Abstract
We prove a number of results about pointwise convergence of eigenfunction expansions of functions on compact manifolds. In particular, we establish that the Pinsky phenomenon holds for piecewise smooth functions on the three-dimensional torus, with jump across the boundary of a ball, in the same form as it was discovered for functions on three-dimensional Euclidean space. Our work on this has been stimulated by recent work of Brandolini and Colzani, and we also discuss some variants of their results.
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References
Bergh, J. and Löfstrom, J. (1976).Interpolation Spaces, an Introduction, Springer-Verlag, New York.
Brandolini, L. and Colzani, L. (1998). Localization and convergence of eigenfunction expansions, Preprint.
Colzani, L. and Vignati, M. (1995). The Gibbs phenomenon for multiple Fourier integrals,J. Approx. Theory,80, 119–131.
Duistermaat, J. and Guillemin, V. (1975). The spectrum of positive elliptic operators and periodic bicharacteristics,Invent. Math.,29, 39–79.
Hörmander, L. (1968). The spectral function of an elliptic operator,Acta Math.,121, 193–218.
Kahane, J.-P. (1995). Le phénomène de Pinsky et la géométrie des surfaces,C.R. Acad. Sci. Paris,321, 1027–1029.
Pinsky, M. (1994). Pointwise Fourier inversion and related eigenfunction expansions,Comm. Pure Appl. Math.,47, 653–681.
Pinsky, M., Stanton, N., and Trapa, P. (1993). Fourier series of radial functions in several variables,J. Funct. Anal.,116, 111–132.
Pinsky, M. and Taylor, M. (1997). Pointwise Fourier inversion: a wave equation approach,J. Fourier Anal.,3, 647–703.
Sogge, S. (1988). Concerning theL p norm of spectral clusters for second order elliptic differential operators on compact manifolds,J. Funct. Anal.,77, 123–134.
Taylor, M. (1981). Pseudodifferential Operators, Princeton University Press, Princeton, NJ.
Taylor, M. (1997). Estimates for approximate solutions to acoustic scattering problems, inInverse Problems in Wave Propagation, Chavent, G., et al., Eds., IMA vol. 90, Springer-Verlag, New York.
Taylor, M. (1997). Pseudodifferential operators, paradifferential operators, and layer potentials, Preprint.
Varchenko, A. (1983). Number of lattice points in families of homothetic domains in ℝn,Funk. An.,17, 1–6.
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Communicated by Gerald B. Folland
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Taylor, M. Pointwise Fourier inversion on tori and other compact manifolds. The Journal of Fourier Analysis and Applications 5, 449–463 (1999). https://doi.org/10.1007/BF01261638
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DOI: https://doi.org/10.1007/BF01261638