Abstract
By using some family \(\mathcal{H}\) of convex nondecreasing functions on [0, ∞), we define the space G(\(\mathcal{H}\)) of 2π-periodic infinitely differentiable functions on the real line with given estimates for all derivatives. A description of the space G(\(\mathcal{H}\)) is obtained in terms of the best trigonometric approximations and the rate of decrease of the Fourier coefficients. We find families \(\mathcal{H}\) for which functions from G(\(\mathcal{H}\)) can be extended to analytic functions in the horizontal strip of the complex plane. An internal description of the space of such extensions is obtained. Examples of a family \(\mathcal{H}\) of convex functions are given.
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REFERENCES
D. Jackson, “Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung,” Gekronte Preisschrift und Inaugural-Dissertation (Göttingen Univ., Göttingen, 1911).
D. Jackson, “The general theory of approximation by polynomials and trigonometric sums,” Bull. Am. Math. Soc. 27, 9–10 (1921).
D. Jackson, The Theory of Approximation, Am. Math. Soc. Colloquim Publications, Vol. 11 (Am. Math. Soc., 1930).
S. N. Bernshtein, “On the best approximation of continuous functions by means of polynomials of a given degree. II,” Soobshch. Khar’k. Mat. O-va. Vtoraya Ser. 13 (4–5), 145–194 (1912).
S. N. Bernstein, “Sur l’ordre de la meilleure approximation des fonctions continues par les polynômes de degré donné,” Mem. Cl. Sci. Acad. R. Belg. 4, 1–103 (1912).
S. N. Bernshtein, “On new investigations relative to the best approximation of continuous functions by polynomials,” Usp. Mat. Nauk 5 (4), 121–131 (1950).
Ch. J. de la Vallo Poussin, Leçons sur l’approximation des fonctions d’une variable réellee (Gauthier-Villars, Paris, 1919).
P. L. Ul’yanov, “On classes of infinitely differentiable functions,” Sov. Math. Dokl. 39, 298–301 (1989).
P. L. Ul’yanov, “Properties of functions in Gevrey classes,” Sov. Math. Dokl. 42, 550–554 (1991).
P. L. Ul’yanov, “On classes of infinitely differentiable functions,” Math. USSR Sb. 70, 11–30 (1991). https://doi.org/10.1070/SM1991v070n01ABEH001251
R. W. Braun, R. Meise, and B. A. Taylor, “Ultradifferentiable functions and Fourier analysis,” Results Math. 17, 206–237 (1990). https://doi.org/10.1007/BF03322459
A. F. Timan, Approximation Theory for Functions of Real Variable, Moscow: Fizmatgiz, 1960.
I. P. Natanson, Constructive Theory of Functions (Gos. Izd-vo Tekh. Teor. Lit., Moscow, 1949).
R. T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics and Physics, Vol. 13 (Princeton Univ. Press, Princeton, N.J., 1970).
I. Kh. Musin, “Surjectivity of linear differential operators in a weighted space of infinitely differentiable functions,” Math. Notes 71, 649–660 (2002). https://doi.org/10.1023/A:1015835904747
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Translated by I. Tselishcheva
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Musin, I.K. Constructive Description of a Class of Periodic Functions on the Real Line. Russ Math. 67, 32–42 (2023). https://doi.org/10.3103/S1066369X23020032
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DOI: https://doi.org/10.3103/S1066369X23020032