Sharp Bernstein–Nikolskii inequalities for polynomials and entire functions of exponential type

The classical Bernstein–Nikolskii inequalities of the form ‖𝐷𝑓‖𝑞 6 𝒞𝑝𝑞‖𝑓‖𝑝 for 𝑓 ∈ 𝑌 , give estimates for the 𝑝𝑞-norms of the differential operators 𝐷 on classes 𝑌 of polynomials and
entire functions of exponential type. These inequalities play an important role in harmonic analysis, approximation theory and find applications in number theory and metric geometry.
Both order inequalities and inequalities with sharp constants are studied. The last case is especially interesting because the extremal functions depend on the geometry of the manifold
and this fact helps in solving geometric problems.

Historically, Bernstein’s inequalities are referred to the case 𝑝 = 𝑞, and Nikolskii’s inequalities to the estimate of the identity operator for 𝑝 < 𝑞. For the first time, an estimate for the derivative of a trigonometric polynomial for 𝑝 = ∞ was given by S.N. Bernstein (1912), although earlier A.A. Markov (1889) gave its algebraic version. Bernstein’s inequality was refined by E. Landau,
M. Riess, and A. Sigmund (1933) proved it for all 𝑝 > 1. For 𝑝 < 1, the Bernstein order inequality was found by V.I. Ivanov (1975), E.A. Storozhenko, V.G. Krotov and P. Oswald (1975), and the sharp inequality by V.V. Arestov (1981). For entire functions of exponential type, the sharp Bernstein inequality was proved by N.I. Akhiezer, B.Ya. Levin (𝑝 > 1, 1957), Q.I. Rahman and G. Schmeisser (𝑝 < 1, 1990).
The first one-dimensional Nikolskii inequalities for 𝑞 = ∞ were established by D. Jackson (1933) for trigonometric polynomials and J. Korevaar (1949) for entire functions of exponential
type. In all generality for 𝑞 6 ∞ and 𝑑-dimensional space, this was done by S.M. Nikolskii (1951). The estimates of Nikolskii constants were refined by I.I. Ibragimov (1959), D. Amir and
Z. Ziegler (1976), R.J. Nessel and G. Wilmes (1978), and many others. Bernstein–Nikolskii order inequalities for different intervals were studied by N.K. Bari (1954). Variants of inequalities for general multiplier differential operators and weighted manifolds can be found in the works of P.I. Lizorkin (1965), A.I. Kamzolov (1984), A.G. Babenko (1992), A.I. Kozko (1998),
K.V. Runovsky and H.-J. Schmeisser (2001), F. Dai and Y. Xu (2013), V.V. Arestov and P.Yu. Glazyrina (2014) and other authors.
For a long time, the theory of Bernstein–Nikolskii inequalities for polynomials and entire functions of exponential type developed in parallel until E. Levin and D. Lubinsky (2015) established that for all 𝑝 > 0 the Nikolskii constant for functions is the limit of trigonometric constants. For the Bernstein–Nikolskii constants, this fact was proved by M.I. Ganzburg and S.Yu. Tikhonov (2017) and refined by the author together with I.A. Martyanov (2018, 2019).
Multidimensional results of the Levin–Lyubinsky type were proved by the author together with F. Dai and S.Yu. Tikhonov (the sphere, 2020), M.I. Ganzburg (the torus, 2019 and the
cube, 2021).
Until now, the sharp Nikolskii constants are known only for (𝑝, 𝑞) = (2,∞). The case of the Nikolskii constant for 𝑝 = 1 is intriguing. Advancement in this area was obtained by Ya.L. Geronimus (1938), S.B. Stechkin (1961), L.V. Taikov (1965), L. H¨ormander and
B. Bernhardsson (1993), N.N. Andreev, S.V. Konyagin and A.Yu. Popov (1996), author (2005), author and I.A. Martyanov (2018), I.E. Simonov and P.Yu. Glazyrina (2015). E. Carneiro,
M.B. Milinovich and K. Soundararajan (2019) pointed out applications in the theory of the Riemann zeta function. V.V. Arestov, M.V. Deikalova et al (2016, 2018) characterized extremal
polynomials for general weighted Nikolskii constants using duality. Here, S.N. Bernshtein, L.V. Taikov (1965, 1993) and others stood at the origins.
A new direction is the proof of Nikolskii’s sharp inequalities on classes of functions with constraints. It reveals a connection with the extremal problems of harmonic analysis of Turan, Delsarte, the uncertainty principle by J. Bourgain, L. Clozel and J.-P. Kahane (2010) and others. For example, the author and coauthors (2020) showed that the sharp Nikolskii constant for nonnegative spherical polynomials gives an estimate for spherical designs by P. Delsarte, J.M. Goethals and J.J. Seidel (1977). Variants of problems for functions lead to famous estimates for the density of spherical packing, and order results are closely related to Fourier inequalities.
These results are presented in the framework of the general theory of Bernstein–Nikolskii inequalities, applications in approximation theory, number theory, metric geometry are presented, open problems are proposed.

1. Akhiezer, N.I. 1965. “Lectures in the theory of approximation”, Second revised and enlarged

2. edition, Nauka, Moscow. (In Russ.)

3. Amir, D. & Ziegler, Z. 1976. “Polynomials of extremal 𝐿𝑝-norm on the 𝐿∞-unit sphere”,

4. J. Approx. Theory, vol. 18, pp. 86–98.

5. Andreev, N.N., Konyagin, S.V. & Popov, A.Yu. 1996. “Extremum problems for functions with

6. small support”, Math. Notes, vol. 60, no. 3, pp. 241–247.

7. Andreev, N.N., Konyagin, S.V. & Popov, A.Yu. 2000. “Letter to the editor: Extremum

8. problems for functions with small support”, Math. Notes, vol. 68, no. 3, pp. 415.

9. Andreev, N.N. & Yudin, V.A. 2001. “Polynomials of least deviation from zero and Chebyshevtype

10. cubature formulas”, Proc. Steklov Inst. Math., vol. 232, pp. 39–51.

11. Arestov, V.V. 1982. “On integral inequalities for trigonometric polynomials and their

12. derivatives”, Math. USSR-Izv., vol. 18, no. 1, pp. 1–17.

13. Arestov, V., & Babenko, A., & Deikalova, M., & Horv𝑎´th, 𝐴´. 2018. “Nikol’skii inequality

14. between the uniform norm and integral norm with Bessel weight for entire functions of

15. exponential type on the half-line”, Anal. Math., vol. 44, no. 1, pp. 21–42.

16. Arestov, V.V. & Berdysheva, E.E. 2002. “The Tur´𝑎n problem for a class of polytopes”, East

17. J. Approx., vol. 8, no. 3, pp. 381–388.

18. Arestov, V.V. & Glazyrina, P.Yu. 2015. “Bernstein–Szeg¨o inequality for fractional derivatives

19. of trigonometric polynomials”, Proc. Steklov Inst. Math. (Suppl.), vol. 288, no. suppl. 1, pp. 13–

20.

21. Arestov, V.V. & Deikalova, M.V. 2014. “Nikol’skii inequality for algebraic polynomials on a

22. multidimensional Euclidean sphere”, Proc. Steklov Inst. Math. (Suppl.), vol. 284, no. suppl. 1,

23. pp. 9–23.

24. Arestov, V. & Deikalova, M. 2015. “Nikol’skii inequality between the uniform norm and 𝐿𝑞-

25. norm with ultraspherical weight of algebraic polynomials on an interval”, Comput. Methods

26. Funct. Theory, vol. 15, no. 4, pp. 689–708.

27. Arestov, V. & Deikalova, M. 2016. “Nikol’skii inequality between the uniform norm and 𝐿𝑞-

28. norm with Jacobi weight of algebraic polynomials on an interval”, Anal. Math., vol. 42, no. 2,

29. pp. 91–120.

30. Ash, J.M. & Ganzburg, M. 1999. “An extremal problem for trigonometric polynomials”, Proc.

31. Amer. Math. Soc., vol. 127, no. 1, pp. 211–216.

32. Attila, M. & Nevai, P.G. 1980. “Bernstein’s inequality in 𝐿𝑝 for 0 < 𝑝 < 1 and (𝐶, 1) bounds

33. for orthogonal polynomials”, Ann. of Math., vol. 111, no. 1, pp. 145–54.

34. Babenko, A.G. 1992. “Weak-type inequalities for trigonometric polynomials”, Trudy Inst. Mat.

35. i Mekh. UrO RAN, vol. 2, pp. 34–41. (In Russ.)

36. Bari, N.K. 1954. “Generalization of inequalities of S.N. Bernshtein and A.A. Markov”, Izv.

37. Akad. Nauk SSSR Ser. Mat., vol. 18, no. 2, pp. 159–176. (In Russ.)

38. Belinsky, E., Dai, F. & Ditzian, Z. 2003. “Multivariate approximating averages”, J. Approx.

39. Theory, vol. 125, no. 1, pp. 85–105.

40. Benyamini, Y., & Kro´𝑜, A. & Pinkus, A. 2012. “𝐿1-approximation and finding solutions with

41. small support”, Constr. Approx., vol. 36, no. 3, pp. 399–431.

42. Bianchi, G. & Kelly, M. 2015. “A Fourier analytic proof of the Blaschke–Santalo inequality”,

43. Proc. Amer. Math. Soc., vol. 143, no. 11, pp. 4901–4912.

44. Boas, R.P. 1954. “Entire functions”, Academic Press, N.Y.

45. Bogatyrev, A.B. 1999. “Effective computation of Chebyshev polynomials for several intervals”,

46. Sb. Math., vol. 190, no. 11, pp. 1571–1605.

47. Bondarenko, A., Radchenko, D. & Viazovska, M. 2015. “Well-separated spherical designs”,

48. Constr. Approx., vol. 41, no. 1, pp. 93–112.

49. Bourgain, J., & Clozel, L., & Kahane, J.-P. 2010. “Principe d’Heisenberg et fonctions positives”,

50. Ann. Inst. Fourier (Grenoble), vol. 60, no. 4, pp. 1215–1232.

51. Brown, L.G. & Lucier, B.J. 1994. “Best approximations in 𝐿1 are near best in 𝐿𝑝, 𝑝 < 1”,

52. Proc. Amer. Math. Soc., vol. 120, no. 1, pp. 97–100.

53. Carneiro, E., Milinovich, M.B. & Soundararajan, K. 2019. “Fourier optimization and prime

54. gaps”, Comment. Math. Helv., vol. 94, pp. 533–568.

55. Cohn, H. & Elkies, N. 2003. “New upper bounds on sphere packings. I”, Ann. of Math. (2),

56. vol. 157, no. 2, pp. 689–714.

57. Cohn, H., & Goncalves, F. 2019. “An optimal uncertainty principle in twelve dimensions via

58. modular forms”, Invent. math., vol. 217, pp. 799–831.

59. Cohn, H., & Kumar, A., & Miller, S.D., Radchenko, D. & Viazovska, M. 2017. “The sphere

60. packing problem in dimension 24”, Ann. of Math., vol. 185, no. 3, pp. 1017–1033.

61. Conway, J.H. & Sloane, N.J.A. 1999. “Sphere packings, lattices and groups”, Third edition,

62. Springer-Verlag, N.Y.

63. Dai, F. 2006. “Multivariate polynomial inequalities with respect to doubling weights and 𝐴∞

64. weights”, J. Funct. Anal., vol. 235, no. 1, pp. 137–170.

65. Dai, F., Feng, H. & Tikhonov, S. 2016. “Reverse H¨older’s inequality for spherical harmonics”,

66. Proc. Amer. Math. Soc., vol. 144, no. 3, pp. 1041–1051.

67. Dai, F., Gorbachev, D. & Tikhonov, S. 2020. “Nikolskii inequality for lacunary spherical

68. polynomials”, Proc. Amer. Math. Soc., vol. 148, no. 3, pp. 1169–1174.

69. Dai, F., Gorbachev, D. & Tikhonov, S. 2020. “Nikolskii constants for polynomials on the unit

70. sphere”, J. d’Anal. Math., vol. 140, no. 1, pp. 161–185.

71. Dai, F., Gorbachev, D. & Tikhonov, S. 2021. “Estimates of the asymptotic Nikolskii constants

72. for spherical polynomials”, Journal of Complexity, vol. 65, 101553.

73. Dai, F. & Tikhonov, S. 2016. “Weighted fractional Bernstein’s inequalities and their applications”,

74. J. d’Anal. Math., vol. 129, pp. 33–68.

75. Dai, F. & Xu, Yu. 2013. “Approximation theory and harmonic analysis on spheres and balls”,

76. Springer, N.Y.

77. Deikalova, M.V. 2009. “About the sharp Jackson–Nikol’skii inequality for algebraic polynomials

78. on a multidimensional Euclidean sphere”, Proc. Steklov Inst. Math. (Suppl.), vol. 265,

79. no. suppl. 1, pp. S129–S142.

80. Erdelyi, T. 2020. “Arestov’s theorems on Bernstein’s inequality”, J. Approx. Theory, vol. 250,

81.

82. Goncalves, F., & Oliveira e Silva, D. & Ramos, J.P.G. 2021. “On regularity and mass

83. concentration phenomena for the sign uncertainty principle”, J. Geom. Anal., vol. 31, pp. 6080–

84.

85. Ganzburg, M.I. 2019. “Sharp constants of approximation theory. II. Invariance theorems and

86. certain multivariate inequalities of different metrics”, Constr. Approx., vol. 50, pp. 543–577.

87. Ganzburg, M.I. 2020. “Sharp constants of approximation theory. I. Multivariate Bernstein–

88. Nikolskii type inequalities”, J. Fourier Anal. Appl., vol. 26, no. 11.

89. Ganzburg, M.I. 2020. “Sharp constants of approximation theory. III. Certain polynomial

90. inequalities of different metrics on convex sets”, J. Approx. Theory, vol. 252, 105351.

91. Ganzburg, M.I. 2021. “Sharp constants of approximation theory. V. An asymptotic equality

92. related to polynomials with given Newton polyhedra”, J. Math. Anal. Appl., vol. 499, no. 1,

93.

94. Ganzburg, M.I. 2021. “Asymptotics of sharp constants in Markov–Bernstein–Nikolskii type

95. inequalities with exponential weights”, J. Approx. Theory, vol. 265, 105550.

96. Ganzburg, M. 2021. “Sharp constants of approximation theory. VI. Multivariate inequalities

97. of different metrics for polynomials and entire functions”, arXiv:2103.09368.

98. Ganzburg, M. & Tikhonov, S. 2017. “On sharp constants in Bernstein–Nikolskii inequalities”,

99. Constr. Approx., vol. 45, no. 3, pp. 449–466.

100. Genchev, T.G. 1977. “Entire functions of exponential type with polynomial growth on R𝑛𝑥

101. ”,

102. J. Math. Anal. Appl., vol. 60, pp. 103–119.

103. Geronimus, J. 1938. “Sur un probl`eme extr´𝑒mal de Tchebycheff”, Izv. Akad. Nauk SSSR Ser.

104. Mat., vol. 2, no. 4, pp. 445–456. (In Russ.)

105. Gorbachev, D.V. 2000. “Extremal problem for entire functions of exponential spherical type,

106. connected with the Levenshtein bound on the sphere packing density in R𝑛”, Izvestiya of the

107. Tula State University Ser. Mathematics, vol. 6, no. 1, pp. 71–78. (In Russ.)

108. Gorbachev, D.V. 2001. “Extremum problem for periodic functions supported in a ball”, Math.

109. Notes, vol. 69, no. 3, pp. 313–319.

110. Gorbachev, D.V. 2005. “An integral problem of Konyagin and the (𝐶,𝐿)-constants of

111. Nikol’skii”, Proc. Steklov Inst. Math., vol. Suppl. 2, pp. S117–S138.

112. Gorbachev, D.V. & Dobrovolskii, N.N. 2018. “Nikolskii constants in 𝐿𝑝(R, |𝑥|2𝛼+1 𝑑𝑥) spaces”,

113. Chebyshevskii Sbornik, vol. 19, no. 2, pp. 67–79. (In Russ.)

114. Gorbachev, D.V. & Ivanov, V.I. 2015. “Gauss and Markov quadrature formulae with nodes at

115. zeros of eigenfunctions of a Sturm–Liouville problem, which are exact for entire functions of

116. exponential type”, Sbornik: Math., vol. 206, no. 8, pp. 1087–1122.

117. Gorbachev, D.V. & Ivanov, V.I. 2018. “Tur´𝑎n’s and Fej´𝑒r’s extremal problems for Jacobi

118. transform”, Anal. Math., vol. 44, no. 4, pp. 419–432.

119. Gorbachev, D.V. & Ivanov, V.I. 2019. “Tur´𝑎n, Fej´𝑒r and Bohman extremal problems for the

120. multivariate Fourier transform in terms of the eigenfunctions of a Sturm–Liouville problem”,

121. Sb. Math., vol. 210, no. 6, pp. 809–835.

122. Gorbachev, D.V. & Ivanov, V.I. 2019. “Nikol’skii–Bernstein constants for entire functions of

123. exponential spherical type in weighted spaces”, Trudy Inst. Mat. i Mekh. UrO RAN, vol. 25,

124. no. 2, pp. 75–87. (In Russ.)

125. Gorbachev, D.V. & Ivanov, V.I. 2019. “Fractional smoothness in 𝐿𝑝 with Dunkl weight and

126. its applications”, Math. Notes, vol. 106, no. 4, pp. 537–561.

127. Gorbachev, D.V., Ivanov, V.I. & Tikhonov, S.Yu. 2020. “Sharp approximation theorems and

128. Fourier inequalities in the Dunkl setting”, J. Approx. Theory, vol. 258, 105462.

129. Gorbachev, D.V., Ivanov, V.I. & Tikhonov, S.Yu. 2019. “Positive 𝐿𝑝-bounded Dunkl-type

130. generalized translation operator and its applications”, Constr. Approx., vol. 49, no. 3, pp. 555–

131.

132. Gorbachev, D., Ivanov, V. & Tikhonov, S. 2020. “Uncertainty principles for eventually constant

133. sign bandlimited functions”, SIAM J. Math. Anal., vol. 52, no. 5, pp. 4751–4782.

134. Gorbachev, D.V. & Martyanov, I.A. 2018. “On interrelation of Nikolskii constants for

135. trigonometric polynomials and entire functions of exponential type”, Chebyshevskii Sbornik,

136. vol. 19, no. 2, pp. 80–89. (In Russ.)

137. Gorbachev, D.V. & Martyanov, I.A. 2019. “Interrelation between Nikolskii–Bernstein constants

138. for trigonometric polynomials and entire functions of exponential type”, Chebyshevskii Sbornik,

139. vol. 20, no. 3, pp. 143–153. (In Russ.)

140. Gorbachev, D.V. & Martyanov, I.A. 2020. “Letter to the Editor”, Chebyshevskii Sbornik, vol. 21,

141. no. 3, pp. 336–338. (In Russ.)

142. Gorbachev, D.V. & Martyanov, I.A. 2020. “Markov–Bernstein–Nikol’skii constants for polynomials

143. in 𝐿𝑝-space with the Gegenbauer weight”, Chebyshevskii Sbornik, vol. 21, no. 4, pp. 29–44.

144. (In Russ.)

145. Gorbachev, D.V. & Martyanov, I.A. 2020. “Novel bounds of algebraic Nikol’skii constant”,

146. Chebyshevskii Sbornik, vol. 21, no. 4, pp. 45–55. (In Russ.)

147. Gorbachev, D.V., & Mart’yanov, I.A. 2020. “Bounds of the Nikol’skii polynomial constants in

148. 𝐿𝑝 with Gegenbauer weight”, Trudy Inst. Mat. i Mekh. UrO RAN, vol. 26, no. 4, pp. 126–137.

149. (In Russ.)

150. Gorbachev, D.V. & Tikhonov, S.Y. 2018. “Wiener’s problem for positive definite functions”,

151. Math. Zeit., vol. 289, no. 3–4, pp. 859–874.

152. Gorbachev, D. & Tikhonov, S. 2019. “Doubling condition at the origin for non-negative positive

153. definite functions”, Proc. Amer. Math. Soc., vol. 147, pp. 609–618.

154. Delsarte, P., Goethals, J.M. & Seidel, J.J. 1977. “Spherical codes and design”, Geom. Dedicata,

155. vol. 6, no. 3, pp. 363–388.

156. Helgason, S. 1962. “Differential geometry and symmetric spaces”, Academic Press, N.Y.–

157. London.

158. H¨ormander, L. & Bernhardsson, B. 1993. “An extension of Bohr’s inequality”, Boundary value

159. problems for partial differential equations and applications, RMA Res. Notes Appl. Math.,

160. vol. 29, pp. 179–194.

161. Ibragimov, I.I. 1958. “Extremum problems in the class of trigonometric polynomials”, Dokl.

162. Akad. Nauk SSSR, vol. 121, no. 3, pp. 415–417.

163. Ibragimov, I.I. & Dzhafarov, A.S. 1961. “Some inequalities for an entire function of finite degree

164. and its derivatives”, Dokl. Akad. Nauk SSSR, vol. 138, no. 4, pp. 755–758. (In Russ.)

165. Ivanov, V.A. 1983. “On the Bernstein–Nikol’skii and Favard inequalities on compact homogeneous

166. spaces of rank 1”, Russian Math. Surveys, vol. 38, no. 3, pp. 145–146.

167. Ivanov, V.A. 1993. “Precise results in the problem of the Bernstein–Nikol’skij inequality on

168. compact symmetric Riemannian spaces of rank 1”, Proc. Steklov Inst. Math., vol. 194, pp. 115–

169.

170. Ivanov, V.I. 1975. “Certain inequalities in various metrics for trigonometric polynomials and

171. their derivatives”, Math. Notes, vol. 18, no. 4, pp. 880–885.

172. Jackson, D. 1933. “Certain problems of closest approximation”, Bull. Am. Math. Soc., vol. 39,

173. pp. 889–906.

174. Kamzolov, A.I. 1974. “On Riesz’s interpolational formula and Bernshtein’s inequality for

175. functions on homogeneous spaces”, Math. Notes, vol. 15, no. 6, pp. 576–582.

176. Kamzolov, A.I. 1984. “Bernstein’s inequality for fractional derivatives of polynomials in

177. spherical harmonics”, Russian Math. Surveys, vol. 39, no. 2, pp. 163–164.

178. Kamzolov, A.I. 1984. “Approximation of functions on the sphere 𝑆𝑛”, Serdica, vol. 84, no. 1,

179. pp. 3–10. (In Russ.)

180. Kolountzakis, M.N., & R´𝑒v´𝑒sz, Sz.Gy. 2003. “On a problem of Tur´𝑎n about positive definite

181. functions”, Proc. Amer. Math. Soc., vol. 131, pp. 3423–3430.

182. Konyagin, S.V. 1978. “Bounds on the derivatives of polynomials”, Dokl. Akad. Nauk SSSR,

183. vol. 243, no. 5, pp. 1116–1118. (In Russ.)

184. Koornwinder, T.H. 1984. “Jacobi functions and analysis on noncompact semisimple Lie

185. groups”, In “Special functions: Group theoretical aspects and applications”, R.A. Askey,

186. T.H. Koornwinder and W. Schempp (eds.), Reidel, Dordrecht, pp. 1–85.

187. Korevaar, J. 1949. “An inequality for entire functions of exponential type”, Nieuw Arch.

188. Wiskunde (2), vol. 23, pp. 55–62.

189. Korneichuk, N.P. 1976. “Extremal problems of approximation theory”, Nauka, Moscow. (In

190. Russ.)

191. Lebedev, V.I. 2004. “Extremal polynomials and methods of optimization of numerical

192. algorithms”, Sb. Math., vol. 195, no. 10, pp. 1413–1459.

193. Levenshtein, V.I. 1998. “Universal bounds for codes and designs”, In “Handbook of coding

194. theory”, V.S. Pless and W.C. Huffman Eds. Elsevier, Amsterdam.

195. Levin, B.Ya. 1980. “Distribution of zeros of entire functions”, Providence, RI, Amer. Math.

196. Soc.

197. Levin, B.Ya. 1996. “Lectures on entire functions”, English revised edition, Amer. Math. Soc.,

198. Providence, RI.

199. Levitan, B.M. 1973. “Theory of generalized shift operators”, Nauka, Moscow. (In Russ.)

200. Littmann, F. & Spanier, M. 2018. “Extremal signatures”, Constr. Approx., vol. 47, pp. 339–356.

201. Logunov, A. 2018. “Nodal sets of Laplace eigenfunctions: Polynomial upper estimates of the

202. Hausdorff measure”, Ann. of Math., vol. 187, no. 1, pp. 221–39.

203. Lubinsky, D.S. 2014. “On sharp constants in Marcinkiewicz–Zygmund and Plancherel-Polya

204. inequalities”, Proc. Amer. Math. Soc., vol. 142, no. 10, pp. 3575–3584.

205. Lubinsky, D.S. 2014. “Weighted Markov–Bernstein inequalities for entire functions of exponential

206. type”, Publications de l’Institut Math´𝑒matique, vol. 96 (110), pp. 181–192.

207. Levin, E. & Lubinsky, D. 2015. “𝐿𝑝 Chritoffel functions, 𝐿𝑝 universality, and Paley–Wiener

208. spaces”, J. d’Anal. Math., vol. 125, pp. 243–283.

209. Levin, E. & Lubinsky, D. 2015. “Asymptotic behavior of Nikolskii constants for polynomials

210. on the unit circle”, Comput. Methods Funct. Theory, vol. 15, no. 3, pp. 459–468.

211. Lizorkin, P.I. 1965. “Bounds for trigonometrical integrals and an inequality of Bernstein for

212. fractional derivatives”, Izv. Akad. Nauk SSSR Ser. Mat., vol. 29, no. 1, pp. 109–126. (In Russ.)

213. Malykhin, Yu.V. & Ryutin, K.S. 2014. “Concentration of the 𝐿1-norm of trigonometric

214. polynomials and entire functions”, Sb. Math., vol. 205, no. 11, pp. 1620–1649.

215. Martyanov, I.A. 2020. “Nikolskii constant for trigonometric polynomials with periodic

216. Gegenbauer weight”, Chebyshevskii Sbornik, vol. 21, no. 1, pp. 247–258. (In Russ.)

217. Milovanovi´𝑐, G.V., & Mitrinovi´𝑐, D.S. & Rassias, Th.M. 1994. “Topics in polynomials:

218. Extremal problems, inequalities, zeros”, World Scientific Publ. Co., Singapore.

219. Nessel, R. & Wilmes, G. 1978. “Nikolskii-type inequalities for trigonometric polynomials and

220. entire functions of exponential type”, J. Austral. Math. Soc., vol. 25, no. 1, pp. 7–18.

221. Nikol’skii, S.M. 1951. “Inequalities for entire functions of finite degree and their application in

222. the theory of differentiable functions of several variables”, Trudy Mat. Inst. Steklov, vol. 38,

223. pp. 244–278. (In Russ.)

224. Nikolskii, S.M. 1975. “Approximation of functions of several variables and imbedding

225. theorems”, Springer, Berlin–Heidelberg–N.Y.

226. Nursultanov, E.D., Ruzhansky, M.V. & Tikhonov, S.Y. 2015. “Nikolskii inequality and

227. functional classes on compact lie groups”, Funct. Anal. Its Appl. vol. 49, pp. 226–229.

228. Pesenson, I.Z. 1991. “The Bernstein inequality in representations of Lie groups”, Dokl. Math.,

229. vol. 42, no. 1, pp. 87–90.

230. Pesenson, I. 2008. “Bernstein–Nikolskii inequalities and Riesz interpolation formula on compact

231. homogeneous manifolds”, J. Approx. Theory, vol. 150, no. 2, pp. 175–198.

232. Pesenson, I. 2009. “Bernstein–Nikolskii and Plancherel–Polya inequalities in 𝐿𝑝-norms on noncompact

233. symmetric spaces”, Math. Nachr., vol. 282, no. 2, pp. 253–269.

234. Pinkus, A. & Ziegler, Z. 1979. “Interlacing properties of the zeros of the error functions in best

235. 𝐿𝑝-approximations”, J. Approx. Theory, vol. 27, no. 1, pp. 1–18.

236. Platonov, S.S. 2007. “Bessel harmonic analysis and approximation of functions on the half-line”,

237. Izvestiya: Math., vol. 71, no. 5, pp. 1001–1048.

238. Queff´𝑒lec, H. & Zarouf, R. 2019. “On Bernstein’s inequality for polynomials”, Anal. Math.

239. Phys., vol. 9, pp. 1181–1207.

240. Rahman, Q.I. & Schmeisser, G. 1990. “𝐿𝑝 inequalities for entire functions of exponential type”,

241. Trans. Amer. Math. Soc., vol. 320, pp. 91–103.

242. Runovski, K., & Schmeisser, H.-J. 2001. “Inequalities of Calderon–Zygmund type for trigonometric

243. polynomials”, Georgian J. of Math., vol. 8, no. 1, pp. 165–179.

244. Szeg¨o, G. 1975. “Orthogonal polynomials”, 4th ed., Providence, RI, Amer. Math. Soc.

245. Shapiro, H. 1971. “Topics in approximation theory”, Lecture notes in mathematics, vol. 187,

246. Springer-Verlag, Berlin–Heidelberg.

247. Siegel, C.L. 1935. “ ¨Uber gitterpunkte in convexen k¨orpern and ein damit zusammenh¨angendes

248. extremalproblem”, Acta Math., vol. 65, pp. 307–323.

249. Simonov, I.E. & Glazyrina, P.Y. 2015. “Sharp Markov–Nikolskii inequality with respect to the

250. uniform norm and the integral norm with Chebyshev weight”, J. Approx. Theory, vol. 192,

251. pp. 69–81.

252. Stein, E. & Weiss, G. 1971. “Introduction to Fourier analysis on Euclidean spaces”, Princeton

253. Univ. Press.

254. Storozhenko, ´

255. 𝐸

256. .A., & Krotov, V.G. & Oswald, P. 1975. “Direct and converse theorems of

257. Jackson type in 𝐿𝑝 spaces, 0 < 𝑝 < 1”, Math. USSR-Sb., vol. 27, no. 3, pp. 355–374.

258. Taikov, L.V. 1965. “A group of extremal problems for trigonometric polynomials”, Uspekhi

259. Mat. Nauk, vol. 20, no. 3 (123), pp. 205—211. (In Russ.)

260. Taikov, L.V. 1967. “A generalization of an inequality of S.N. Bernshtein”, Proc. Steklov Inst.

261. Math., vol. 78, pp. 43–48.

262. Taikov, L.V. 1993. “On the best approximation of Dirichlet kernels”, Math. Notes, vol. 53,

263. no. 6, pp. 640–643.

264. Temlyakov, V.N. 1989. “Approximation of functions with bounded mixed derivative”, English

265. transl. in Proc. Steklov Inst. Math., vol. 1, pp. 1–112.

266. Temlyakov, V. & Tikhonov, S. 2017. “Remez-type and Nikol’skii-type inequalities: General

267. relations and the hyperbolic cross polynomials”, Constr. Approx., vol. 46, pp. 593–615.

268. Tikhonov, S. & Yuditskii, P. 2020. “Sharp Remez inequality”, Constr. Approx., vol. 52.

269. Timan, A.F. 1963. “Theory of approximation of functions of a real variable”, Pergamon Press,

270. MacMillan, N.Y.

271. Vaaler, J.D. 1985. “Some extremal functions in Fourier analysis”, Bull. Amer. Math. Soc. (New

272. Series), vol. 12, no. 2, pp. 183–216.

273. Viazovska, M.S. 2017. “The sphere packing problem in dimension 8”, Ann. of Math., vol. 185,

274. no. 3, pp. 991–1015.

275. Vilenkin, N.J. 1978. “Special functions and the theory of group representations”, Translations

276. of mathematical monographs, vol. 22, Providence, RI, Amer. Math. Soc.

277. Vinogradov, O.L. & Gladkaya, A.V. 2015. “Entire functions with the least deviation from zero

278. in the uniform and the integral metrics with a weight”, St. Petersburg Math. J., vol. 26, no. 6,

279. pp. 867–879.

280. Yudin, V.A. 2002. “Positive values of polynomials”, Math. Notes, vol. 72, no. 3, pp. 440–443.

281. Yudin, V.A. 2004. “On positive values of spherical harmonics and trigonometric polynomials”,

282. Math. Notes, vol. 75, no. 3, pp. 447–450.

283. Zastavnyi, V.P. & Manov, A. 2018. “Positive definiteness of complex piecewise linear functions

284. and some of its applications”, Math. Notes, vol. 103, no. 4, pp. 550–564.

285. Ziegler, Z. 1977. “Minimizing the 𝐿𝑝,∞-distortion of trigonometric polynomials”, J. Math. Anal.

286. Appl., vol. 61, no. 2, pp. 426–431