Exponentials Form a Basis of Discrete Holomorphic Functions on a Compact
Bulletin de la Société Mathématique de France, Volume 132 (2004) no. 2, p. 305-326

We show that discrete exponentials form a basis of discrete holomorphic functions on a finite critical map. On a combinatorially convex set, the discrete polynomials form a basis as well.

Nous montrons que les exponentielles forment une base des fonctions holomorphes discrètes sur une carte critique compacte. Sur un convexe, les polynômes discrets forment également une base.

DOI : https://doi.org/10.24033/bsmf.2467
Classification:  30G25,  52C26,  31C20,  39A12
Keywords: discrete holomorphic functions, discrete analytic functions, monodriffic functions, exponentials
@article{BSMF_2004__132_2_305_0,
     author = {Mercat, Christian},
     title = {Exponentials Form a Basis of Discrete Holomorphic Functions on a Compact},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {132},
     number = {2},
     year = {2004},
     pages = {305-326},
     doi = {10.24033/bsmf.2467},
     zbl = {1080.30042},
     mrnumber = {2076370},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2004__132_2_305_0}
}
Mercat, Christian. Exponentials Form a Basis of Discrete Holomorphic Functions on a Compact. Bulletin de la Société Mathématique de France, Volume 132 (2004) no. 2, pp. 305-326. doi : 10.24033/bsmf.2467. http://www.numdam.org/item/BSMF_2004__132_2_305_0/

[1] S. Agafonov & A. Bobenko - « Discrete Z γ and Painlevé equations », Internat. Math. Res. Notices (2000), no. 4, p. 165-193. | MR 1747617 | Zbl 0969.39015

[2] A. Bobenko - « Discrete conformal maps and surfaces », Symmetries and integrability of difference equations (Canterbury, 1996), Cambridge Univ. Press, Cambridge, 1999, p. 97-108. | MR 1705222 | Zbl 1001.53001

[3] A. Bobenko & U. Pinkall - « Discrete isothermic surfaces », J. reine angew. Math. 475 (1996), p. 187-208. | MR 1396732 | Zbl 0845.53005

[4] A. Bobenko & R. Seiler (éds.) - Discrete integrable geometry and physics, Oxford Lecture Series in Mathematics and its Applications, vol. 16, The Clarendon Press Oxford University Press, New York, 1999. | MR 1676681 | Zbl 0936.37027

[5] A. Bobenko & Y. Suris - « Integrable systems on quad-graphs », http://arXiv.org/abs/nlin.SI/0110004, 2001. | MR 1890049 | Zbl 1004.37053

[6] Y. Colin De Verdière, I. Gitler & D. Vertigan - « Réseaux électriques planaires. II », Comm. Math. Helv. 71 (1996), no. 1, p. 144-167. | MR 1371682 | Zbl 0853.05074

[7] R. Duffin - « Basic properties of discrete analytic functions », Duke Math. J. 23 (1956), p. 335-363. | MR 78441 | Zbl 0070.30503

[8] -, « Potential theory on a rhombic lattice », J. Comb. Theory 5 (1968), p. 258-272. | MR 232005 | Zbl 0247.31003

[9] U. Hertrich-Jeromin - Introduction to Möbius Differential Geometry, London Mathematical Society Lecture Note Series, vol. 300, Cambridge University Press, Cambridge, 2003. | MR 2004958 | Zbl 1040.53002

[10] R. Kenyon - « The Laplacian and Dirac operators on critical planar graphs », Invent. Math. 150 (2002), no. 2, p. 409-439, http://arXiv.org/abs/math-ph/0202018. | MR 1933589 | Zbl 1038.58037

[11] C. Mercat - « Discrete Period Matrices and Related Topics », http://arXiv.org/abs/math-ph/0111043.

[12] -, « Discrete Polynomials and Discrete Holomorphic Approximation », http://arXiv.org/abs/math-ph/0206041.

[13] -, « Discrete Riemann surfaces and the Ising model », Comm. Math. Phys. 218 (2001), no. 1, p. 177-216. | MR 1824204 | Zbl 1043.82005