A first-order boundary value problem with boundary condition on a countable set of points

A. M. Minkin

Matematičeskie zametki, Tome 62 (1997) no. 3, pp. 418-424.

Résumé

Let $E=\{E_n\}$ be the family of subspaces spanning the eigenfunctions and adjoint functions of the boundary-value problem $$ -i\frac{dy}{dx}=\lambda y,\quad -a\le x\le a,\qquad U(y)\equiv\int_{-a}^ay(t)d\sigma(t)=0, $$ that correspond to “close” eigenvalues (in the sense of the distance defined as the maximal of the Euclidean and the hyperbolic metrics). For a purely discrete measure $d\sigma$ it is shown that the system $E$ does not form an unconditional basis of subspaces in $L^2(-a,a)$ if at least one of the end points $\pm a$ is mass-free.

Bibliographie

[1] Levin B. Ya., “O bazisakh pokazatelnykh funktsii v $L_2$”, Zapiski matem. otd. fiz.-matem. fak-ta Kharkovkogo universiteta i Kharkovskogo matem. obschestva, 27:4 (1961), 39–48 | MR

[2] Golovin V. D., “Biortogonalnye razlozheniya v $L^2$ po lineinym kombinatsiyam pokazatelnykh funktsii”, Zapiski matem. otd. fiz.-matem. fak-ta Kharkovkogo universiteta i Kharkovskogo matem. obschestva, 30:4 (1963), 18–29

[3] Pavlov B. S., “Spektralnyi analiz differentsialnogo operatora s “razmazannym” granichnym usloviem”, Problemy matem. fiziki, no. 6, LGU, L., 1973, 101–109

[4] Molodenkov V. A., “Razlozhenie po sobstvennym funktsiyam odnoi kraevoi zadachi”, Differents. uravneniya i vychislitelnaya matem., no. 2, Saratov, 1975, 56–65 | MR | Zbl

[5] Molodenkov V. A., Khromov A. P., “Razlozhenie po sobstvennym funktsiyam odnoi kraevoi zadachi dlya operatora differentsirovaniya”, Differents. uravneniya i vychislitelnaya matem., no. 1, Saratov, 1972, 17–26 | MR | Zbl

[6] Sedletskii A. M., “Biortogonalnye razlozheniya funktsii v ryady eksponent na konechnom intervale veschestvennoi osi”, UMN, 37:5 (227) (1982), 51–95 | MR

[7] Sedletskii A. M., “Razlozhenie po sobstvennym funktsiyam operatora differentsirovaniya s razmazannym kraevym usloviem”, Differents. uravneniya, 30:1 (1994), 70–76 | MR

[8] Sedletskii A. M., “O ravnomernoi skhodimosti negarmonicheskikh ryadov Fure”, Tr. MIAN, 200, Nauka, M., 1991, 299–309

[9] Levin B. Ya., Raspredelenie kornei tselykh funktsii, GITTL, M., 1956

[10] Gubreev G. M., “Spektralnyi analiz biortogonalnykh razlozhenii funktsii v ryady eksponent”, Izv. AN SSSR. Ser. matem., 53:6 (1989), 1236–1268

[11] Pavlov B. S., “Bazisnost sistemy eksponent i uslovie Makenkhaupta”, Dokl. AN SSSR, 247:1 (1979), 37–40 | MR | Zbl

[12] Nikolskii N. K., “Bazisy iz eksponent i znachenii vosproizvodyaschikh yader”, Dokl. AN SSSR, 252:6 (1980), 1316–1320 | MR | Zbl

[13] Khruschëv S. V., “Teoremy vozmuscheniya dlya bazisov iz eksponent i uslovie Makenkhaupta”, Dokl. AN SSSR, 247:1 (1979), 44–48 | MR | Zbl

[14] Pavlov B. S., Nikolskii N. K., Hrushev S. V., “Unconditional bases of exponentials and reproducing kernels”, Lecture Notes in Math., 864, 1981, 214–335 | Zbl

[15] Minkin A. M., “Otrazhenie pokazatelei i bezuslovnye bazisy iz eksponent”, Algebra i analiz, 3:5 (1991), 109–134 | MR | Zbl

[16] Avdonin S. A., Joó I., “Riesz bases of exponentials and sine-type functions”, Acta Math. Hungar., 51:1–2 (1988), 3–14 | DOI | MR | Zbl

[17] Avdonin S. A., Io I., Khorvat M., “Bazisy Rissa iz elementov vida $x^ke^{i\lambda_nx}$”, Vestn. LGU. Matem., mekh., astron., 1989, no. 4 (22), 3–7 | MR

[18] Gubreev G. M., Ignatenko T. R., “Ob odnom klasse ortogonalizatorov semeistv eksponent s veschestvennymi chastotami”, Ukr. matem. zh., 44:8 (1992), 1031–1044 | MR

[19] Eremenko A. E., Sodin M. L., “Parametrization of entire functions of sine-type by their critical values”, Adv. Soviet Math., 11 (1992), 237–242 | MR | Zbl

[20] Minkin A. M., “Metod proektirovaniya i bezuslovnye bazisy”, Algebraicheskie i topologicheskie metody v matematicheskoi fizike (1994, Katsiveli, Ukraina)

[21] Nikolskii N. K., Lektsii ob operatore sdviga, Nauka, M., 1980