Extremal problems on Pompeiu sets

V. V. Volchkov

Sbornik. Mathematics, Tome 189 (1998) no. 7, pp. 955-976

Résumé

The local Pompeiu problem and related questions about functions with vanishing integrals over parallelepipeds are studied. One of the central results of the paper is the solution of the problem about the ball of smallest radius in which parallelepipeds of fixed size have the Pompeiu property. Several results of interest for complex analysis and approximation theory are obtained as consequences.

Bibliographie

[1] Zalcman L., “A bibliographic survey of the Pompeiu problem”, Approximation by solutions of partial differential equations, eds. B. Fuglede et al., 1992, 185–194 | MR | Zbl

[2] Berenstein K. A., Struppa D., “Kompleksnyi analiz i uravneniya v svertkakh”, Itogi nauki i tekhn. Sovrem. probl. matem. Fundam. napravleniya, 54, VINITI, M., 1989, 5–111 | MR

[3] Volchkov V. V., “Novye teoremy o srednem dlya reshenii uravneniya Gelmgoltsa”, Matem. sb., 184:7 (1993), 71–78 | MR | Zbl

[4] Garofalo N., Segala F., “Univalent functions and the Pompeiu problem”, Trans. Amer. Math. Soc., 346 (1994), 137–146 | DOI | MR | Zbl

[5] Ullrich D. C., “More on the Pompeiu problem”, Amer. Math. Monthly, 101 (1994), 165–168 | DOI | MR | Zbl

[6] Berenstein C. A., Gay R., “Le probleme de Pompeiu locale”, J. Anal. Math., 52 (1989), 133–166 | DOI | MR | Zbl

[7] Berenstein C. A., Gay R., “A local version of the two-circles theorem”, Israel J. Math., 55 (1986), 267–288 | DOI | MR | Zbl

[8] Volchkov V. V., “Ekstremalnye varianty problemy Pompeiyu”, Matem. zametki, 59:5 (1996), 671–680 | MR | Zbl

[9] Volchkov V. V., “Ob odnoi ekstremalnoi zadache, svyazannoi s teoremoi Morery”, Matem. zametki, 60:6 (1996), 804–809 | MR | Zbl

[10] Volchkov V. V., “Okonchatelnyi variant lokalnoi teoremy o dvukh radiusakh”, Matem. sb., 186:6 (1995), 15–34 | MR | Zbl

[11] Volchkov V. V., “Okonchatelnyi variant teoremy o srednem v teorii garmonicheskikh funktsii”, Matem. zametki, 59:3 (1996), 351–358 | MR | Zbl

[12] Volchkov V. V., “Novye teoremy o dvukh radiusakh v teorii garmonicheskikh funktsii”, Izv. RAN. Seriya matem., 58:1 (1994), 182–194 | Zbl

[13] Volchkov V. V., “Teoremy o srednem dlya odnogo klassa polinomov”, Sib. matem. zhur., 35:4 (1994), 737–745 | MR | Zbl

[14] Volchkov V. V., “Teoremy o dvukh radiusakh na ogranichennykh oblastyakh evklidovykh prostranstv”, Dif. uravneniya, 30:10 (1994), 1719–1724 | MR | Zbl

[15] Volchkov V. V., “Teoremy o dvukh radiusakh na prostranstvakh postoyannoi krivizny”, Dokl. RAN, 347:3 (1996), 300–302 | MR | Zbl

[16] Volchkov V. V., “Ob odnoi probleme Zaltsmana i ee obobscheniyakh”, Matem. zametki, 53:2 (1993), 30–36 | MR | Zbl

[17] Volchkov V. V., “Novye teoremy o srednem dlya polianaliticheskikh funktsii”, Matem. zametki, 56:3 (1994), 20–28 | MR | Zbl

[18] Volchkov V. V., “Morera type theorems on the unit disk”, Anal. Math., 20 (1994), 49–63 | DOI | MR | Zbl

[19] Volchkov V. V., “Teoremy tipa Morery na oblastyakh so slabym usloviem konusa”, Izv. vuzov. Matematika, 1993, no. 10, 15–20 | MR | Zbl

[20] Berenstein C. A., Gay R., Yger A., “The three squares theorem, a local version”, Analysis and Partial Differential Equations, eds. C. Sadosky, Marcel Deccer, New York, 1990, 35–50 | MR

[21] Szabo G., “On functions having the same integral on congruent semidiscs”, Ann. Univ. Sci. Budapest. Sect. Comput., 3 (1982), 3–9 | MR | Zbl

[22] Volchkov V. V., “Approksimatsiya funktsii na ogranichennykh oblastyakh v $\mathbb R^n$ lineinymi kombinatsiyami sdvigov”, Dokl. RAN, 334:5 (1994), 560–561 | MR | Zbl

[23] Dzyadyk V. K., “Geometricheskoe opredelenie analiticheskikh funktsii”, UMN, 15:1 (1960), 191–194 | MR

[24] Zalcman L., “Analyticity and the Pompeiu problem”, Arch. Rational Mech. Anal., 47 (1972), 237–254 | DOI | MR | Zbl

[25] Aizenberg L. A., “Variatsii na temu teoremy Morera i problemy Pompeiyu”, Dokl. RAN, 337:6 (1994), 709–712 | Zbl

[26] Berenstein C. A., Chang D. C., Pascuas D., Zalcman L., “Variations on the theorem of Morera”, Contemp. Math., 137 (1992), 63–78 | MR | Zbl

[27] Agranovsky M., Berenstein C. A., Chang D. C., “Morera theorem for holomorphic $H^p$ spaces in the Heisenberg group”, J. Reine Angew. Math., 443 (1993), 49–89 | MR | Zbl

[28] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[29] Vilenkin N. Ya., Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1991 | MR | Zbl

[30] Leng S., $SL_2(\mathbb R)$, Mir, M., 1977

[31] Khelgason S., Gruppy i geometricheskii analiz, Mir, M., 1987 | MR

[32] Khermander L., Lineinye differentsialnye operatory s chastnymi proizvodnymi, Mir, M., 1965 | MR

[33] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR