Abstract
Let μ Σ be the natural measure on R N (N≥3) supported by a compact oriented analytic hypersurface Σ, ψ a smooth function on R N and P(D) a differential operator in N variables of order m. We determine a sufficient condition on the number λ such that the Fourier integral of the distribution P(D)ψ μ Σ be summable by Cesàro means of order λ to zero in a point outside the hypersurface. This condition depends on m and on the position of the point with respect to the caustic of the hypersurface.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arnol’d, V.I.: Catastrophe theory. In: Dynamical Systems. Encyclopaedia of Mathematical Sciences, vol. V, pp. 207–271. Springer, Berlin (1994)
Arnol’d, V.I., Guseĭn-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps. Vol. II (Monodromy and Asymptotics of Integrals). Birkhäuser, Boston (1988)
Brauner, H.: Differentialgeometrie. Vieweg, Braunschweig (1981)
Chapman, S., Hardy, G.H.: A general view of the theory of summable series. Q. J. Math. 42, 181–215 (1911)
Fedoryuk, M.V.: The stationary phase method for multidimensional integrals. U.S.S.R. Comput. Math. Math. Phys. 2, 152–157 (1962)
González Vieli, F.J.: Inversion de Fourier ponctuelle des distributions à support compact. Arch. Math. 75, 290–298 (2000)
González Vieli, F.J.: Fourier inversion of distributions with support on a plane curve. Integral Transform. Spec. Funct. 13, 93–100 (2002)
Hardy, G.H.: Divergent Series. Clarendon, Oxford (1949)
Hewitt, E.: Remarks on the inversion of Fourier-Stieltjes transforms. Ann. Math. (2) 57, 458–474 (1953)
Pinsky, M.A.: Fourier inversion for piecewise smooth functions in several variables. Proc. Am. Math. Soc. 118, 903–910 (1993)
Pinsky, M.A., Taylor, M.E.: Pointwise Fourier inversion: a wave equation approach. J. Fourier Anal. Appl. 3, 647–703 (1997)
Popov, D.A.: Spherical convergence of the Fourier integral of the indicator function of an N-dimensional domain. Sb. Math. 189, 1101–1113 (1998)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)
Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Clarendon, Oxford (1948)
Varchenko, A.N.: Newton polyhedra and estimation of oscillating integrals. Funct. Anal. Appl. 10, 175–196 (1976)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1944)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Tom Körner.
Rights and permissions
About this article
Cite this article
González Vieli, F.J., Seifert, E. Fourier Inversion of Distributions Supported by a Hypersurface. J Fourier Anal Appl 16, 34–51 (2010). https://doi.org/10.1007/s00041-009-9073-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-009-9073-1