Abstract
A constructive analytic function f is defined as a pair of form (A,Ω), where. A is a fundamental sequence in some constructive metric space and Ω is a regulator of its convergence into itself. The pointwise-defined function f corresponding to function f* turns out to be Bishop-differentiable [2], while the domain of f* is the limit of a growing sequence of compacta. The derivative of a constructive analytic function and the integral along a curve are defined approximatively. It is proved that the fundamental theorems of constructive complex analysis are valid for such functions. Eight items of literature are cited.
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Literature cited
B. A. Kushner, “On the existence of unbounded constructive analytic functions,” Dokl. Akad. Nauk SSSR,160, No. 1, 29–31 (1965).
E. Bishop, Foundations of Constructive Analysis, McGraw-Hill Book Co., New York (1967).
R. L. Goodstein, Recursive Analysis, North-Holland Publ. Co., Amsterdam (1961).
N. A. Shanin, “Constructive real numbers and constructive functional spaces,” Tr. Mat. Inst. Akad. Nauk SSSR,67, 15–294 (1962).
J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 2nd ed., Amer. Math. Soc., Providence, Rhode Island (1960).
R. L. Goodstein, “A constructivist theory of plane curves,” Fund. Math.,43, No. 1, 23–35 (1956).
F. Hausdorff, Set Theory, 2nd. ed., Chelsea Publ. Co., New York (1962).
V. P. Orevkov, “New proof of the uniqueness theorem for constructive differential functions of a real variable,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,40, 119–126 (1974).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 50, pp. 49–58, 1976. Result announced November 1, 1973.
The author thanks N. A. Shanin, who suggested the consideration of the approximative approach to the definition of a constructive analytic function.
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Dantsin, E.Y. On an approximative version of the notion of constructive analytic function. J Math Sci 14, 1457–1463 (1980). https://doi.org/10.1007/BF01693977
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DOI: https://doi.org/10.1007/BF01693977