Family of Bessel operator functions

A. V. Glushak

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 187 (2020), pp. 36-43.

Résumé

A family of Bessel operator functions and a generator of this family are introduced and considered. Their properties are studied, a criterion of the uniform well-posedness of the Cauchy problem for the Euler–Poisson–Darboux equation is obtained, and the connections of this family with a series of other solving operators are found.

Détail

Keywords: Bessel operator function, generator, Cauchy problem, uniform well-posedness criterion.
Mots-clés : Euler–Poisson–Darboux equation

Bibliographie

[1] Vasilev V. V., Krein S. G., Piskarev S. I., “Polugruppy operatorov, kosinus-operator funktsii i lineinye differentsialnye uravneniya”, Itogi nauki i tekhn. Mat. analiz., 28 (1990), 87–202 | MR | Zbl

[2] Glushak A. V., “O vozmuschenii abstraktnogo uravneniya Eilera—Puassona—Darbu”, Mat. zametki., 60:3 (1996), 363–369 | MR | Zbl

[3] Glushak A. V., “Operatornaya funktsiya Besselya”, Dokl. RAN., 352:5 (1997), 587–589 | MR | Zbl

[4] Glushak A. V., “Regulyarnoe i singulyarnoe vozmuscheniya abstraktnogo uravneniya Eilera—Puassona—Darbu”, Mat. zametki., 66:3 (1999), 364–371 | MR | Zbl

[5] Glushak A. V., “Operatornaya funktsiya Besselya i svyazannye s neyu polugruppy i modifitsirovannoe preobrazovanie Gilberta”, Differ. uravn., 35:1 (1999), 128–130 | MR | Zbl

[6] Glushak A. V., “Operatornaya funktsiya Lezhandra”, Izv. RAN. Ser. mat., 65:6 (2001), 3–14 | MR | Zbl

[7] Glushak A. V., “O svyazi prointegrirovannoi kosinus-operator-funktsii s operatornoi funktsiei Besselya”, Differ. uravn., 42:5 (2006), 583–589 | MR | Zbl

[8] Glushak A. V., “Abstraktnaya zadacha Koshi dlya uravneniya Besselya—Struve”, Differ. uravn., 53:7 (2017), 891–905 | MR | Zbl

[9] Glushak A. V., “Kriterii razreshimosti vesovoi zadachi Koshi dlya abstraktnogo uravneniya Eilera—Puassona—Darbu”, Differ. uravn., 54:5 (2018), 627–-637 | Zbl

[10] Glushak A. V., “Operatornaya formula sdviga resheniya zadachi Koshi dlya abstraktnogo uravneniya Eilera—Puassona—Darbu”, Mat. zametki., 105:5 (2019), 656–665 | MR | Zbl

[11] Glushak A. V., Kononenko V. I., Shmulevich S. D., “Ob odnoi singulyarnoi abstraktnoi zadache Koshi”, Izv. vuzov. Mat., 1986, no. 6, 55–56 | MR | Zbl

[12] Glushak A. V., Pokruchin O. A., “Kriterii razreshimosti zadachi Koshi dlya abstraktnogo uravneniya Eilera—Puassona—Darbu”, Differ. uravn., 52:1 (2016), 41–59 | MR | Zbl

[13] Goldstein Dzh., Polugruppy lineinykh operatorov i ikh prilozheniya, Vischa shkola, Kiev, 1989 | MR

[14] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967

[15] Lebedev N. N., Spetsialnye funktsii i ikh prilozheniya, Fizmatgiz, M., 1963

[16] Levitan B. M., “Razlozhenie po funktsiyam Besselya v ryady i integraly Fure”, Usp. mat. nauk., 1:2 (42) (1951), 102–143

[17] Melnikova I. V., Filinkov A. I., “Integrirovannye polugruppy i $C$-polugruppy. Korrektnost i regulyarizatsiya differentsialno-operatornykh zadach”, Usp. mat. nauk., 49:6 (300) (1994), 111–150 | MR | Zbl

[18] Orlov V. P., “O slabo vyrozhdayuschikhsya giperbolicheskikh uravneniyakh”, Differ. uravn., 39:10 (2003), 1409–1419 | MR | Zbl

[19] Bresters D. W., “On the Euler–Poisson–Darboux equation”, SIAM J. Math. Anal., 4:1 (1973), 31–41 | DOI | MR | Zbl

[20] Da Prato G., Giusti E., “Una caratterizzazione dei generatori di funzioni coseno astratte”, Boll. Union. Mat. Ital., 22 (1967), 357–362 | MR | Zbl

[21] Fattorini H. O., “Ordinary differential equations in linear topological space, II”, J. Differ. Equ., 6 (1969), 50–70 | DOI | MR | Zbl

[22] Fattorini H. O., “A note on fractional derivatives of semigroups and cosine functions”, Pac. J. Math., 109:2 (1983), 335–347 | DOI | MR | Zbl

[23] Sova M., “Cosine operator functions”, Rozpr. mat., 49 (1966), 1–47 | MR