In this article, we study the pointwise asymptotic behavior of iterated convolutions on the one-dimensional lattice . We generalize the so-called local limit theorem in probability theory to complex-valued sequences. A sharp rate of convergence toward an explicitly computable attractor is proved together with a generalized Gaussian bound for the asymptotic expansion up to any order of the iterated convolution.
BerryA. C. (1941). The accuracy of the Gaussian approximation to the sum of independent variates. Transactions of the American Mathematical Society, 49, 122–136. 10.2307/1990053
2.
ComtetL. (1974). Advanced combinatorics: The art of finite and infinite expansions enlarged edition. D. Reidel Publishing Co.
3.
ConwayJ. B. (1990). A course in functional analysis. Graduate texts in mathematics (Vol. 96, pp. xvi+399). Springer-Verlag.
CoulombelJ.-F.FayeG. (2022). Generalized Gaussian bounds for discrete convolution powers. Revista Matemática Iberoamericana, 38(5), 1553–1604. https://ems.press/journals/rmi/articles/5093715
6.
CoulombelJ.-F.FayeG. (2023). Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions. IMA Journal of Numerical Analysis, 43(1), 187–224. 10.1093/imanum/drab088
DesprésB. (2009). Uniform asymptotic stability of strang’s explicit compact schemes for linear advection. Journal on Numerical Analysis, 47(5), 3956–3976. https://epubs.siam.org/doi/10.1137/080734571
9.
DiaconisP.Saloff-CosteL. (2014). Convolution powers of complex functions on . Mathematische Nachrichten, 287(10), 1106–1130. 10.1002/mana.201200163
10.
EsseenC.-G. (1942). On the Liapounoff limit of error in the theory of probability. Arkiv för Matematik, Astronomi och Fysik, 28A(9), 19.
11.
GodillonP. (2003). Green’s function pointwise estimates for the modified Lax-Friedrichs scheme. Mathematical Modelling and Numerical Analysis (ESAIM: M2AN), 37(1), 1–39 DOI: 10.1051/m2an:2003022.
12.
GrevilleT. N. E. (1966). On stability of linear smoothing formulas. SIAM Journal on Numerical Analysis, 3(1), 157–170. 10.1137/0703011
13.
GustafssonB.KreissH.-O.OligerJ. (1995). Time dependent problems and difference methods (Pure and applied mathematics). A Wiley-Interscience Publication, John Wiley & Sons, Inc.
14.
KatoT. (1995). Perturbation theory for linear operators (classics in mathematics). Reprint of the 1980 edition. Springer-Verlag.
15.
KreissH.-O. (1968). Stability theory for difference approximations of mixed initial boundary value problems. I. Mathematics of Computation, 22, 703–714 . 10.2307/2004572
16.
NewmanD. J. (1975). A simple proof of Wiener’s theorem. Proceedings of the American Mathematical Society, 48, 264–265. 10.2307/2040730
17.
PetrovV. V. (1975). Sums of independent random variables (Ergebnisse der mathematik und ihrer grenzgebiete, Band 82). Springer-Verlag.
18.
RandlesE.Saloff-CosteL. (2015). On the convolution powers of complex functions on . Journal of Fourier Analysis and Applications, 21(4), 754–798. 10.1007/s00041-015-9386-1
19.
RobinsonD. W. (1991). Elliptic operators and Lie groups (Oxford mathematical monographs). Oxford Science Publications, The Clarendon Press, Oxford University Press.
20.
RudinW. (1987). Real and complex analysis (3rd ed.). McGraw-Hill Book Co.
21.
SchoenbergI. J. (1953). On smoothing operations and their generating functions. The Bulletin of the American Mathematical Society, 59, 199–230. 10.1090/S0002-9904-1953-09695-1
22.
StrangG. (1968). On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis, 5, 506–517. 10.1137/0705041
23.
TadmorE. (1986). Complex symmetric matrices with strongly stable iterates. Linear Algebra and its Applications, 78, 65–77. 10.1016/0024-3795(86)90016-9
24.
ThoméeV. (1965). Stability of difference schemes in the maximum-norm. Journal of Differential Equations, 1, 273–292. 10.1016/0022-0396(65)90008-2
25.
ZumbrunK.HowardP. (1998). Pointwise semigroup methods and stability of viscous shock waves. Indiana University Mathematics Journal, 47(3), 741–871. 10.1512/iumj.1998.47.1604