Equiconvergence of eigenfunction expansions for Sturm-Liouville operators with a distributional potential

I. V. Sadovnichaya

Sbornik. Mathematics, Tome 201 (2010) no. 9, pp. 1307-1322

Résumé

The Sturm-Liouville operator $L=-d^2/dx^2+q(x)$ in the space $L_2[0,\pi]$ under Dirichlet boundary conditions is investigated. It is assumed that $q(x)=u'(x)$, $u(x)\in L_2[0,\pi]$ (here, differentiation is used in the distributional sense). The problem of when the expansion of a function $f(x)$ in terms of a series of eigenfunctions and associated functions of the operator $L$ is uniformly equiconvergent on the whole of the interval $[0,\pi]$ with its Fourier sine series expansion is considered. It is shown that such uniform convergence holds for any function $f(x)$ in the space $L_2[0,\pi]$. Bibliography: 22 titles.

Détail

Keywords: Sturm-Liouville operator, singular potential
Mots-clés : uniform equiconvergence.

Bibliographie

[1] A. M. Savchuk, A. A. Shkalikov, “Sturm–Liouville operators with singular potentials”, Math. Notes, 66:6 (1999), 741–753 | DOI | MR | Zbl

[2] A. M. Savchuk, A. A. Shkalikov, “Sturm–Liouville operators with distribution potentials”, Trans. Moscow Math. Soc., 2003, 143–192 | MR | Zbl

[3] A. M. Savchuk, A. A. Shkalikov, “On the eigenvalues of the Sturm–Liouville operator with potentials from Sobolev spaces”, Math. Notes, 80:5–6 (2006), 814–832 | DOI | MR | Zbl

[4] U. Dini, Fondamenti per la teorica delle funzioni di variabili reali, Piza, 1878 | Zbl

[5] V. A. Steklov, “Sur les expressions asymptotiques de certaines fonctions, définies par les équations différentielles linéaires du second ordre, et leurs applications au problem̀e du développement d'une fonction arbitraire en séries procédant suivant lesdites fonctions”, Soobsch. Khark. matem. ob-va (2), 10:2–6 (1909), 97–200 | Zbl

[6] A. Haar, “Zur Theorie der orthogonalen Funktionensysteme”, Math. Ann., 69:3 (1910), 331–371 ; 71:1 (1911), 38–53 | DOI | MR | Zbl | DOI | MR

[7] M. H. Stone, “A comparison of the series of Fourier and Birkhoff”, Trans. Amer. Math. Soc., 28:4 (1926), 695–761 | DOI | MR | Zbl

[8] V. A. Marchenko, Sturm–Liouville operators and applications, Oper. Theory Adv. Appl., 22, Birkhäuser, Basel, 1986 | MR | MR | Zbl | Zbl

[9] V. A. Il'in, “Equiconvergence, with trigonometric series, of expansions in root functions of a one-dimensional Schrödinger operator with a complex potential in the class $L_1$”, Differential Equations, 27:4 (1991), 401–416 | MR | Zbl

[10] A. M. Gomilko, G. V. Radzievskiǐ, “Equiconvergence of series in eigenfunctions of ordinary functional-differential operators”, Soviet Math. Dokl., 43:1 (1991), 47–52 | MR | Zbl

[11] A. P. Khromov, “O ravnoskhodimosti razlozhenii po sobstvennym funktsiyam differentsialnykh operatorov vtorogo poryadka”, Differents. uravneniya i vych. matem., 5:2 (1975), 3–20 | MR | Zbl

[12] E. Wermuth, Konvergenzuntersuchungen bei Eigenfunktionsentwicklungen zu Randeigenwertproblemen $n$-ter Ordnung mit parameterbhängigen Randbedingugen, RWTH, Mathematisch-Naturwissenschaft. fak., Aachen, 1984

[13] A. M. Minkin, “Equiconvergence theorems for differential operators”, J. Math. Sci. (New York), 96:6 (1999), 3631–3715 | DOI | MR | Zbl

[14] V. A. Vinokurov, V. A. Sadovnichii, “Uniform equiconvergence of a Fourier series in eigenfunctions of the first boundary value problem and of a Fourier trigonometric series”, Dokl. Math., 64:2 (2001), 248–252 | MR | Zbl

[15] P. Djakov, B. Mityagin, “Spectral gaps of Schrodinger operators with periodic singular potentials”, Dyn. Partial Differ. Equ., 6:2 (2009), 95–165 | MR | Zbl

[16] P. Djakov, B. Mityagin, “Bari–Markus property for Riesz projections of Hill operators with singular potentials”, Functional analysis and complex analysis (Istanbul, Turkey, 2007), Contemp. Math., 481, Amer. Math. Soc., Providence, RI, 2009, 59–80 | MR | Zbl

[17] I. V. Sadovnichaya, “On the equiconvergence rate of trigonometric series expansions and eigenfunction expansions for the Sturm–Liouville operator with a distributional potential”, Differ. Equ., 44:5 (2008), 675–684 | DOI | MR | Zbl

[18] I. V. Sadovnichaya, “Equiconvergence of series expansions in terms of trigonometric functions and eigenfunctions of the Sturm–Liouville operator with a distribution potential”, Dokl. Math., 68:2 (2003), 191–193 | MR | Zbl

[19] M. A. Naimark, Lineinye differentsialnye operatory, 2-e izd., Nauka, M., 1969 | MR | MR | Zbl

[20] A. M. Savchuk, O sobstvennykh funktsiyakh operatora Shturma–Liuvillya s potentsialami iz prostranstv Soboleva, arXiv: 1003.3172

[21] N. K. Bari, Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[22] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1934 | MR | MR | Zbl