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On harmonic functions which take integer values on integer lattices

  • D. H. Armitage (a1)

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Throughout this paper α denotes a number larger than 1 and C denotes the complex plane. Pó1ya(15),(16) (see also Hardy(10) and Landau(13)) proved the following results.

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References

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(1)Armitage, D. H.A linear function from a space of polynomials onto a space of harmonic polynomials. J. London Math. Soc. (2) 5 (1972), 529538.
(2)Armitage, D. H.On harmonic polynomials. Proc. London Math. Soc. (3) 34 (1977), 173192.
(3)Armitage, D. H.Uniqueness theorems for harmonic function which vanish at lattice points. J. Approximation Theory (submitted).
(4)Armitage, D. H.On the derivatives at the origin of entire harmonic functions. Proc. Glasgow Math. Assoc. (in the Press).
(5)Baker, A.A note on integral integer-valued function of several variables. Proc. Cambridge Philos. Soc. 63 (1967), 715720.
(6)Brelot, M.Éléments de la théorie classique du potentiel (3rd ed.Paris, 1965).
(7)Buck, R. C.Integral valued entire functions. Duke Math. J. 15 (1948), 879891.
(8)Gelfond, A. O.Sur les propriétés arithmétiques des fonctions entières. Tôhoku Math. J. 30 (1929), 280285.
(9)Gramain, F.Sur les fonctions entières de plusieurs variables complexes prenant des valeurs algébriques aux points entiers. C. R. Acad. Sci. Paris 284 (1977), 1719.
(10)Hardy, G. H.On a theorem of Mr G. Pó1ya. Proc. Cambridge Philos. Soc. 19 (1920), 6063.
(11)Hayman, W. K.Power series expansions for harmonic functions. Bull. London Math. Soc. 2 (1970), 152158.
(12)Kuran, Ü.On Brelot–Choquet axial polynomials. J. London Math. Soc. (2) 4 (1971), 1526.
(13)Landau, E.On Mr. Hardy's extension of a theorem of Mr. Pólya. Proc. Cambridge Philos. Soc. 20 (1920), 1415.
(14)Lang, S.Algebraic values of meromorphic functions. Topology 3 (1965), 183191.
(15)Pólya, G.Über ganzwertige ganze Funktionen. Rend. Circ. Mat. Palermo 40 (1915), 116.
(16)Pólya, G.Über ganze ganzwertige Funktionen. Göttingen Nachrichten (1920), 110.
(17)Selberg, A.Über einen Satz von A. Gelfond. Arch. Math. Naturvid. 44 (1941), 159170.
(18)Straus, E. G.On entire functions with algebraic derivatives at certain algebraic points. Ann. of Math. (2) 52 (1950), 188198.
(19)Whittaker, J. M.Interpolatory function theory (Cambridge, 1935).

On harmonic functions which take integer values on integer lattices

  • D. H. Armitage (a1)

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