For functions f ∈ L 1 (ℝ+) with cosine (sine) Fourier transforms \( {{\hat{f}}_c} \) \( \left( {{{\hat{f}}_s}} \right) \) in L 1 (ℝ), we give necessary and sufficient conditions in terms of \( {{\hat{f}}_c} \) \( \left( {{{\hat{f}}_s}} \right) \) for f to belong to generalized Lipschitz classes H ω,m and h ω,m Conditions for the uniform convergence of the Fourier integral and for the existence of the Schwartz derivative are also obtained.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 616–627, May, 2012.
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Golubov, B.I., Volosivets, S.S. Fourier cosine and sine transforms and generalized Lipschitz classes in the uniform metric. Ukr Math J 64, 693–710 (2012). https://doi.org/10.1007/s11253-012-0672-7
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DOI: https://doi.org/10.1007/s11253-012-0672-7