Sharp Jackson-type inequalities for approximations of classes of convolutions by entire functions of exponential type

Vinogradov, O.

doi:10.1090/S1061-0022-06-00922-8

Sharp Jackson-type inequalities for approximations of classes of convolutions by entire functions of exponential type
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by O. L. Vinogradov
Translated by: the author

St. Petersburg Math. J. 17 (2006), 593-633

DOI: https://doi.org/10.1090/S1061-0022-06-00922-8

Published electronically: May 3, 2006

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Abstract:

In this paper, a new method is introduced for the proof of sharp Jackson-type inequalities for approximation of convolution classes of functions defined on the real line. These classes are approximated by linear operators with values in sets of entire functions of exponential type. In particular, a sharp Jackson-type inequality for the even-order derivatives of the conjugate function is proved. For the uniform and the integral norm, the estimates are sharp even if their left-hand sides are replaced by the best approximation. Sharp inequalities for approximations of periodic functions by trigonometric polynomials and of almost-periodic functions by generalized trigonometric polynomials are special cases of the inequalities mentioned above.

References

N. I. Ahiezer, Lektsii po teorii approksimatsii, Second, revised and enlarged edition, Izdat. “Nauka”, Moscow, 1965 (Russian). MR 0188672
J. Favard, Sur les meilleurs procédés d’approximation de certaines classes des fonctions par des polynômes trigonométriques, Bull. Sci. Math. (2) 61 (1937), 209–224, 243–256.
N. I. Akhiezer and M. G. Kreĭn, Best approximation of differentiable periodic functions by trigonometric sums, Dokl. Akad. Nauk SSSR 15 (1937), no. 3, 107–112. (Russian)
N. I. Akhiezer, Best approximation of a class of continuous periodic functions, Dokl. Akad. Nauk SSSR 17 (1937), no. 9, 451–453. (Russian)
—, Best approximation of analytic functions, Dokl. Akad. Nauk SSSR 18 (1938), no. 4–5, 241–244. (Russian)
M. G. Kreĭn, On best approximation of periodic functions, Dokl. Akad. Nauk SSSR 18 (1938), no. 4–5, 245–249. (Russian)
B. Sz.-Nagy, Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. I. Periodischer Fall, Ber. Verh. Sächs. Akad. Wiss. Leipzig 90 (1938), 103–134.
V. K. Dzjadyk, Best approximation on classes of periodic functions that are defined by integrals of a linear combination of absolutely monotonic kernels, Mat. Zametki 16 (1974), 691–701 (Russian). MR 380212
Nguen Tkhi Tkh′eu Khoa, Some extremal problems on classes of functions defined by linear differential operators, Mat. Sb. 180 (1989), no. 10, 1355–1395, 1440 (Russian); English transl., Math. USSR-Sb. 68 (1991), no. 1, 213–255. MR 1025687
Nguen Tkhi Tkh′eu Khoa, The operator $D(D^2+1^2)\cdots (D^2+n^2)$ and trigonometric interpolation, Anal. Math. 15 (1989), no. 4, 291–306 (Russian, with English summary). MR 1032110
S. Nikolsky, Approximation of functions in the mean by trigonometrical polynomials, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 10 (1946), 207–256 (Russian, with English summary). MR 0017402
Sun Yung-sheng, On the best approximation of classes of functions representable in the convolution form, Dokl. Akad. Nauk SSSR (N.S.) 118 (1958), 247–250 (Russian). MR 0103372
M. G. Kreĭn, Best approximation of continuous differentiable functions on the real axis, Dokl. Akad. Nauk SSSR 18 (1938), no. 9, 619–623. (Russian)
B. Sz.-Nagy, Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. \rom{II}. Nichtperiodischer Fall, Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 3–24.
E. S. Hvostenko, An analogue of A. A. Markov’s theorem for the linear-fractional kernels of M. G. Kreĭn, Studies in contemporary problems of summability and approximation of functions and their applications (Russian), Dnepropetrovsk. Gos. Univ., Dnepropetrovsk, 1976, pp. 57–63, 157 (Russian). MR 0586563
V. V. Žuk, Certain exact inequalities between best approximations and moduli of continuity, Sibirsk. Mat. Ž. 12 (1971), 1283–1291 (Russian). MR 0291710
A. A. Ligun, The exact constants of approximation of differentiable periodic functions, Mat. Zametki 14 (1973), 21–30 (Russian). MR 330880
A. Ju. Gromov, Exact constants in the approximation of differentiable functions by entire functions, Studies in contemporary problems of summability and approximation of functions and their applications (Russian), Dnepropetrovsk. Gos. Univ., Dnepropetrovsk, 1976, pp. 17–21, 153 (Russian). MR 0614756
V. F. Babenko and A. Yu. Gromov, Exact estimates in the approximation of classes of differentiable functions by entire functions, Studies in Contemporary Problems of Summability and Approximation of Functions and their Applications, Dnepropetrovsk. Gos. Univ., Dnepropetrovsk, 1977, pp. 3–6. (Russian)
A. A. Ligun, Exact constants in inequalities of Jackson type, Mat. Zametki 38 (1985), no. 2, 248–256, 349 (Russian). MR 808893
N. I. Merlina, Sharp estimates of seminorms and best approximations by entire functions. (Manuscript dep. VINITI 10.04.79, no. 1257–79). (Russian)
—, On sharp estimates of seminorms and best approximations by entire functions, Theory of Functions of Complex Variable and Boundary Problems, No. 3, Cheboksary, 1979, pp. 20–26. (Russian)
V. V. Žuk, On the question of the constants in direct theorems of approximation theory for differentiable functions, Vestnik Leningrad. Univ. 19, Mat. Meh. Astronom. Vyp. 4 (1976), 51–57, 155 (Russian, with English summary). MR 0430650
V. V. Zhuk, Approksimatsiya periodicheskikh funktsiĭ, Leningrad. Univ., Leningrad, 1982 (Russian). MR 665432
O. L. Vinogradov and V. V. Zhuk, Sharp inequalities connected with estimates for approximations of periodic functions by means of moduli of continuity of their derivatives of odd order with different steps, J. Math. Sci. (New York) 101 (2000), no. 2, 2914–2929. Nonlinear equations and mathematical analysis. MR 1784684, DOI 10.1007/BF02672177
O. L. Vinogradov and V. V. Zhuk, Jackson-type exact inequalities for differentiable functions and the minimization of the step of the modulus of continuity, Proceedings of the St. Petersburg Mathematical Society, Vol. 8 (Russian), Tr. St.-Peterbg. Mat. Obshch., vol. 8, Nauchn. Kniga, Novosibirsk, 2000, pp. 29–51 (Russian). MR 1868027
B. M. Levitan, Počti-periodičeskie funkcii, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953 (Russian). MR 0060629
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0061695
I. M. Ryžik and I. S. Gradšteĭn, Tablicy integralov, summ, ryadov i proizvedeniĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1951 (Russian). 3d ed. MR 0052590
A. F. Timan, Teorij pribli+enij funkciĭ deĭstvitel’nogo peremennogo, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960 (Russian). MR 0117478
N. P. Korneĭchuk, Tochnye konstanty v teorii priblizheniya, “Nauka”, Moscow, 1987 (Russian). MR 926687
V. V. Zuk and G. I. Natanson, The approximation of differentiable periodic functions by linear methods, Vestnik Leningrad. Univ. 19 Mat. Meh. Astronom. vyp. 4 (1977), 16–21, 146 (Russian, with English summary). MR 0467136
S. N. Bernšteĭn, Sobranie sočinenii. Tom I. Konstruktivnaya teoriya funkciĭ [1905–1930], Izdat. Akad. Nauk SSSR, Moscow, 1952 (Russian). MR 0048360
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
V. V. Zhuk and V. F. Kuzyutin, Approksimatsiya funktsiĭ i chislennoe integrirovanie, Izdatel′stvo Sankt-Peterburgskogo Universiteta, St. Petersburg, 1995 (Russian, with Russian summary). MR 1664064
Allan Pinkus, $n$-widths in approximation theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 7, Springer-Verlag, Berlin, 1985. MR 774404, DOI 10.1007/978-3-642-69894-1