Distributional asymptotic expansions of spectral functions and of the associated Green kernels

Electronic Journal of Differential Equations

Volumn 1999, Issue , 1999, Pages 1-37

  Estrada, R ^a   Fulling, S A ^b  

^a Texas A and M University ^*  (Costa Rica)

^b TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Distributions; Heat kernel; Riesz means; Spectral asymptotics

Indexed keywords

EID: 0242321111     PISSN: 10726691     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (40)

1. F. H. Brownell, An extension of Weyl's law for eigenvalues, Pac. J. Math. 5, 483-489 (1955).
2. F. H. Brownell, Extended asymptotic eigenvalue distributions for bounded domains in n-space, J. Math. Mech. 6, 119-166 (1957).
3. A. H. Chamseddine and A. Connes, Universal formula for noncommutative geometry actions: Unification of gravity and the standard model, Phys. Rev. Lett. 77, 4868-4871 (1996).
4. E.M. De Jager, Theory of distributions, in Mathematics Applied to Physics, ed. by E. Roubine, (Springer-Verlag, New York, 1970), pp. 52-110.
5. S. Desjardins and P. Gilkey, Heat content asymptotics for operators of Laplace type with Neumann boundary conditions, Math. Z. 215, 251-268 (1994).
6. B. S. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965).
7. R. Estrada, The asymptotic expansion of certain series considered by Ramanujan, Appl. Anal. 43, 191-228 (1992).
8. R. Estrada, Summability of cardinal series and of localized Fourier series, Appl. Anal., in press.
9. R. Estrada, The Cesàro behavior of distributions, Proc. Roy. Soc. London A 454, 2425-2443 (1998).
10. R. Estrada, J. M. Gracia-Bondía, and J. C. Várilly, On summability of distributions and spectral geometry, Commun. Math. Phys. 191, 219-248 (1998).
11. R. Estrada and R. P. Kanwal, A distributional theory for asymptotic expansions, Proc. Roy. Soc. London A 428,We 399-430 (1990).
12. R. Estrada and R. P. Kanwal, Asymptotic Analysis: A Distributional Approach (Birkhäuser, Boston, 1994).
13. S. A. Fulling, The local geometric asymptotics of continuum eigenfunction expansions. I, SIAM J. Math. Anal. 13, 891-912 (1982).
14. S. A. Fulling, The local geometric asymptotics of continuum eigenfunction expansions. II, SIAM J. Math. Anal. 14, 780-795 (1983).
15. S. A. Fulling, The local geometric asymptotics of continuum eigenfunction expansions. III, J. Phys. A 16, 2615-2631 (1983).
16. S. A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time (Cambridge University Press, Cambridge, 1989).
17. S. A. Fulling and R. A. Gustafson, Some properties of Riesz means and spectral expansions, Electron. J. Diff. Eqns., 1999 No. 06, 1-39 (1999).
18. S. A. Fulling, F. J. Narcowich, and R. M. Wald, Singularity structure of the two-point function in quantum field theory in curved space-time. II, Ann. Phys. (N.Y.) 136, 243-272 (1981).
19. S. A. Fulling and and S. N. M. Ruijsenaars, Temperature, periodicity, and horizons, Phys. Reports 152, 135-176 (1987).
20. I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. 1 (Academic Press, New York, 1964).
21. P. B. Gilkey, Heat equation and heat content asymptotics, in Heat Kernel Techniques and Quantum Gravity, ed. by S. A. Fulling (Mathematics Dept., Texas A&M University, College Station, 1994), pp. 1-10.
22. P. Greiner, An asymptotic expansion for the heat equation, Arch. Rat. Mech. Anal. 41, 163-218 (1971).
23. A. Grossman, G. Loupias, and E. M. Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space, Ann. Inst. Fourier 18, 343-368 (1968).
24. G. H. Hardy, The second theorem of consistency for summable series, Proc. London Math. Soc. (2) 15, 72-88 (1916).
25. G. H. Hardy, Divergent Series (Clarendon Press, Oxford, 1949).
26. G. H. Hardy and M. Riesz, The General Theory of Dirichlet's Series (Cambridge University Press, Cambridge, 1915).
27. L. Hörmander, On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators, in Belfer Graduate School of Science Annual Science Conference Proceedings: Some Recent Advances in the Basic Sciences, ed. by A. Gelbart, Vol. 2 (1965-1966) (Yeshiva University, New York, 1969), pp. 155-202.
28. L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121, 193-218 (1968).
29. R. P. Kanwal, Generalized Functions: Theory and Technique (Academic Press, New York, 1983).
30. K. Kodaira, The eigenvalue problem for ordinary differential equations of second order and Heisenberg's theory of S-matrices, Amer. J. Math. 71, 921-945 (1949).
31. S. Łojasiewicz, Sur la valeur et la limite d'une distribution en un point, Studia Math. 16, 1-36 (1957).
32. D. M. McAvity, Heat kernel asymptotics for mixed boundary conditions, Class. Quantum Grav. 9, 1983-1997 (1992).
33. D. M. McAvity, Surface energy from heat content asymptotics, J. Phys. A 36, 823-830 (1993).
34. S. Minakshisundaram and Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Can. J. Math. 1, 242-256 (1949).
35. S. Pilipovic, Quasi-asymptotic expansion and the Laplace transform, Appl. Anal. 35, 247-261 (1990).
36. L. Schwartz, Théory des Distributions (Hermann, Paris, 1966).
37. J. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664-679 (1951).
38. E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-order Differential Equations, Part One, 2nd ed. (Oxford University Press, Oxford, 1962).
39. M. van den Berg and P. B. Gilkey, Heat content asymptotics of a Riemannian manifold with boundary, J. Func. Anal. 120, 48-71 (1994).
40. V. S. Vladimirov, Y. N. Drozhinov, and B. I. Zavyalov, Multidimensional Tauberian Theorems for Generalized Functions (Nauka, Moscow, 1986).