Decay of Fourier Transforms and Generalized Besov Spaces

Year 2020, Volume: 3 Issue: 1, 20 - 35, 01.03.2020

Thaís Jordão

https://doi.org/10.33205/cma.646557

Cited By: 3

Abstract

A characterization of the generalized Lipschitz and Besov spaces in terms of decay of Fourier transforms is given.\ In particular, necessary and sufficient conditions of Titchmarsh type are obtained.\ The method is based on two-sided estimate for the rate of approximation of a $\beta$-admissible family of multipliers operators in terms of decay properties of Fourier transforms.

Keywords

generalized Besov spaces, Fourier transforms, Titchmarsh type theorem

References

• Bari, N. K.; Stechkin, S. B., Best approximations and differential properties of two conjugate functions. Trudy Moskov. Mat. Ob$\check{s}\check{c}$ 5 (1956), 483--522.
• Belinsky, E.; Dai, F.; Ditzian, Z., Multivariate approximating averages. J. Approx. Theory 125 (2003), no. 1, 85--105.
• Benedetto, J.J., Heinig, H.P., Weighted Fourier inequalities: new proofs and generalizations. J. Math. Anal. Appl. 9 (2003), no. 1, 1--37.
• Bingham, N. H.,Goldie, C. M., Teugels, J. L., Regular Variation.\ Cambridge University Press, Cambridge, 1987.
• Butzer, P. L.; Dyckhoff, H.; G\"{o}rlich, E.; Stens, R. L., Best trigonometric approximation, fractional order derivatives and Lipschitz classes. Canad. J. Math. 29 (1977), no. 4, 781--793.
• De Carli, L.; Gorbachev, D.; Tikhonov, S., Pitt inequalities and restriction theorems for the Fourier transform. Rev. Mat. Iberoam. 33, no. 3 (2017), 789--808.
• Cline, Daren B. H., Regularly varying rates of decrease for moduli of continuity and Fourier transforms of functions on $\mathbb{R}^d$. J. Math. Anal. Appl. 159 (1991), no. 2, 507--519.
• Daher, R.; Delgado, J.; Ruzhansky, M., Titchmarsh theorems for Fourier transforms of H\"{o}lder-Lipschitz functions on compact homogeneous manifolds. arXiv:1702.05731v1
• Dai, F.; Ditzian, Z., Combinations of multivariate averages. J. Approx. Theory 131 (2004), no. 2, 268--283.
• Ditzian, Z., On the Marchaud-type inequality. Proc. Amer. Math. Soc. 103 (1988), no. 1, 198--202.
• Erd\'{e}lyi, A.; Magnus, W., Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions. McGraw-Hill, New York (1953).
• Gorbachev, D.; Liflyand, E.; Tikhonov, S., Weighted Fourier inequalities: Boas' Conjecture. J. Analyse Math\`{e}matique 114 (2011), no. x, 99--120.
• Gorbachev, D.; Tikhonov, S., Moduli of smoothness and growth properties of Fourier transforms: two-sided estimates. J. Approx. Theory 164 (2012), no. 9, 1283--1312.
• Jord\~{a}o, T.; Besov spaces and generalized smoothness. Preprint.
• Jord\~{a}o, T.; Sun, X., General types of spherical mean operators and K-functionals of fractional orders. Commun. Pure Appl. Anal. 14 (2015), no. 3, 743--757.
• Karamata, J., Sur un mode de croissance r\'{e}guli\`{e}re Th\'{e}or\`{e}mes fon damentaux. Bull. Soc. Math. France 61 (1933) 55--62.
• Liflyand, E.; Tikhonov, S. A concept of general monotonicity and applications. Math. Nachr. 284 (2011), no. 8-9, 1083--1098.
• Lorentz, G. G., Fourier-Koeffizienten und Funktionenklassen. Mathematische Zeitschrift 51 (1948) 2, 135--149.
• Platonov, S. S., The Fourier transform of functions satisfying a Lipschitz condition on symmetric spaces of rank 1. (Russian) Sibirsk. Mat. Zh. 46 (2005), no. 6, 1374--1387; translation in Siberian Math. J. 46 (2005), no. 6, 1108--1118.
• Platonov, S. S., An analogue of the Titchmarsh theorem for the Fourier transform on the group of p-adic numbers. p-Adic Numbers Ultrametric Anal. Appl. 9 (2017), no. 2, 158--164.
• Platonov, S. S.. An analogue of the Titchmarsh theorem for the Fourier transform on locally compact Vilenkin groups. p-Adic Numbers Ultrametric Anal. Appl. 9 (2017), no. 4, 306--313.
• Simonov, B.; Tikhonov, S., Sharp Ul'yanov-type inequalities using fractional smoothness. 162 (2010), no. 9, 1654--1684.
• Stein, E. M., Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. (1970).
• Tikhonov., S., Best approximation and moduli of smoothness: computation and equivalence theorems. Journal of Approx. Theory, 153 (2008) 19--39.
• Tikhonov, S., Trigonometric series of Nikol'skii classes.\emph{Acta Math. Hungar.}, 114 (2007) no 1-2, 61--78.
• Tikhonov, Sergey, Embedding results in questions of strong approximation by Fourier series. (English summary) {\emph Acta Sci. Math.} 72 (2006), no. 1--2, 117--128.\ Published first as S.Tikhonov, Embedding theorems of function classes, IV. November 2005, CRM preprint.
• Tikhonov, S., On generalized Lipschitz classes and Fourier series. Z. Anal. Anwend. 23 (2004), no. 4, 745--764.
• Tikhonov, S., Trigonometric series with general monotone coefficients. J. Math. Anal. Appl. 326 (2007), no. 1, 721--735.
• Titchmarsh, E. C., Introduction to the theory of Fourier integrals. Second edition. Oxford University Press (1984).
• Triebel, H., Limits of Besov norms. Arch. Math. (Basel) 96 (2011), no. 2, 169--175.
• Volosivets, S. S., Fourier transforms and generalized Lipschitz classes in uniform metric. J. Math. Anal. Appl. 383 (2011), no. 2, 344--352.
• Weiss, Guido; Stein, Elias M., Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., (1971).
• Younis, M. S., Fourier transforms of Lipschitz functions on certain Lie groups. Int. J. Math. Math. Sci. 27 (2001), no. 7, 439--448.
• Younis, M. S., Fourier transforms of Dini-Lipschitz functions. Internat. J. Math. Math. Sci} 9 (1986), no. 2, 301--312.