. G., Petunin Yu. I., Semenov E. I., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[3] Khardi G. G., Litlvud D. E., Polia G., Neravenstva, IL, M., 1948

[4] Titchmarsh E., Vvedenie v teoriyu integralov Fure, Gostekhizdat, M.-L., 1948

[5] Bellman R., “A note on a theorem of Hardy on Fourier constants”, Bull. Amer. Math. Soc., 50 (1944), 741–744 | DOI | MR | Zbl

[6] Golubov B. I., “Ob odnoi teoreme Bellmana o koeffitsientakh Fure”, Matem. sb., 185:11 (1994), 31–40 | Zbl

[7] Fine N., “The generalised Walsh functions”, Trans. Amer. Math. Soc., 69 (1950), 66–77 | DOI | MR | Zbl

[8] Schipp F., Wade W. R., Simon P., Walsh series, Akademiai Kiadó, Budapest, 1990 | MR

[9] Golubov B. I., Efimov A. V., Skvortsov V. A., Ryady i preobrazovaniya Uolsha. Teoriya i primeneniya, Nauka, M., 1987 | MR | Zbl

[10] Bespalov M. S., Multiplikativnye preobrazovaniya Fure v $L^p$, Rukopis. deponir. v VINITI, No. 100-82 Dep., Mosk. in-t elektronnoi tekhniki, 21 pp.

[11] Billard P., “Sur la convergence presque partout des series de Fourier–Walsh des fonctions de l'espace $L^2(0,1)$”, Studia Math., 28:3 (1967), 363–388 | MR | Zbl