Abstract
We consider the algebra C u = C u (ℝ) of uniformly continuous bounded complex functions on the real line ℝ with pointwise operations and sup-norm. Let I be a closed ideal in C u invariant with respect to translations, and let ah I (f) denote the minimal real number (if it exists) satisfying the following condition. If λ > ah I (f), then \(\left. {\left( {\hat f - \hat g} \right)} \right|_V = 0\) for some g ∈ I, where V is a neighborhood of the point λ. The classical Titchmarsh convolution theorem is equivalent to the equality ah I (f 1 · f 2) = ah I (f 1) + ah I (f 2), where I = {0}. We show that, for ideals I of general form, this equality does not generally hold, but ah I (f n) = n · ah I (f) holds for any I. We present many nontrivial ideals for which the general form of the Titchmarsh theorem is true.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. V. Treschev, “Oscillator and thermostat,” Discrete Contin. Dyn. Syst. (DCDS-A), 28:4 (2010), 1693–1712; http://arxiv.org/abs/0912.1186.
B. Ya. Levin, Lectures on Entire Functions, Translations of Math. Monographs, vol. 150, Amer. Math. Soc., Providence, RI, 1996.
N. Bourbaki, Theories spectrales, Hermann, Paris, 1967.
E. A. Gorin, “G. E. Silov’s investigations in the theory of commutative Banach algebras and their subsequent development,” Uspekhi Mat. Nauk, 33:4 (1978), 169–188; English transl.: Russian Math. Surveys, 33:4 (1978), 197–217.
L. Hormander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Second ed., Grundlehren der Mathematischen Wissenschaften, vol. 256, Springer-Verlag, Berlin, 1990.
J. L. Lions, “Supports de produits de compositions,” C. R. Acad. Sci., 232 (1951), 1530–1532.
J. L. Lions, “Supports de produits de compositions. I,” C. R. Acad. Sci., 232 (1951), 1622–1624.
J. L. Lions, “Supports dans la transfomations de Laplace. II,” J. Anal. Math., 2 (1953), 369–380.
L. Hormander, The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients. Grundlehren der Mathematischen Wissenschaften, vol. 257, Springer-Verlag, Berlin, 1983.
B. Weiss, “Titchmarsh’s convolution theorem on groups,” Proc. Amer. Math. Soc., 19 (1968), 75–79.
Y. Domar, “Convolution theorems of Titchmarsh type on discrete R n,” Proc. EdinburghMath. Soc., 32:3 (1989), 449–457.
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 46, No. 1, pp. 31–38, 2012
Original Russian Text Copyright © by E. A. Gorin and D. V. Treschev
D. V. Treschev acknowledges the partial support of the Presidium of the Russian Academy of Science, the program “Mathematical Control Theory.”
Rights and permissions
About this article
Cite this article
Gorin, E.A., Treschev, D.V. Relative version of the Titchmarsh convolution theorem. Funct Anal Its Appl 46, 26–32 (2012). https://doi.org/10.1007/s10688-012-0003-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-012-0003-7