SzegTaikov inequality for conjugate polynomials Polina Glazyrina - - PowerPoint PPT Presentation

Szegö–Taikov inequality for conjugate polynomials

Polina Glazyrina Ural Federal University Yekaterinburg, Russia Sixth Workshop on Fourier Analysis and Related Fields Hungary, 2017

Problem

Fn : fn(t) = a0+

n

• k=1

(ak cos kt + bk sin kt), ak, bk ∈ C the conjugate to fn:

• fn(t) =

n

• k=1

(ak sin kt − bk cos kt) We study the operator Λθ,Afn(t) = Aa0 +

n

• k=1

(ak cos(kt + θ) + bk sin(kt + θ)) = Aa0 + cos θ(fn(t) − a0) − sin θ fn(t), θ ∈ R, A ∈ C

• Λθ,Afn
• ∞ Cn(θ, A)fn∞,

fn ∈ Fn.

Some values of Λθ,A

Λθ,Afn(t) = Aa0 +

n

• k=1

(ak cos(kt + θ) + bk sin(kt + θ)) = Aa0 + cos θ(fn(t) − a0) − sin θ fn(t) Λ0,1fn = fn, Λ0,0fn = fn − a0, Λ3π/2,0fn = fn Λθ,cos θfn(t) = D0 cos θ fn(t) − sin θ fn(t)

• Weyl derivative

Dαfn(t) =

n

• k=1

kα ak cos(kt + απ/2) + bk sin(kt + απ/2)

• .

Known estimates for Cn(θ, A)

Λθ,Afn(t) = Aa0 + cos θ(fn(t) − a0) − sin θ fn(t)

• Λθ,Afn
• ∞ Cn(θ, A)fn∞,

fn ∈ Fn. Cn(θ, A) = 2 π| sin θ| ln n + O(1) as n → ∞. C2m−1(3π/2, 0) is Lebesgue constant for Lagrange interpolation polynomial based on the zeros of the Chebyshev polynomial of the first kind of degree m. At present, two values of Cn(θ, A) are known: Fejer, 1913: fn − a0∞ Cn(0, 0)fn∞, Szegö, 1943:

• fn∞ Cn(3π/2, 0)fn∞.

The result of Fejer for C(0, 0)

In 1913, Fejér proved that for any fn 0, a0 > 0 fn a0(n + 1). He also established the relation M nm, M is the maximum, −m is the minimum of a real polynomial fn with a0 = 0. These results imply that fn − a0∞ 2n n + 1fn∞, the extremal polynomial is Kn(t) − Kn∞/2, Kn is the Fejér kernel Kn(t) = 1 2 +

n

• k=1
• 1 −

k n + 1

• cos kt =

1 2(n + 1) sin(n + 1)t/2 sin t/2 2 .

The Fejér kernel

Kn(0) = n + 1 2 Kn 2πℓ n + 1

• = 0,

± ℓ = 1, . . . , n + 1 2

The result of Szegö for C(3π/2, 0)

In 1943, Szegö proved that

• fn∞

2 n + 1 ⌊ n−1

2 ⌋

• ℓ=0

cot π + 2πℓ 2(n + 1)

• fn∞.

Szegö found all extremal polynomials with real coefficients. More precisely, for odd n, an extremal polynomial is unique (up to a non-zero constant factor and a shift of argument) and it is f ∗

n (t) =

⌊ n−1

2 ⌋

• ℓ=0

Kn

• t − π + 2πℓ

n + 1

• − Kn
• t + π + 2πℓ

n + 1

• .

For even n, extremal polynomials are given by the relation f ∗∗

n (t) = f ∗ n (t) + γKn(t − π),

γ ∈ R, |γ| 1.

The extremal polynomial for n = 4, γ = 0.7

Interpolation formula

For any θ ∈ R, A ∈ C, and fn ∈ Fn Λθ,Afn(t) = 1 n + 1

n

• ℓ=0

(qℓ + A) · fn (t + tℓ) , where tℓ = 2θ + 2πℓ n + 1 , ℓ = 0, . . . , n, qℓ = −sin(tℓ/2 − θ) sin(tℓ/2) = − cos θ + sin θ cot tℓ/2, θ = 0 mod π, q0 = n cos θ, qℓ = − cos θ, ℓ = 1, . . . , n, θ = 0 mod π. If θ = 0 mod π or A = cos θ, then at least n of the coefficients qℓ + A do not vanish. In particular, qℓ = 0 if and only if 1 ℓ n − 1 and θ = πℓ n or 0 ℓ n − 2 and ℓ = θn π − n − 1.

Theorem

Let θ ∈ [0, 2π) and A ∈ C, then

• Λθ,Afn
• ∞

1 n + 1

n

• ℓ=0

|qℓ + A|fn∞, fn ∈ Fn. (1) If θ = 0 mod π or A = cos θ, then inequality (1) turns into an equality only for polynomials f ∗

n (t) = c n

• ℓ=0

sℓKn(t − t∗ − tℓ), t∗ ∈ R, c ∈ C, where sℓ = sign(qℓ + A) if qℓ + A = 0 and sℓ is an arbitrary complex number such that |sℓ| 1 if qℓ + A = 0.

Thank you for your attention!

References

Fejér, L.: Sur les polynomes harmoniques quelconques, C. R. 157, 506–509 (1913). Fejér, L.: Sur les polynomes trigonométriques, C. R. 157, 571–574 (1913). Günttner, R.: On the norms of conjugate trigonometric polynomials, Acta Math. Hungar. 66 (4), 269–273 (1995). Jiang, T.: Asymptotic expansion of norm associated with conjugate trigonometric polynomial, Periodica Mathematica Hungarica 27 (2), 89–93 (1993). Kozko, A.I.: The exact constants in the Bernstein–Zygmund–Szegö inequalities with fractional derivatives and the Jackson–Nikolskii inequality for trigonometric polynomials, East J. Approx. 4 (3), 391–416 (1998). Szegö, G.: On conjugate trigonometric polynomials, American J.

• Math. 65 (4), 532–536 (1943).