Approximation and Boundary Properties of Polyanalytic Functions

K. Yu. Fedorovskiy

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and geometric issues of complex analysis, Tome 235 (2001), pp. 262-271.

Résumé

We give a review of some recent results concerning the uniform approximation of functions by polyanalytic functions and polyanalytic polynomials on planar compact sets. We discuss the boundary properties of polyanalytic functions and their relationships with uniform approximation problems. Some problems of approximation by the solutions of homogeneous second-order elliptic equations with constant complex coefficients are also considered.

Bibliographie

[1] Balk M. B., Polyanalytic functions, Math. Res., 63, Akad. Verl., Berlin, 1991 | MR | Zbl

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

[3] Trent T., Wang J. L., “Uniform approximation by rational modules on nowhere dense sets”, Proc. Amer. Math. Soc., 81 (1981), 62–64 | DOI | MR | Zbl

[4] Carmona J. J., “Mergelyan approximation theorem for rational modules”, J. Approx. Theory, 44 (1985), 113–126 | DOI | MR | Zbl

[5] Wang J. L., “A localization operator for rational modules”, Rocky Mount. J. Math., 19:4 (1989), 999–1002 | DOI | MR | Zbl

[6] Verdera J., “On the uniform approximation problem for the square of the Cauchy–Riemann operator”, Pacif. J. Math., 159 (1993), 379–396 | MR | Zbl

[7] Mazalov M. Ya., “Ravnomernoe priblizhenie funktsii, nepreryvnykh na proizvolnom kompakte v $\mathbb{C}$ i analiticheskikh vnutri kompakta, funktsiyami, bianaliticheskimi v ego zamykanii”, Mat. zametki, 69:2 (2001), 245–261 | MR | Zbl

[8] Mergelyan S. N., “Ravnomernoe priblizhenie funktsii kompleksnogo peremennogo”, UMN, 7:2 (1952), 31–122 | MR | Zbl

[9] Fedorovskii K. Yu., “O ravnomernykh priblizheniyakh funktsii $n$-analiticheskimi polinomami na spryamlyaemykh konturakh v $\mathbb{C}$”, Mat. zametki, 59:4 (1996), 604–610 | MR | Zbl

[10] Fedorovski K. Yu., “On uniform approximation by polyanalytic polynomials on compact subsets of the plane”, Ann. Univ. M. Curie-Sklodowska. A, 53:3 (1999), 27–39 | MR | Zbl

[11] Carmona J. J., Fedorovski K. Yu., Paramonov P. V., On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions, Preprint No 415, Centre Rec. Mat., Barcelona, Spain, 1999 | MR

[12] Kusis P., Vvedenie v teoriyu prostranstv $H^p$, Mir, M., 1984 | MR

[13] Douglas R. G., Shapiro H. S., Shields A. L., “Cyclic vectors and invariant subspaces for the backward shift operator”, Ann. Inst. Fourier (Grenoble), 20:1 (1970), 37–76 | MR | Zbl

[14] Johnston E. H., “The boundary modulus of continuity of harmonic functions”, Pacif. J. Math., 90 (1980), 87–98 | MR | Zbl

[15] Paramonov P. V., “$C^m$-priblizheniya garmonicheskimi polinomami na kompaktnykh mnozhestvakh v $\mathbb{R}^n$”, Mat. sb., 184:2 (1993), 105–128 | MR | Zbl

[16] Mazalov M. Ya., “Primer nepostoyannoi bianaliticheskoi funktsii, obraschayuscheisya v nul vsyudu na nigde ne analiticheskoi granitse”, Mat. zametki, 62:3–4 (1997), 524–526 | MR | Zbl

[17] Keldysh M. V., “O razreshimosti i ustoichivosti zadachi Dirikhle”, UMN, 1941, no. 8, 171–231 | MR | Zbl

[18] Deny J., “Systèmes totaux de fonctions harmoniques”, Ann. Inst. Fourier (Grenoble), 1 (1949), 103–113 | MR

[19] Walsh J. L., “The approximation of harmonic functions by polynomials and by harmonic rational functions”, Bull. Amer. Math. Soc., 35 (1929), 499–544 | DOI | MR | Zbl

[20] Lebesgue H., “Sur le probleme de Dirichlet”, Rend. Circ. Mat. Palermo, 29 (1907), 371–402 | DOI

[21] Landkof N. S., Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR | Zbl

[22] Paramonov P. V., Fedorovskii K. Yu., “O ravnomernoi i $C^1$-priblizhaemosti funktsii na kompaktakh v $\mathbb{R}^2$ resheniyami ellipticheskikh uravnenii vtorogo poryadka”, Mat. sb., 190:2 (1999), 123–144 | MR | Zbl

[23] Verchota G. C., Vogel A. L., “Nonsymmetric systems on nonsmooth planar domains”, Trans. Amer. Math. Soc., 349:11 (1997), 4501–4535 | DOI | MR | Zbl

[24] Verdera J., Removability, capacity and approximation. Complex potential theory, NATO ASI. Ser. C., 439, Kluwer Acad. Publ., Dordrecht, 1993 | MR

[25] Paramonov P. V., “O priblizheniyakh garmonicheskimi polinomami v $C^1$-norme na kompaktakh v $\mathbb{R}^2$”, Izv. RAN. Ser. mat., 57:2 (1993), 113–124 | MR | Zbl

[26] Verdera J., “$C^m$-approximation by solutions of elliptic equations and Calderon–Zygmund operators”, Duke Math. J., 55:1 (1987), 157–187 | DOI | MR | Zbl

[27] Melnikov M. S., “Saga o probleme Penleve i analiticheskoi emkosti”, Trudy MIAN, 157–164