Geodesic
Uniform Convergence and Integrability of Multiplicative Fourier Transforms
Matematičeskie zametki, Tome 98 (2015) no. 1, pp. 44-60.

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

For multiplicative Fourier transforms, analogs of results obtained by R. P. Boas, F. Moricz, M. I. D'yachenko, I. P. Liflyand, and S. Yu. Tikhonov and dealing with conditions for the uniform convergence and weighted integrability with power weight of classical Fourier transforms as well as conditions for these transforms to belong to Lipschitz classes are proved. Certain results of C. W. Onneweer concerning conditions for multiplicative Fourier transforms to belong to Lipschitz–Besov and Herz spaces are also generalized.
Mots-clés : multiplicative Fourier transform
Keywords: weighted integrability of Fourier transforms, Lipschitz class, Lipschitz–Besov space, Herz space, Dirichlet kernel, Hölder's inequality, Hardy's inequality, Minkowski's inequality. 
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S. S. Volosivets; B. I. Golubov. Uniform Convergence and Integrability of Multiplicative Fourier Transforms. Matematičeskie zametki, Tome 98 (2015) no. 1, pp. 44-60. http://geodesic.mathdoc.fr/item/MZM_2015_98_1_a3/

[1] M. Dyachenko, E. Liflyand, S. Tikhonov, “Uniform convergence and integrability of Fourier integrals”, J. Math. Anal. Appl., 372:1 (2010), 328–338 | DOI | MR | Zbl

[2] G. Sampson, G. Tuy, “Fourier transforms and their Lipschitz classes”, Pacific J. Math., 75:2 (1978), 519–537 | DOI | MR | Zbl

[3] F. Moricz, “Best possible sufficient conditions for the Fourier transform to satisfy Lipschitz or Zygmund condition”, Studia Math., 199:2 (2010), 199–205 | DOI | MR | Zbl

[4] E. Liflyand, S. Tikhonov, “Extended solution of Boas' conjecture on Fourier transforms”, C.R. Acad. Sci. Paris, Ser.I., 346:21-22 (2008), 1137–1142 | DOI | MR | Zbl

[5] S. S. Volosivets, B. I. Golubov, “Vesovaya integriruemost multiplikativnykh preobrazovanii Fure”, Teoriya funktsii i differentsialnye uravneniya, Tr. MIAN, 269, MAIK, M., 2010, 71–81 | MR | Zbl

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

[7] C. W. Onneweer, “Generalized Lipschitz spaces and Herz spaces on certain totally disconnected groups”, Martingale Theory in Harmonic Analysis and Banach Spaces, Lecture Notes in Math., 939, Springer-Verlag, Berlin, 1982, 106–121 | DOI | MR | Zbl

[8] B. I. Golubov, S. S. Volosivets, “On the integrability and uniform convergence of multiplicative Fourier transforms”, Georgian Math. J., 16:3 (2009), 533–546 | MR | Zbl

[9] G. G. Khardi, Dzh. E. Littlvud, G. Polia, Neravenstva, IL, M., 1948 | MR | Zbl

[10] J. S. Bradley, “Hardy inequalities with mixed norms”, Canad. Math. Bull., 21:4 (1978), 405–408 | DOI | MR | Zbl

[11] E. M. Stein, “Interpolation of linear operators”, Trans. Amer. Math. Soc., 83 (1956), 482–492 | DOI | MR | Zbl

[12] C. W. Onneweer, “The Fourier transform of Herz spaces on certain groups”, Monatsch. Math., 97:4 (1984), 297–310 | DOI | MR | Zbl

[13] S. S. Volosivets, “Fourier transforms and generalized Lipschitz classes in uniform metric”, J. Math. Anal. Appl., 383:2 (2011), 344–352 | DOI | MR | Zbl

[14] S. S. Volosivets, “O modifitsirovannykh multiplikativnykh integrale i proizvodnoi proizvolnogo poryadka na poluosi”, Izv. RAN. Ser. matem., 70:2 (2006), 3–24 | DOI | MR | Zbl

[15] S. Fridli, “On the rate of convergence of Cesaro means of Walsh-Fourier series”, J. Approx. Theory, 76:1 (1994), 31–53 | DOI | MR | Zbl