The 𝐿^{𝑝} norm of sums of translates of a function

Marek, Kanter

doi:10.1090/S0002-9947-1973-0361617-4

The $L^p$ norm of sums of translates of a function
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by Kanter Marek

Trans. Amer. Math. Soc. 179 (1973), 35-47

DOI: https://doi.org/10.1090/S0002-9947-1973-0361617-4

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Abstract:

For p not an even integer, $p > 0$, we prove that knowledge of the ${L^p}$ norm of all linear combinations of translates of a real valued function in ${L^p}(R)$ determines the function up to translation and multiplication by $\pm 1$.

References

Salomon Bochner, Harmonic analysis and the theory of probability, University of California Press, Berkeley-Los Angeles, Calif., 1955. MR 0072370, DOI 10.1525/9780520345294
J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
Kiyosi Itô, The canonical modification of stochastic processes, J. Math. Soc. Japan 20 (1968), 130–150. MR 226719, DOI 10.2969/jmsj/02010130
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751

Théorie de l’addition des variables aléatoires

Michael Schilder, Some structure theorems for the symmetric stable laws, Ann. Math. Statist. 41 (1970), 412–421. MR 254915, DOI 10.1214/aoms/1177697080
Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121