Abstract
We obtain new inequalities for the Fourier transform in the space \(\textrm{L}^{p}(\mathbb {R}^{n})\), \(1<p\le 2\), using a generalized spherical mean operator for proving the estimates in certain classes of functions characterized by a generalized continuity modulus.
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References
Abilov, V.A., Abilova, F.V.: Approximation of functions by Fourier–Bessel sums. IZV. Vyssh. Uchebn. Zaved. Mat. 8, 3–9 (2001)
Abilov, V.A., Abilova, F.V., Kerimov, M.K.: Some remarks concerning the Fourier transform in the space \({\rm L}_{2}({\mathbb{R}}^{n})\). Comp. Math. Math. Phys. 48(12), 2146–2153 (2008)
Bray, W.O., Pinsky, M.A.: Growth proprieties of the Fourier transforms via moduli of continuity. J. Funct. Anal. 255, 2265–2285 (2008)
Bray, W.O., Pinsky, M.A.: Growth properties of the Fourier transforms. Filomat 26(4), 755–760 (2012)
Daher, R., Boujeddaine, M., El Hamma, M.: Generalization of Titchmars’s theorem for the Fourier transform in the space \({\rm L} ^{2}(\mathbb{R}^{n})\). Afr. Mat. 27, 753–758 (2016)
Daher, R., El Hamma, M.: Equivalence of \(K\)-functionals and modulus of smoothness for Fourier transform. Int. J. Nonlinear. Anal. appl. 3(2), 38–43 (2012)
Daher, R., El Hamma, M.: On estimates for the Fourier transform in the space \({\rm L} ^{2}({\mathbb{R} }^{n})\). C. R. Acad. Sci. Paris Ser. I 352, 235–240 (2014)
Daher, R., El Hamma, M.: On estimates for the Fourier transform in the space \({\rm L} ^{p}(\mathbb{R}^{n})\). Afr. Mat. 26, 1215–1220 (2015)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton, NJ (1971)
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El Hamma, M., Daher, R., Mahfoud, A. et al. Some inequalities involving Fourier transforms. Afr. Mat. 34, 80 (2023). https://doi.org/10.1007/s13370-023-01124-x
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DOI: https://doi.org/10.1007/s13370-023-01124-x