## The expected $L_{p}$ norm of random polynomials

HTML articles powered by AMS MathViewer

- by Peter Borwein and Richard Lockhart PDF
- Proc. Amer. Math. Soc.
**129**(2001), 1463-1472

## Abstract:

The results of this paper concern the expected $L_{p}$ norm of random polynomials on the boundary of the unit disc (equivalently of random trigonometric polynomials on the interval $[0, 2\pi ]$). Specifically, for a random polynomial \[ q_{n}(\theta ) = \sum _{0}^{n-1}X_{k}e^{ik\theta }\] let \[ ||q_{n}||_{p}^{p}= \int _{0}^{2\pi } |q_{n}(\theta )|^{p} d\theta /(2\pi ). \] Assume the random variables $X_{k},k\ge 0$, are independent and identically distributed, have mean 0, variance equal to 1 and, if $p>2$, a finite $p^{th}$ moment ${\mathrm E}(|X_{k}|^{p})$. Then \[ \frac {\text { E}(||q_{n}||_{p}^{p})}{n^{p/2}} \to \Gamma (1+p/2) \] and \[ \frac {\text {E}(||q_{n}^{(r)}||_{p}^{p})}{n^{(2r+ 1)p/2}} \to (2r+1)^{-p/2}\Gamma (1+p/2) \] as $n\to \infty$. In particular if the polynomials in question have coefficients in the set $\{+1,-1\}$ (a much studied class of polynomials), then we can compute the expected $L_{p}$ norms of the polynomials and their derivatives \[ \frac {\text { E}(||q_{n}||_{p})}{n^{1/2}} \to (\Gamma (1+p/2))^{1/p} \] and \[ \frac {\text { E}(||q_{n}^{(r)}||_{p})}{n^{(2r+1)/2}} \to (2r+1)^{-1/2}(\Gamma (1+p/2))^{1/p}. \] This complements results of Fielding in the $p:=0$ case, Newman and Byrnes in the $p:=4$ case, and Littlewood et al. in the $p=\infty$ case.## References

- Hong Zhi An, Zhao Guo Chen, and E. J. Hannan,
*The maximum of the periodogram*, J. Multivariate Anal.**13**(1983), no.Â 3, 383â400. MR**716931**, DOI 10.1016/0047-259X(83)90017-9 - JĂłzsef Beck,
*Flat polynomials on the unit circleânote on a problem of Littlewood*, Bull. London Math. Soc.**23**(1991), no.Â 3, 269â277. MR**1123337**, DOI 10.1112/blms/23.3.269 - S. Bernstein,
*Quelques remarques sur le thĂ©orĂšme limite Liapounoff*, Doklad. Akad. Nauk. SSSR**24**(1939), 3â8. - A. T. Bharucha-Reid and M. Sambandham,
*Random polynomials*, Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, 1986. MR**856019** - P. Borwein,
*Some Old Problems on Polynomials with Integer Coefficients*, in Approximation Theory IX, ed. C. Chui and L. Schumacher, Vanderbilt Univ. Press (1998), 51â58. - Peter Borwein and TamĂĄs ErdĂ©lyi,
*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 - David W. Boyd,
*On a problem of Byrnes concerning polynomials with restricted coefficients*, Math. Comp.**66**(1997), no.Â 220, 1697â1703. MR**1433263**, DOI 10.1090/S0025-5718-97-00892-2 - D.R. Brillinger,
*Time Series Data Analysis and Theory*, Holt, Rinehart and Winston, New York, 1975. - B. M. Brown,
*Characteristic functions, moments, and the central limit theorem*, Ann. Math. Statist.**41**(1970), 658â664. MR**261672**, DOI 10.1214/aoms/1177697109 - F. W. Carroll, Dan Eustice, and T. Figiel,
*The minimum modulus of polynomials with coefficients of modulus one*, J. London Math. Soc. (2)**16**(1977), no.Â 1, 76â82. MR**480955**, DOI 10.1112/jlms/s2-16.1.76 - J. Clunie,
*On the minimum modulus of a polynomial on the unit circle*, Quart. J. Math.**10**(1959), 95â98. - P. ErdĆs,
*An inequality for the maximum of trigonometric polynomials*, Ann. Polon. Math.**12**(1962), 151â154. MR**141933**, DOI 10.4064/ap-12-2-151-154 - G. T. Fielding,
*The expected value of the integral around the unit circle of a certain class of polynomials*, Bull. London Math. Soc.**2**(1970), 301â306. MR**280689**, DOI 10.1112/blms/2.3.301 - G. HalĂĄsz,
*On a result of Salem and Zygmund concerning random polynomials*, Studia Sci. Math. Hungar.**8**(1973), 369â377. MR**367545** - Morgan Ward and R. P. Dilworth,
*The lattice theory of ova*, Ann. of Math. (2)**40**(1939), 600â608. MR**11**, DOI 10.2307/1968944 - Jean-Pierre Kahane,
*Sur les polynĂŽmes Ă coefficients unimodulaires*, Bull. London Math. Soc.**12**(1980), no.Â 5, 321â342 (French). MR**587702**, DOI 10.1112/blms/12.5.321 - Jean-Pierre Kahane,
*Some random series of functions*, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR**833073** - S. Konjagin,
*On a problem of Littlewood*, Izv. A. N. SSSR, ser. mat.**45, 2**(1981), 243â265. - T. W. KĂ¶rner,
*On a polynomial of Byrnes*, Bull. London Math. Soc.**12**(1980), no.Â 3, 219â224. MR**572106**, DOI 10.1112/blms/12.3.219 - J. E. Littlewood,
*On the mean values of certain trigonometric polynomials*, J. London Math. Soc.**36**(1961), 307â334. MR**141934**, DOI 10.1112/jlms/s1-36.1.307 - 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), 367â376. 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** - Kurt Mahler,
*On two extremum properties of polynomials*, Illinois J. Math.**7**(1963), 681â701. MR**156950** - Donald J. Newman and J. S. Byrnes,
*The $L^4$ norm of a polynomial with coefficients $\pm 1$*, Amer. Math. Monthly**97**(1990), no.Â 1, 42â45. MR**1034349**, DOI 10.2307/2324003 - A. M. Odlyzko and B. Poonen,
*Zeros of polynomials with $0,1$ coefficients*, Enseign. Math. (2)**39**(1993), no.Â 3-4, 317â348. MR**1252071** - HervĂ© Queffelec and Bahman Saffari,
*On Bernsteinâs inequality and Kahaneâs ultraflat polynomials*, J. Fourier Anal. Appl.**2**(1996), no.Â 6, 519â582. MR**1423528**, DOI 10.1007/s00041-001-4043-2 - L. Robinson,
*Polynomials With ${\pm 1}$ Coefficients: Growth Properties on the Unit Circle*, Simon Fraser University, Masters Thesis (1997). - B. Saffari,
*Barker sequences and Littlewoodâs âtwo sided conjecturesâ on polynomials with $\pm 1$ coefficients*, SĂ©minaire dâAnalyse Harmonique, 1989/90, Univ. Paris XI, Orsay, 1990, 139â151. - Tadasi Nakayama,
*On Frobeniusean algebras. I*, Ann. of Math. (2)**40**(1939), 611â633. MR**16**, DOI 10.2307/1968946 - David C. Ullrich,
*An extension of the Kahane-Khinchine inequality*, Bull. Amer. Math. Soc. (N.S.)**18**(1988), no.Â 1, 52â54. MR**919660**, DOI 10.1090/S0273-0979-1988-15596-6

## Additional Information

**Peter Borwein**- Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
- Email: pborwein@cecm.sfu.ca
**Richard Lockhart**- Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
- Email: lockhart@sfu.ca
- Received by editor(s): December 18, 1998
- Received by editor(s) in revised form: August 31, 1999
- Published electronically: October 25, 2000
- Additional Notes: Research supported in part by the NSERC of Canada.
- Communicated by: Albert Baernstein II
- © Copyright 2000 by the authors
- Journal: Proc. Amer. Math. Soc.
**129**(2001), 1463-1472 - MSC (1991): Primary 26D05
- DOI: https://doi.org/10.1090/S0002-9939-00-05690-2
- MathSciNet review: 1814174