Abstract
The eigenvalue problem for differential operators of arbitrary order with integral constraints is considered. The asymptotics of the eigenvalues is obtained. The results are applied to finding the asymptotics of the probability of small deviations for some detrended processes of nth order.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
X. Ai and W. V. Li, “Karhunen–Loève expansions for them-th order detrended Brownianmotion,” Sci. China Math. 57 (10), 2043–2052 (2014).
A. L. Skubachevskii, “Nonclassical boundary value problems. I,” in CMFD (RUDN, Moscow, 2007), Vol. 26, pp. 3–132 [J. Math. Sci. 155 (2), 199–334 (2008)].
M. Janet, “Les valeurs moyennes des carres de deux dérivées d’ordres consecutifs, et le développement en fraction continue de tang x,” Bull. Sci. Math. (2) 55, 11–23 (1931).
A. I. Nazarov and A. N. Petrova, “On exact constants in some embedding theorems of high order,” Vestnik St. Petersburg Univ. Ser. I Mat. Mekh. Astronom., No. 4, 16–20 (2008) [Vestn. St. Petersbg. Univ., Math. 41 (4), 298–302 (2008)].
M. Janet, “Sur le minimum du rapport de certaines intégrales,” C. R. Acad. Sci. Paris 193, 977–979 (1931).
A. C. Slastenin, Exact Constants in Some One-Dimensional Embedding Theorems, Diploma thesis (St. Petersburg University, St. Petersburg, 2014) [in Russian].
I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products (Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963; Academic Press, New York–London, 1965).
M. A. Naimark, Linear Differential Operators (Nauka, Moscow, 1969) [in Russian].
E. Titchmarsh, The Theory of Functions, 2nd ed., (Oxford Univ. Press, London, 1964; Nauka, Moscow, 1980).
A. I. Nazarov, “On the exact constant in the asymptotics of small deviations in the L2-norm of someGaussian processes,” in Problems of Mathematical Analysis, Vol. 26: Nonlinear Problems and the Theory of Functions (T. Rozhkovskaya, Novosibirsk, 2003), pp. 179–214 [in Russian].
L. Beghin, Ya. Yu. Nikitin, and E. Orsingher, “Exact small ball constants for some Gaussian processes under L2-norm,” in Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI), Vol. 298: Probability and Statistics. 6 (POMI, St. Petersburg., 2003), pp. 5–21 [J. Math. Sci. (New York) 128 (1), 2493–2502 (2005)].
P. Deheuvels, “A Karhunen–Loève expansion for a mean-centered Brownian bridge,” Statist. Probab. Lett. 77 (12), 1190–1200 (2007).
X. Ai, W. V. Li and G. Liu, “Karhunen–Loève expansions for the detrended Brownian motion,” Statist. Probab. Lett. 82 (7), 1235–1241 (2012).
M. A. Lifshits, Lectures on Gaussian Random Processes (Lan’, St. Petersburg, 2016) [in Russian].
W. V. Li, “Comparison results for the lower tail of Gaussian seminorms,” J. Theoret. Probab. 5 (1), 1–31 (1992).
F. Gao, J. Hannig, and F. Torcaso, “Comparison theorems for small deviations of random series,” Electron. J. Probab. 8 (21), 1–17 (2003).
A. I. Nazarov and Ya. Yu. Nikitin, “Exact L2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems,” Probab. Theory Related Fields 129 (4), 469–494 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu. P. Petrova, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 3, pp. 405–414.
Rights and permissions
About this article
Cite this article
Petrova, Y.P. Spectral asymptotics for problems with integral constraints. Math Notes 102, 369–377 (2017). https://doi.org/10.1134/S0001434617090073
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434617090073