Abstract
For the quaternion algebra \({\mathbb {H}}\) and \(f:\mathbb R^2\rightarrow {\mathbb {H}}\), we consider a two-sided quaternion Fourier transform \({\widehat{f}}\). Necessary and sufficient conditions for \({\widehat{f}}\) to belong to generalized uniform Lipschitz spaces are given in terms of behavior of f.
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References
Bary, N.K., Stechkin, S.B.: Best approximation and differential properties of two conjugate functions. Trudy Mosk. Mat. Obs. 5, 483–522 (1956). ((in Russian))
Berkak, E.M., Loualid, E.M., Daher, R.: Boas-type theorems for the \(q\)-Bessel Fourier transform. Anal. Math. Phys. 11(3), 102 (2021). https://doi.org/10.1007/s13324-021-00542-z
Boas, R.P.: Integrability theorems for trigonometric transforms. Springer-Verlag, New York (1967)
Butzer, P.L., Nessel, R.J.: Fourier analysis and approximation. Birkhauser, Basel-Stuttgart (1971)
Fülöp, V., Móricz, F.: Absolutely convergent multiple Fourier series and multiplicative Zygmund classes of functions. Analysis 28(3), 345–354 (2008). https://doi.org/10.1524/anly.2008.0919
Fülöp, V., Móricz, F., Sáfár, Z.: Double Fourier transforms, Lipschitz and Zygmund classes of functions on the plane. East J. Approx. 17(1), 111–125 (2011)
Fülöp, V., Móricz, F.: On double sine and cosine transforms, Lipschitz and Zygmund classes. Anal. Theory Appl. 27(4), 351–364 (2011). https://doi.org/10.1007/s10496-011-0351-9
Hitzer, E.: Quaternion Fourier Transformation on Quaternion Fields and Generalizations. Adv. Appl. Clifford Algebras 17(3), 497–517 (2007). https://doi.org/10.1007/s00006-007-0037-8
Hitzer, E.: Directional Uncertainty Principle for Quaternion Fourier Transforms. Adv. Appl. Clifford Algebras 20(2), 271–284 (2010). https://doi.org/10.1007/s00006-009-0175-2
Loualid, E.M., Elgargati, A., Daher, R.: Quaternion Fourier transform and generalized Lipschitz classes. Adv. Appl. Clifford Algebras 31(1), 14 (2021). https://doi.org/10.1007/s00006-020-01098-0
Loualid, E.M., Elgargati, A., Berkak, E.M., Daher, R.: Boas-type theorems for the Bessel transform. RACSAM. 115(3), 102 (2021). https://doi.org/10.1007/s13398-021-01087-3
Móricz, F.: Absolutely convergent Fourier series and function classes. J. Math. Anal. Appl. 324(2), 1168–1177 (2006). https://doi.org/10.1016/j.jmaa.2005.12.051
Móricz, F.: Higher order Lipschitz classes of functions and absolutely convergent Fourier series. Acta Math. Hung. 120(4), 355–366 (2008). https://doi.org/10.1007/s10474-007-7141-z
Móricz, F.: Absolutely convergent Fourier integrals and classical function spaces. Arch. Math. 91(1), 49–62 (2008). https://doi.org/10.1007/s00013-008-2626-8
Móricz, F.: Absolutely convergent multiple Fourier series and multiplicative Lipschitz classes of functions. Acta Math. Hung. 121(1-2), 1–19 (2008). https://doi.org/10.1007/s10474-008-7164-0
Móricz, F.: Best possible conditions for the Fourier transform to satisfy the Lipschitz or Zygmund conditions. Stud. Math. 199(2), 199–205 (2010). https://doi.org/10.4064/sm199-2-5
Rakhimi, L., Daher, R.: Boas-type theorems for Laguerre type operator. J. Pseudo-Differ. Oper. Appl. 13(3), 42 (2022). https://doi.org/10.1007/s11868-022-00472-9
Tikhonov, S.: On generalized Lipschitz classes and Fourier series. Zeit. Anal. Anwend. 23(4), 745–764 (2004)
Tikhonov, S.: Smoothness conditions and Fourier series. Math. Ineq. Appl. 10(2), 229–242 (2007)
Titchmarsh, E.: Introduction to the theory of Fourier integrals. Clarendon press, Oxford (1937)
Volosivets, S.S.: Fourier transforms and generalized Lipschitz classes in uniform metric. J. Math. Anal. Appl. 383(1), 344–352 (2011). https://doi.org/10.1016/j.jmaa.2011.05.026
Volosivets, S.S., Golubov, B.I.: Fourier transforms in generalized Lipschitz classes. Proc. Steklov Inst. Math. 280, 120–131 (2013). https://doi.org/10.1134/S0081543813010070
Volosivets, S.S.: Multiple Fourier coefficients and generalized Lipschitz classes in uniform metric. J. Math. Anal. Appl. 427(2), 1070–1083 (2015). https://doi.org/10.1016/j.jmaa.2015.02.011
Volosivets, S.S.: Fourier-Bessel transforms and generalized uniform Lipschitz classes. Integral Transforms Spec. Funct. 33(7), 559–569 (2022). https://doi.org/10.1080/10652469.2021.1986815
Volosivets, S.S.: Fourier-Dunkl transforms and generalized symmetric Lipschitz classes. J. Math. Anal. Appl. 520(1), 126895 (2023). https://doi.org/10.1016/j.jmaa.2022.126895
Yu, D.: Double trigonometric series with positive coefficients. Anal. Math. 35(2), 149–167 (2009). https://doi.org/10.1007/s10476-009-0205-2
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Communicated by Vladislav Kravchenko.
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Volosivets, S. Dual Boas Type Results for the Quaternion Transform and Generalized Lipschitz Spaces. Adv. Appl. Clifford Algebras 33, 56 (2023). https://doi.org/10.1007/s00006-023-01301-y
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DOI: https://doi.org/10.1007/s00006-023-01301-y