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Let ζ be a fixed complex number. In this paper, we study the
quantity S(ζ, n) := max_(f∈An)
│f(ζ)│; where Λ_n is the set of all real polynomials
of degree at most n-1 with coefficients in the interval [0, 1]: We
first show how, in principle, for any given ζ∈ C and n ∈ N, the quantity
S(ζ, n) can be calculated. Then we compute the limit lim_(n→∞) S(ζ, n)/n
for every ζ ∈ C of modulus 1. It is equal to 1/π if ζ is not a root of unity.
If ζ = exp(2πik=d), where d ∈ N and k ∈ [1; d-1] is an integer satisfying
gcd(k, d) = 1, then the answer depends on the parity of d: More precisely,
the limit is 1, 1=(d sin(π/d)) and 1=(2d sin(π/2d)) for d = 1, d even and
d > 1 odd, respectively.
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참고문헌 (9)
참고문헌 신청
A. M. Odlyzko / 1993 / Zeros of polynomials with 0; 1 coefficients / Enseign. Math 39 (3) : 317 ~ 348
C. J. Smyth / 1985 / Some results on Newman polynomials / Indiana Univ. Math. J. 34 (1) : 195 ~ 200
D. J. Newman / 1965 / An L1 extremal problem for polynomials / Proc. Amer. Math. Soc. 16 : 1287 ~ 1290
D. M. Campbell / 1983 / Newman polynomials on jzj = 1 / Indiana Univ. Math. J. 32 (4) : 517 ~ 525
D. W. Boyd / 1986 / Large Newman polynomials, Diophantine analysis / London Math. Soc. Lecture Note Ser. 109 : 159 ~ 170
C. J. Smyth / 1985 / Some results on Newman polynomials / Indiana Univ. Math. J. 34 (1) : 195 ~ 200
D. J. Newman / 1965 / An L1 extremal problem for polynomials / Proc. Amer. Math. Soc. 16 : 1287 ~ 1290
D. M. Campbell / 1983 / Newman polynomials on jzj = 1 / Indiana Univ. Math. J. 32 (4) : 517 ~ 525
D. W. Boyd / 1986 / Large Newman polynomials, Diophantine analysis / London Math. Soc. Lecture Note Ser. 109 : 159 ~ 170
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