Linear convolution integral equations with asymptotically almost periodic solutions

Jordan, G. S.; Madych, W. R.; Wheeler, R. L.

doi:10.1090/S0002-9939-1980-0553371-7

Linear convolution integral equations with asymptotically almost periodic solutions
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by G. S. Jordan, W. R. Madych and R. L. Wheeler PDF

Proc. Amer. Math. Soc. 78 (1980), 337-341 Request permission

Abstract:

Let $\mu$ be a bounded Borel measure and f be asymptotically almost periodic. Conditions are found which ensure that certain bounded solutions of the linear convolution integral equation $g \ast \mu = f$ are asymptotically almost periodic. This result is also extended to the case where the measure $\mu$ is replaced by a tempered distribution $\tau$ for which convolution with bounded functions makes sense.

References

Arne Beurling, Un théorème sur les fonctions bornées et uniformément continues sur l’axe réel, Acta Math. 77 (1945), 127–136 (French). MR 12701, DOI 10.1007/BF02392224
Arne Beurling, Sur une classe de fonctions presque-périodiques, C. R. Acad. Sci. Paris 225 (1947), 326–328 (French). MR 21146
S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 2039–2043. MR 145283, DOI 10.1073/pnas.48.12.2039
Raouf Doss, On the almost periodic solutions of a class of integrodifferential-difference equations, Ann. of Math. (2) 81 (1965), 117–123. MR 170169, DOI 10.2307/1970387
R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799
A. M. Fink and W. R. Madych, On certain bounded solutions of $g\ast \mu =f$, Proc. Amer. Math. Soc. 75 (1979), no. 2, 235–242. MR 532143, DOI 10.1090/S0002-9939-1979-0532143-5
G. S. Jordan and Robert L. Wheeler, Linear integral equations with asymptotically almost periodic solutions, J. Math. Anal. Appl. 52 (1975), no. 3, 454–464. MR 410311, DOI 10.1016/0022-247X(75)90072-4
J. J. Levin and D. F. Shea, On the asymptotic behavior of the bounded solutions of some integral equations. I, II, III, J. Math. Anal. Appl. 37 (1972). MR 306849, DOI 10.1016/0022-247X(72)90258-2
B. Lewitan, On an integral equation with an almost periodic solution, Bull. Amer. Math. Soc. 43 (1937), no. 10, 677–679. MR 1563616, DOI 10.1090/S0002-9904-1937-06624-9