George Lorentz and inequalities in approximation
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- by T. Erdélyi
- St. Petersburg Math. J. 21 (2010), 365-405
- DOI: https://doi.org/10.1090/S1061-0022-10-01099-X
- Published electronically: February 24, 2010
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Abstract:
George Lorentz influenced the author’s research on inequalities in approximation in many ways. This is the connecting thread of this survey paper. The themes of the survey are listed at the very beginning of the paper.References
- Francesco Amoroso, Sur le diamètre transfini entier d’un intervalle réel, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 4, 885–911 (1991) (French, with English summary). MR 1096596
- V. V. Arestov, Integral inequalities for trigonometric polynomials and their derivatives, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 1, 3–22, 239 (Russian). MR 607574
- Joseph Bak and Donald J. Newman, Rational combinations of $x^{\lambda k}$, $\lambda _{k}\geq 0$ are always dense in $C[0, 1]$, J. Approximation Theory 23 (1978), no. 2, 155–157. MR 487180, DOI 10.1016/0021-9045(78)90101-6
- J. Bastero, $l^{q}$-subspaces of stable $p$-Banach spaces, $0<p\leq 1$, Arch. Math. (Basel) 40 (1983), no. 6, 538–544. MR 710019, DOI 10.1007/BF01192821
- József Beck, Flat polynomials on the unit circle—note on a problem of Littlewood, Bull. London Math. Soc. 23 (1991), no. 3, 269–277. MR 1123337, DOI 10.1112/blms/23.3.269
- David Benko and Tamás Erdélyi, Markov inequality for polynomials of degree $n$ with $m$ distinct zeros, J. Approx. Theory 122 (2003), no. 2, 241–248. MR 1988302
- David Benko, Tamás Erdélyi, and József Szabados, The full Markov-Newman inequality for Müntz polynomials on positive intervals, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2385–2391. MR 1974635, DOI 10.1090/S0002-9939-03-06980-6
- S. N. Bernstein, Sur la représentation des polynômes positifs, Soobshcheniya Khar′kov. Mat. Obshch. Ser. 2 14 (1915), 227–228.
- Frank Beaucoup, Peter Borwein, David W. Boyd, and Christopher Pinner, Multiple roots of $[-1,1]$ power series, J. London Math. Soc. (2) 57 (1998), no. 1, 135–147. MR 1624809, DOI 10.1112/S0024610798005857
- A. Bloch and G. Pólya, On the roots of certain algebraic equations, Proc. London Math. Soc. (2) 33 (1932), 102–114.
- Enrico Bombieri and Jeffrey D. Vaaler, Polynomials with low height and prescribed vanishing, Analytic number theory and Diophantine problems (Stillwater, OK, 1984) Progr. Math., vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 53–73. MR 1018369
- Peter Borwein, Markov’s inequality for polynomials with real zeros, Proc. Amer. Math. Soc. 93 (1985), no. 1, 43–47. MR 766524, DOI 10.1090/S0002-9939-1985-0766524-4
- Peter Borwein, Computational excursions in analysis and number theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 10, Springer-Verlag, New York, 2002. MR 1912495, DOI 10.1007/978-0-387-21652-2
- P. Borwein and T. Erdélyi, Markov-Bernstein-type inequalities for classes of polynomials with restricted zeros, Constr. Approx. 10 (1994), no. 3, 411–425. MR 1291056, DOI 10.1007/BF01212567
- Peter Borwein and Tamás Erdélyi, Markov and Bernstein type inequalities in $L_p$ for classes of polynomials with constraints, J. London Math. Soc. (2) 51 (1995), no. 3, 573–588. MR 1332893, DOI 10.1112/jlms/51.3.573
- Peter Borwein and Tamás Erdélyi, Upper bounds for the derivative of exponential sums, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1481–1486. MR 1232137, DOI 10.1090/S0002-9939-1995-1232137-X
- Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
- Peter Borwein and Tamás Erdélyi, Müntz spaces and Remez inequalities, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 38–42. MR 1273395, DOI 10.1090/S0273-0979-1995-00553-7
- Peter Borwein and Tamás Erdélyi, Sharp extensions of Bernstein’s inequality to rational spaces, Mathematika 43 (1996), no. 2, 413–423 (1997). MR 1433285, DOI 10.1112/S0025579300011876
- Peter Borwein and Tamás Erdélyi, The full Müntz theorem in $C[0,1]$ and $L_1[0,1]$, J. London Math. Soc. (2) 54 (1996), no. 1, 102–110. MR 1395070, DOI 10.1112/jlms/54.1.102
- Peter Borwein and Tamás Erdélyi, Newman’s inequality for Müntz polynomials on positive intervals, J. Approx. Theory 85 (1996), no. 2, 132–139. MR 1385812, DOI 10.1006/jath.1996.0034
- Peter Borwein and Tamás Erdélyi, The $L_p$ version of Newman’s inequality for lacunary polynomials, Proc. Amer. Math. Soc. 124 (1996), no. 1, 101–109. MR 1285974, DOI 10.1090/S0002-9939-96-03022-5
- Peter Borwein and Tamás Erdélyi, A sharp Bernstein-type inequality for exponential sums, J. Reine Angew. Math. 476 (1996), 127–141. MR 1401698
- Peter Borwein and Tamás Erdélyi, The integer Chebyshev problem, Math. Comp. 65 (1996), no. 214, 661–681. MR 1333305, DOI 10.1090/S0025-5718-96-00702-8
- Peter Borwein and Tamás Erdélyi, Generalizations of Müntz’s theorem via a Remez-type inequality for Müntz spaces, J. Amer. Math. Soc. 10 (1997), no. 2, 327–349. MR 1415318, DOI 10.1090/S0894-0347-97-00225-7
- Peter Borwein and Tamás Erdélyi, On the zeros of polynomials with restricted coefficients, Illinois J. Math. 41 (1997), no. 4, 667–675. MR 1468873
- Peter Borwein and Tamás Erdélyi, Pointwise Remez- and Nikolskii-type inequalities for exponential sums, Math. Ann. 316 (2000), no. 1, 39–60. MR 1735078, DOI 10.1007/s002080050003
- Peter Borwein and Tamás Erdélyi, Markov- and Bernstein-type inequalities for polynomials with restricted coefficients, Ramanujan J. 1 (1997), no. 3, 309–323. MR 1606930, DOI 10.1023/A:1009761214134
- P. Borwein and T. Erdélyi, Littlewood-type problems on subarcs of the unit circle, Indiana Univ. Math. J. 46 (1997), no. 4, 1323–1346. MR 1631600, DOI 10.1512/iumj.1997.46.1435
- Peter Borwein and Tamás Erdélyi, Markov-Bernstein type inequalities under Littlewood-type coefficient constraints, Indag. Math. (N.S.) 11 (2000), no. 2, 159–172. MR 1813157, DOI 10.1016/S0019-3577(00)89074-6
- Peter Borwein and Tamás Erdélyi, Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes, J. Approx. Theory 125 (2003), no. 2, 190–197. MR 2019608, DOI 10.1016/j.jat.2003.11.004
- Peter Borwein and Tamás Erdélyi, Nikolskii-type inequalities for shift invariant function spaces, Proc. Amer. Math. Soc. 134 (2006), no. 11, 3243–3246. MR 2231907, DOI 10.1090/S0002-9939-06-08533-9
- Peter Borwein and Tamás Erdélyi, Lower bounds for the number of zeros of cosine polynomials in the period: a problem of Littlewood, Acta Arith. 128 (2007), no. 4, 377–384. MR 2320719, DOI 10.4064/aa128-4-5
- Peter Borwein, Tamás Erdélyi, and Géza Kós, Littlewood-type problems on $[0,1]$, Proc. London Math. Soc. (3) 79 (1999), no. 1, 22–46. MR 1687555, DOI 10.1112/S0024611599011831
- Peter Borwein, Tamás Erdélyi, and John Zhang, Müntz systems and orthogonal Müntz-Legendre polynomials, Trans. Amer. Math. Soc. 342 (1994), no. 2, 523–542. MR 1227091, DOI 10.1090/S0002-9947-1994-1227091-4
- Peter Borwein, Tamás Erdélyi, and John Zhang, Chebyshev polynomials and Markov-Bernstein type inequalities for rational spaces, J. London Math. Soc. (2) 50 (1994), no. 3, 501–519. MR 1299454, DOI 10.1112/jlms/50.3.501
- Peter Borwein, Tamás Erdélyi, and Friedrich Littmann, Polynomials with coefficients from a finite set, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5145–5154. MR 2415068, DOI 10.1090/S0002-9947-08-04605-9
- P. Borwein, T. Erdélyi, R. Ferguson, and R. Lockhart, On the zeros of cosine polynomials: solution to a problem of Littlewood, Ann. of Math. (2) 167 (2008), no. 3, 1109–1117. MR 2415396, DOI 10.4007/annals.2008.167.1109
- P. B. Borwein, C. G. Pinner, and I. E. Pritsker, Monic integer Chebyshev problem, Math. Comp. 72 (2003), no. 244, 1901–1916. MR 1986811, DOI 10.1090/S0025-5718-03-01477-7
- David W. Boyd, On a problem of Byrnes concerning polynomials with restricted coefficients, Math. Comp. 66 (1997), no. 220, 1697–1703. MR 1433263, DOI 10.1090/S0025-5718-97-00892-2
- Dietrich Braess, Nonlinear approximation theory, Springer Series in Computational Mathematics, vol. 7, Springer-Verlag, Berlin, 1986. MR 866667, DOI 10.1007/978-3-642-61609-9
- J. A. Clarkson and P. Erdös, Approximation by polynomials, Duke Math. J. 10 (1943), 5–11. MR 7813
- B. Conrey, A. Granville, B. Poonen, and K. Soundararajan, Zeros of Fekete polynomials, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 3, 865–889 (English, with English and French summaries). MR 1779897
- Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635
- Z. Ditzian and D. S. Lubinsky, Jackson and smoothness theorems for Freud weights in $L_p\ (0<p\leq \infty )$, Constr. Approx. 13 (1997), no. 1, 99–152. MR 1424365, DOI 10.1007/s003659900034
- Tamás Erdélyi, The Remez inequality on the size of polynomials, Approximation theory VI, Vol. I (College Station, TX, 1989) Academic Press, Boston, MA, 1989, pp. 243–246. MR 1090997
- T. Erdélyi, Markov-type estimates for certain classes of constrained polynomials, Constr. Approx. 5 (1989), no. 3, 347–356. MR 996935, DOI 10.1007/BF01889614
- Tamás Erdélyi, Estimates for the Lorentz degree of polynomials, J. Approx. Theory 67 (1991), no. 2, 187–198. MR 1133059, DOI 10.1016/0021-9045(91)90017-5
- Tamás Erdélyi, Bernstein-type inequalities for the derivatives of constrained polynomials, Proc. Amer. Math. Soc. 112 (1991), no. 3, 829–838. MR 1036985, DOI 10.1090/S0002-9939-1991-1036985-7
- Tamás Erdélyi, Bernstein and Markov type inequalities for generalized nonnegative polynomials, Canad. J. Math. 43 (1991), no. 3, 495–505. MR 1118006, DOI 10.4153/CJM-1991-030-3
- Tamás Erdélyi, Remez-type inequalities on the size of generalized polynomials, J. London Math. Soc. (2) 45 (1992), no. 2, 255–264. MR 1171553, DOI 10.1112/jlms/s2-45.2.255
- Tamás Erdélyi, Remez-type inequalities and their applications, J. Comput. Appl. Math. 47 (1993), no. 2, 167–209. MR 1237312, DOI 10.1016/0377-0427(93)90003-T
- Tamás Erdélyi, Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients, J. Anal. Math. 74 (1998), 165–181. MR 1631654, DOI 10.1007/BF02819449
- Tamás Erdélyi, Markov-type inequalities for constrained polynomials with complex coefficients, Illinois J. Math. 42 (1998), no. 4, 544–563. MR 1648584
- Tamás Erdélyi, Markov- and Bernstein-type inequalities for Müntz polynomials and exponential sums in $L_p$, J. Approx. Theory 104 (2000), no. 1, 142–152. MR 1753516, DOI 10.1006/jath.1999.3437
- Tamás Erdélyi, The resolution of Saffari’s phase problem, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 10, 803–808 (English, with English and French summaries). MR 1807192, DOI 10.1016/S0764-4442(00)01709-2
- Tamás Erdélyi, On the zeros of polynomials with Littlewood-type coefficient constraints, Michigan Math. J. 49 (2001), no. 1, 97–111. MR 1827077, DOI 10.1307/mmj/1008719037
- Tamás Erdélyi, How far is an ultraflat sequence of unimodular polynomials from being conjugate-reciprocal?, Michigan Math. J. 49 (2001), no. 2, 259–264. MR 1852302, DOI 10.1307/mmj/1008719772
- Tamás Erdélyi, The phase problem of ultraflat unimodular polynomials: the resolution of the conjecture of Saffari, Math. Ann. 321 (2001), no. 4, 905–924. MR 1872534, DOI 10.1007/s002080100259
- Tamás Erdélyi, Proof of Saffari’s near-orthogonality conjecture for ultraflat sequences of unimodular polynomials, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 7, 623–628 (English, with English and French summaries). MR 1868226, DOI 10.1016/S0764-4442(01)02116-4
- Tamás Erdélyi, The “full Clarkson-Erdős-Schwartz theorem” on the closure of non-dense Müntz spaces, Studia Math. 155 (2003), no. 2, 145–152. MR 1961190, DOI 10.4064/sm155-2-4
- Tamás Erdélyi, Extremal properties of the derivatives of the Newman polynomials, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3129–3134. MR 1992853, DOI 10.1090/S0002-9939-03-06986-7
- Tamás Erdélyi, On the real part of ultraflat sequences of unimodular polynomials: consequences implied by the resolution of the phase problem, Math. Ann. 326 (2003), no. 3, 489–498. MR 1992274, DOI 10.1007/s00208-003-0432-y
- Tamás Erdélyi, The “full Müntz theorem” revisited, Constr. Approx. 21 (2005), no. 3, 319–335. MR 2122313, DOI 10.1007/s00365-004-0573-6
- Tamás Erdélyi, Bernstein-type inequalities for linear combinations of shifted Gaussians, Bull. London Math. Soc. 38 (2006), no. 1, 124–138. MR 2201611, DOI 10.1112/S0024609305018035
- Tamás Erdélyi, Markov-Nikolskii type inequalities for exponential sums on finite intervals, Adv. Math. 208 (2007), no. 1, 135–146. MR 2304313, DOI 10.1016/j.aim.2006.02.003
- Tamás Erdélyi, The Remez inequality for linear combinations of shifted Gaussians, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 3, 523–530. MR 2496341, DOI 10.1017/S0305004108001849
- Tamás Erdélyi, Newman’s inequality for increasing exponential sums, Number theory and polynomials, London Math. Soc. Lecture Note Ser., vol. 352, Cambridge Univ. Press, Cambridge, 2008, pp. 127–141. MR 2428519, DOI 10.1017/CBO9780511721274.010
- T. Erdélyi, Inequalities for exponential sums via interpolation and Turán-type reverse Markov inequalities, Frontiers in interpolation and approximation, Pure Appl. Math. (Boca Raton), vol. 282, Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 119–144. MR 2274176
- Tamás Erdélyi, An improvement of the Erdős-Turán theorem on the distribution of zeros of polynomials, C. R. Math. Acad. Sci. Paris 346 (2008), no. 5-6, 267–270 (English, with English and French summaries). MR 2414166, DOI 10.1016/j.crma.2008.01.020
- Tamás Erdélyi, Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials, J. Théor. Nombres Bordeaux 20 (2008), no. 2, 281–287 (English, with English and French summaries). MR 2477504
- Tamás Erdélyi and William B. Johnson, The “full Müntz theorem” in $L_p[0,1]$ for $0<p<\infty$, J. Anal. Math. 84 (2001), 145–172. MR 1849200, DOI 10.1007/BF02788108
- T. Erdélyi and J. Szabados, On polynomials with positive coefficients, J. Approx. Theory 54 (1988), no. 1, 107–122. MR 951032, DOI 10.1016/0021-9045(88)90119-0
- Paul Nevai, Tamás Erdélyi, and Alphonse P. Magnus, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), no. 2, 602–614. MR 1266580, DOI 10.1137/S0036141092236863
- Tamás Erdélyi and Paul Nevai, Lower bounds for derivatives of polynomials and Remez type inequalities, Trans. Amer. Math. Soc. 349 (1997), no. 12, 4953–4972. MR 1407486, DOI 10.1090/S0002-9947-97-01875-8
- Tamás Erdélyi, Attila Máté, and Paul Nevai, Inequalities for generalized nonnegative polynomials, Constr. Approx. 8 (1992), no. 2, 241–255. MR 1152881, DOI 10.1007/BF01238273
- T. Erdélyi and J. Szabados, On a generalization of the Bernstein-Markov inequality, Algebra i Analiz 14 (2002), no. 4, 36–53; English transl., St. Petersburg Math. J. 14 (2003), no. 4, 563–576. MR 1935916
- P. Erdös, On extremal properties of the derivatives of polynomials, Ann. of Math. (2) 41 (1940), 310–313. MR 1945, DOI 10.2307/1969005
- Paul Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), 291–300. MR 98702
- P. Erdös and P. Turán, On the distribution of roots of polynomials, Ann. of Math. (2) 51 (1950), 105–119. MR 33372, DOI 10.2307/1969500
- J. Erőd, Bizonyos polinomok maximumának alsó korlátjáról, Matematikai és Fizikai Lapok 46 (1939), 83–84.
- C. Frappier, Quelques problèmes extrémaux pour les polynômes et les fonctions entières de type exponentiel, Ph. D. Dissertation, Univ. Montréal, 1982.
- G. Freud, Orthogonal polynomials, Pergamon Press, Oxford, 1971.
- C. Sinan Güntürk, Approximation by power series with $\pm 1$ coefficients, Int. Math. Res. Not. 26 (2005), 1601–1610. MR 2148266, DOI 10.1155/IMRN.2005.1601
- G. Halász, Markov-type inequalities for polynomials with restricted zeros, J. Approx. Theory 101 (1999), no. 1, 148–155. MR 1724030, DOI 10.1006/jath.1999.3363
- K. G. Hare and C. J. Smyth, The monic integer transfinite diameter, Math. Comp. 75 (2006), no. 256, 1997–2019. MR 2240646, DOI 10.1090/S0025-5718-06-01843-6
- Felix Hausdorff, Summationsmethoden und Momentfolgen. I, Math. Z. 9 (1921), no. 1-2, 74–109 (German). MR 1544453, DOI 10.1007/BF01378337
- Loo Keng Hua, Introduction to number theory, Springer-Verlag, Berlin-New York, 1982. Translated from the Chinese by Peter Shiu. MR 665428
- M. Kac, On the average number of real roots of a random algebraic equation. II, Proc. London Math. Soc. (2) 50 (1949), 390–408. MR 30713, DOI 10.1112/plms/s2-50.5.390
- Jean-Pierre Kahane, Sur les polynômes à coefficients unimodulaires, Bull. London Math. Soc. 12 (1980), no. 5, 321–342 (French). MR 587702, DOI 10.1112/blms/12.5.321
- T. W. Körner, On a polynomial of Byrnes, Bull. London Math. Soc. 12 (1980), no. 3, 219–224. MR 572106, DOI 10.1112/blms/12.3.219
- Sergei Konyagin, On a question of Pichorides, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 4, 385–388 (English, with English and French summaries). MR 1440953, DOI 10.1016/S0764-4442(97)80072-9
- S. V. Konyagin, On the Littlewood problem, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 2, 243–265, 463 (Russian). MR 616222
- Sergei V. Konyagin and Vsevolod F. Lev, Character sums in complex half-planes, J. Théor. Nombres Bordeaux 16 (2004), no. 3, 587–606 (English, with English and French summaries). MR 2144960
- Norman Levenberg and Evgeny A. Poletsky, Reverse Markov inequality, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 1, 173–182. MR 1884358
- J. E. Littlewood, On the mean values of certain trigonometric polynomials, J. London Math. Soc. 36 (1961), 307–334. MR 141934, DOI 10.1112/jlms/s1-36.1.307
- J. E. Littlewood, On the real roots of real trigonometrical polynomials. II, J. London Math. Soc. 39 (1964), 511–532. MR 171117, DOI 10.1112/jlms/s1-39.1.511
- J. E. Littlewood, On polynomials $\sum ^{n}\pm z^{m}$, $\sum ^{n}e^{\alpha _{m}i}z^{m}$, $z=e^{\theta _{i}}$, J. London Math. Soc. 41 (1966), 367–376. MR 196043, DOI 10.1112/jlms/s1-41.1.367
- John E. Littlewood, Some problems in real and complex analysis, D. C. Heath and Company Raytheon Education Company, Lexington, Mass., 1968. MR 0244463
- J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. II, Proc. Cambridge Philos. Soc. 35 (1939), 133–148.
- A. Kroó and J. Szabados, Constructive properties of self-reciprocal polynomials, Analysis 14 (1994), no. 4, 319–339. MR 1310618, DOI 10.1524/anly.1994.14.4.319
- G. G. Lorentz, The degree of approximation by polynomials with positive coefficients, Math. Ann. 151 (1963), 239–251. MR 155135, DOI 10.1007/BF01398235
- G. G. Lorentz, Notes on approximation, J. Approx. Theory 56 (1989), no. 3, 360–365. MR 990350, DOI 10.1016/0021-9045(89)90125-1
- G. G. Lorentz, Approximation of functions, 2nd ed., Chelsea Publishing Co., New York, 1986. MR 917270
- George G. Lorentz, Manfred v. Golitschek, and Yuly Makovoz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 304, Springer-Verlag, Berlin, 1996. Advanced problems. MR 1393437, DOI 10.1007/978-3-642-60932-9
- G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1298187, DOI 10.1142/1284
- O. Carruth McGehee, Louis Pigno, and Brent Smith, Hardy’s inequality and the $L^{1}$ norm of exponential sums, Ann. of Math. (2) 113 (1981), no. 3, 613–618. MR 621019, DOI 10.2307/2007000
- Idris David Mercer, Unimodular roots of special Littlewood polynomials, Canad. Math. Bull. 49 (2006), no. 3, 438–447. MR 2252265, DOI 10.4153/CMB-2006-043-x
- H. N. Mhaskar, Introduction to the theory of weighted polynomial approximation, Series in Approximations and Decompositions, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1469222
- —, When is approximation by Gaussian networks necessarily a linear process? Neural Networks 17 (2004), 989–1001.
- Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543, DOI 10.1090/cbms/084
- C. Müntz, Über den Approximationsatz von Weierstrass, H. A. Schwartz Festschrift, Berlin, 1914.
- F. L. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Algebra i Analiz 5 (1993), no. 4, 3–66 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 4, 663–717. MR 1246419
- D. J. Newman, Derivative bounds for Müntz polynomials, J. Approximation Theory 18 (1976), no. 4, 360–362. MR 430604, DOI 10.1016/0021-9045(76)90007-1
- Donald J. Newman, Approximation with rational functions, CBMS Regional Conference Series in Mathematics, vol. 41, Conference Board of the Mathematical Sciences, Washington, D.C., 1979. Expository lectures from the CBMS Regional Conference held at the University of Rhode Island, Providence, R.I., June 12–16, 1978. MR 539314
- S. M. Nikol′skiĭ, Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables, Trudy Mat. Inst. Steklov. 38 (1951), 244–278 (Russian). MR 0048565
- A. M. Odlyzko and B. Poonen, Zeros of polynomials with $0,1$ coefficients, Enseign. Math. (2) 39 (1993), no. 3-4, 317–348. MR 1252071
- P. P. Petrushev and V. A. Popov, Rational approximation of real functions, Encyclopedia of Mathematics and its Applications, vol. 28, Cambridge University Press, Cambridge, 1987. MR 940242
- S. K. Pichorides, Notes on trigonometric polynomials, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 84–94. MR 730060
- Qazi Ibadur Rahman and Gerhard Schmeisser, Les inégalités de Markoff et de Bernstein, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 86, Presses de l’Université de Montréal, Montreal, QC, 1983 (French). MR 729316
- Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, London Mathematical Society Monographs. New Series, vol. 26, The Clarendon Press, Oxford University Press, Oxford, 2002. MR 1954841
- E. J. Remez, Sur une propriété extrémale des polynômes de Tchebyscheff, Zap. Nauchn.-Issled. Inst. Mat. i Mekh. Khar′kov. Univ. i Khar′kov. Mat. Obshch. (4) 13 (1936), vyp. 1, 93–95.
- Szilárd Gy. Révész, Turán type reverse Markov inequalities for compact convex sets, J. Approx. Theory 141 (2006), no. 2, 162–173. MR 2252096, DOI 10.1016/j.jat.2006.03.002
- Hervé Queffélec and Bahman Saffari, Unimodular polynomials and Bernstein’s inequalities, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 3, 313–318 (English, with English and French summaries). MR 1346133
- Hervé Queffelec and Bahman Saffari, On Bernstein’s inequality and Kahane’s ultraflat polynomials, J. Fourier Anal. Appl. 2 (1996), no. 6, 519–582. MR 1423528, DOI 10.1007/s00041-001-4043-2
- B. Saffari, The phase behaviour of ultraflat unimodular polynomials, Probabilistic and stochastic methods in analysis, with applications (Il Ciocco, 1991) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 372, Kluwer Acad. Publ., Dordrecht, 1992, pp. 555–572. MR 1187327, DOI 10.1007/978-94-011-2791-2_{2}6
- John T. Scheick, Inequalities for derivatives of polynomials of special type, J. Approximation Theory 6 (1972), 354–358. MR 342909, DOI 10.1016/0021-9045(72)90041-x
- Eckard Schmidt, Zur Kompaktheit bei Exponentialsummen, J. Approximation Theory 3 (1970), 445–454 (German). MR 271588, DOI 10.1016/0021-9045(70)90045-6
- I. Schur, Untersuchungen über algebraische Gleichungen, Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. 1933, 403–428.
- T. Sheil-Small, Complex polynomials, Cambridge Studies in Advanced Mathematics, vol. 75, Cambridge University Press, Cambridge, 2002. MR 1962935, DOI 10.1017/CBO9780511543074
- G. Somorjai, A Müntz-type problem for rational approximation, Acta Math. Acad. Sci. Hungar. 27 (1976), no. 1-2, 197–199. MR 430617, DOI 10.1007/BF01896775
- Boris Solomyak, On the random series $\sum \pm \lambda ^n$ (an Erdős problem), Ann. of Math. (2) 142 (1995), no. 3, 611–625. MR 1356783, DOI 10.2307/2118556
- G. Szegő, Bemerkungen zu einem Satz von E. Schmidt uber algebraische Gleichungen, Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. 1934, 86–98.
- G. Szegő and A. Zygmund, On certain mean values of polynomials, J. Analyse Math. 3 (1954), 225–244. MR 64910, DOI 10.1007/BF02803592
- Vilmos Totik and Péter P. Varjú, Polynomials with prescribed zeros and small norm, Acta Sci. Math. (Szeged) 73 (2007), no. 3-4, 593–611. MR 2380067
- P. Turan, Über die Ableitung von Polynomen, Compositio Math. 7 (1939), 89–95 (German). MR 228
- Paul Turán, On a new method of analysis and its applications, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. With the assistance of G. Halász and J. Pintz; With a foreword by Vera T. Sós; A Wiley-Interscience Publication. MR 749389
- Qiang Wu, A new exceptional polynomial for the integer transfinite diameter of $[0,1]$, J. Théor. Nombres Bordeaux 15 (2003), no. 3, 847–861 (English, with English and French summaries). MR 2142240
Bibliographic Information
- T. Erdélyi
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- Email: terdelyi@math.tamu.edu
- Received by editor(s): April 14, 2008
- Published electronically: February 24, 2010
- © Copyright 2010 American Mathematical Society
- Journal: St. Petersburg Math. J. 21 (2010), 365-405
- MSC (2000): Primary 41-02
- DOI: https://doi.org/10.1090/S1061-0022-10-01099-X
- MathSciNet review: 2588761