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 n � F n : f n ( t ) = a 0 + ( a k cos kt + b k sin kt ) , a k , b k ∈ C k = 1 n � � the conjugate to f n : f n ( t ) = ( a k sin kt − b k cos kt ) k = 1 We study the operator n � Λ θ, A f n ( t ) = Aa 0 + ( a k cos ( kt + θ ) + b k sin ( kt + θ )) k = 1 = Aa 0 + cos θ ( f n ( t ) − a 0 ) − sin θ � f n ( t ) , θ ∈ R , A ∈ C � � � Λ θ, A f n � ∞ � C n ( θ, A ) � f n � ∞ , f n ∈ F n .

Some values of Λ θ, A n � Λ θ, A f n ( t ) = Aa 0 + ( a k cos ( kt + θ ) + b k sin ( kt + θ )) k = 1 = Aa 0 + cos θ ( f n ( t ) − a 0 ) − sin θ � f n ( t ) Λ 3 π/ 2 , 0 f n = � Λ 0 , 1 f n = f n , Λ 0 , 0 f n = f n − a 0 , f n Λ θ, cos θ f n ( t ) = D 0 � � cos θ f n ( t ) − sin θ � f n ( t ) Weyl derivative n � k α � � D α f n ( t ) = a k cos ( kt + απ/ 2 ) + b k sin ( kt + απ/ 2 ) . k = 1

Known estimates for C n ( θ, A ) Λ θ, A f n ( t ) = Aa 0 + cos θ ( f n ( t ) − a 0 ) − sin θ � f n ( t ) � � � Λ θ, A f n � ∞ � C n ( θ, A ) � f n � ∞ , f n ∈ F n . C n ( θ, A ) = 2 π | sin θ | ln n + O ( 1 ) as n → ∞ . C 2 m − 1 ( 3 π/ 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 C n ( θ, A ) are known: � f n − a 0 � ∞ � C n ( 0 , 0 ) � f n � ∞ , Fejer, 1913: � � Szegö, 1943: f n � ∞ � C n ( 3 π/ 2 , 0 ) � f n � ∞ .

The result of Fejer for C ( 0 , 0 ) In 1913, Fejér proved that for any f n � 0 , a 0 > 0 f n � a 0 ( n + 1 ) . He also established the relation M � nm , M is the maximum, − m is the minimum of a real polynomial f n with a 0 = 0. These results imply that 2 n � f n − a 0 � ∞ � n + 1 � f n � ∞ , the extremal polynomial is K n ( t ) − � K n � ∞ / 2 , K n is the Fejér kernel � � � sin ( n + 1 ) t / 2 � 2 n � K n ( t ) = 1 k 1 2 + 1 − cos kt = . n + 1 2 ( n + 1 ) sin t / 2 k = 1

The Fejér kernel K n ( 0 ) = n + 1 2 � 2 πℓ � K n = 0 , n + 1 � n + 1 � ± ℓ = 1 , . . . , 2

The result of Szegö for C ( 3 π/ 2 , 0 ) In 1943, Szegö proved that ⌊ n − 1 2 ⌋ � π + 2 πℓ � � 2 � � f n � ∞ � cot � f n � ∞ . n + 1 2 ( n + 1 ) ℓ = 0 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 ⌊ n − 1 2 ⌋ � � � � � t − π + 2 πℓ t + π + 2 πℓ f ∗ n ( t ) = K n − K n . n + 1 n + 1 ℓ = 0 For even n , extremal polynomials are given by the relation f ∗∗ n ( t ) = f ∗ n ( t ) + γ K n ( t − π ) , γ ∈ R , | γ | � 1 .

The extremal polynomial for n = 4, γ = 0 . 7

Interpolation formula For any θ ∈ R , A ∈ C , and f n ∈ F n n � 1 Λ θ, A f n ( t ) = ( q ℓ + A ) · f n ( t + t ℓ ) , n + 1 ℓ = 0 where t ℓ = 2 θ + 2 πℓ , ℓ = 0 , . . . , n , n + 1 q ℓ = − sin ( t ℓ / 2 − θ ) = − cos θ + sin θ cot t ℓ / 2 , θ � = 0 mod π, sin ( t ℓ / 2 ) q 0 = n cos θ, q ℓ = − 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 q ℓ + A do not vanish. In particular, q ℓ = 0 if and only if θ = πℓ 1 � ℓ � n − 1 and n or ℓ = θ n 0 � ℓ � n − 2 and π − n − 1 .

Theorem Let θ ∈ [ 0 , 2 π ) and A ∈ C , then n � � � 1 � Λ θ, A f n � | q ℓ + A |� f n � ∞ , f n ∈ F n . (1) ∞ � n + 1 ℓ = 0 If θ � = 0 mod π or A � = cos θ, then inequality (1) turns into an equality only for polynomials n � s ℓ K n ( t − t ∗ − t ℓ ) , t ∗ ∈ R , c ∈ C , f ∗ n ( t ) = c ℓ = 0 where s ℓ = sign ( q ℓ + A ) if q ℓ + A � = 0 and s ℓ is an arbitrary complex number such that | s ℓ | � 1 if q ℓ + A = 0.

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