Geodesic
Cauchy and Poisson formulas for polyanalytic functions and applications
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2016), pp. 15-26.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain new Cauchy and Poisson integral formulas for polyanalytic functions. As an application, we establish mean value theorems for functions polyanalytic and real polyharmonic in a disk. We also give applications to sharp estimates of generalized maximum modulus principle type for associated functions, and, in particular, to estimates for rational functions (components) in the problem of singularity separation for polyrational functions.
Keywords: polyanalytic and polyrational functions, Cauchy and Poisson integral formulas, mean value theorems, generalized maximum modulus principle. 
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V. I. Danchenko. Cauchy and Poisson formulas for polyanalytic functions and applications. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2016), pp. 15-26. http://geodesic.mathdoc.fr/item/IVM_2016_1_a1/

[1] Balk M. B., “Polianaliticheskie funktsii i ikh obobscheniya”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem. Fundament. napravleniya, 85, 1991, 187–246 | MR | Zbl

[2] Balk M. B., Zuev M. F., “O polianaliticheskikh funktsiyakh”, UMN, 25:5 (1970), 203–226 | MR | Zbl

[3] Teodorescu N., La derivee areolaire et ses applications la physique mathematique, Paris, 1931

[4] Fedorov V. S., “Osnovnye svoistva obobschennykh monogennykh funktsii”, Izv. vuzov. Matem., 1958, no. 6, 257–265 | MR | Zbl

[5] Fedorov V. S., “Ob odnom vide monogennykh giperkompleksnykh funktsii”, Matem. sb., 50(92):1 (1960), 101–108 | MR | Zbl

[6] Ramazanov A.-R. K., “Otsenka normy polianaliticheskoi funktsii cherez normu ee poligarmonicheskoi sostavlyayuschei”, Matem. zametki, 75:4 (2004), 608–613 | DOI | MR | Zbl

[7] Ramazanov A.-R. K., “O stroenii prostranstv polianaliticheskikh funktsii”, Matem. zametki, 72:5 (2002), 750–764 | DOI | MR | Zbl

[8] Ramazanov A.-R. K., “Predstavlenie prostranstva polianaliticheskikh funktsii v vide pryamoi summy ortogonalnykh podprostranstv. Prilozhenie k ratsionalnym approksimatsiyam”, Matem. zametki, 66:5 (1999), 741–759 | DOI | MR | Zbl

[9] Nicolesco M., Les fonctions polyharmoniques, Hermann, Paris, 1936 | Zbl

[10] Nicolesco M., “Nouvelles recherches sur les fonctions polyharmoniques”, Disq. Math. Phys., 1 (1940), 43–56 | MR | Zbl

[11] Karmona D. D., Paramonov P. V., Fedorovskii K. Yu., “O ravnomernoi approksimatsii polianaliticheskimi mnogochlenami i zadache Dirikhle dlya bianaliticheskikh funktsii”, Matem. sb., 193:10 (2002), 75–98 | DOI | MR | Zbl

[12] Buave A., Gote P. M., Paramonov P. V., “O ravnomernoi approksimatsii $n$-analiticheskimi funktsiyami na zamknutykh mnozhestvakh”, Izv. RAN. Ser. matem., 68:3 (2004), 15–28 | DOI | MR | Zbl

[13] Mazalov M. Ya., Paramonov P. V., Fedorovskii K. Yu., “Usloviya $C^m$-priblizhaemosti funktsii resheniyami ellipticheskikh uravnenii”, UMN, 67:6 (2012), 53–100 | DOI | MR | Zbl

[14] Mazalov M. Ya., “O ravnomernykh priblizheniyakh bianaliticheskimi funktsiyami na proizvolnykh kompaktakh v $\mathbb C$”, Matem. sb., 195:5 (2004), 79–102 | DOI | MR | Zbl

[15] Mazalov M. Ya., “O zadache Dirikhle dlya polianaliticheskikh funktsii”, Matem. sb., 200:10 (2009), 59–80 | DOI | MR | Zbl

[16] Carmona J. J., Fedorovskiy K. Yu., “New conditions for uniform approximation by polyanalytic polynomials”, Tr. MIAN, 279, 2012, 227–241 | MR | Zbl

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

[18] Perelman N. R., Rasulov K. M., “Trekhelementnaya zadacha tipa Karlemana dlya bianaliticheskikh funktsii v kruge”, Izv. Saratovsk. un-ta, Nov. ser., Ser. Matem. Mekhan. Informatika, 12:2 (2012), 18–26 | Zbl

[19] Bogdanova N. N., Rasulov K. M., “O reshenii nevyrozhdennoi chetyrekhelementnoi kraevoi zadachi tipa Karlemana dlya bianaliticheskikh funktsii v kruge”, Izv. Saratovsk. un-ta, Nov. ser., Ser. Matem. Mekhan. Informatika, 9:2 (2009), 6–12

[20] Dolzhenko E. P., Danchenko V. I., “O granichnom povedenii reshenii obobschennogo uravneniya Koshi–Rimana”, Vestn. MGU. Matem. Mekhan., 1998, no. 3, 16–25 | MR | Zbl

[21] Dolzhenko E. P., Danchenko V. I., “O granichnykh svoistvakh reshenii obobschennogo uravneniya Koshi–Rimana”, Tr. MIAN, 236, 2002, 142–152 | MR | Zbl

[22] Danchenko V. I., Dolzhenko E. P., “On mean integral values of solutions of the generalized Cauchy–Riemann equations”, J. Math. Sci., 145:5 (2007), 5188–5191 | DOI | MR | Zbl

[23] Danchenko V. I., Dolzhenko E. P., “On boundary behavior of holomorphic components of bianalytic functions”, J. Math. Sci., 126:6 (2005), 1586–1592 | DOI | MR | Zbl

[24] Vitushkin A. G., “Analiticheskaya emkost mnozhestv v zadachakh teorii priblizhenii”, UMN, 22:6 (1967), 141–199 | MR | Zbl

[25] Khavin V. P., “Razdelenie osobennostei analiticheskikh funktsii s sokhraneniem ogranichennosti”, Algebra i analiz, 16:1 (2004), 293–319 | MR | Zbl

[26] Bochtein A. M., Katsnelson V. E., “Ob otsenke normy proektora v odnom prostranstve analiticheskikh funktsii”, Teoriya funktsii, funkts. analiz i ikh prilozheniya, 12, Kharkov, 1970, 81–85

[27] Gonchar A. A., Grigoryan L. D., “Ob otsenkakh normy golomorfnoi sostavlyayuschei meromorfnoi funktsii”, Matem. sb., 99(141):4 (1976), 634–638 | MR | Zbl

[28] Pekarskii A. A., “Otsenki proizvodnoi integrala tipa Koshi s meromorfnoi plotnostyu i ikh prilozheniya”, Matem. zametki, 31:3 (1982), 389–402 | MR | Zbl

[29] Danchenko V. I., “O razdelenii osobennostei meromorfnykh funktsii”, Matem. sb., 125(167):2 (1984), 181–198 | MR | Zbl