An extremal problem for polynomials

Authors:
E. Beller and D. J. Newman

Journal:
Proc. Amer. Math. Soc. **29** (1971), 474-481

MSC:
Primary 30.10; Secondary 42.00

MathSciNet review:
0280688

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the maximum of the norm, , of all *n*th degree polynomials satisfying for . We prove that is asymptotic to , by exhibiting polynomials (which are partial sums of certain Fourier series), whose norm is asymptotic to .

**[1]**E. Beller,*Polynomial extremal problems in 𝐿^{𝑝}*, Proc. Amer. Math. Soc.**30**(1971), 249–259. MR**0281884**, 10.1090/S0002-9939-1971-0281884-9**[2]**Paul Erdős,*Some unsolved problems*, Michigan Math. J.**4**(1957), 291–300. MR**0098702****[3]**D. J. Newman,*An 𝐿¹ extremal problem for polynomials*, Proc. Amer. Math. Soc.**16**(1965), 1287–1290. MR**0185119**, 10.1090/S0002-9939-1965-0185119-4**[4]**Walter Rudin,*Some theorems on Fourier coefficients*, Proc. Amer. Math. Soc.**10**(1959), 855–859. MR**0116184**, 10.1090/S0002-9939-1959-0116184-5**[5]**H. S. Shapiro, Thesis for S.M. degree, M.I.T., Cambridge, Mass., 1957.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
30.10,
42.00

Retrieve articles in all journals with MSC: 30.10, 42.00

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0280688-0

Keywords:
Close to constant polynomials,
coefficients of close to constant modulus,
extremal polynomials,
norm of polynomials,
partial sums of Fourier series,
upper bound for *k*th derivative

Article copyright:
© Copyright 1971
American Mathematical Society