The Dirichlet-Bohr radius Manuel Maestre April 13, 2014 Kent State - PowerPoint PPT Presentation

The Dirichlet-Bohr radius Manuel Maestre April 13, 2014 – Kent State University

Content Dirichlet series Manuel Maestre The Dirichlet-Bohr radius

Content Dirichlet series Dirichlet series and complex analysis on polydiscs Manuel Maestre The Dirichlet-Bohr radius

Content Dirichlet series Dirichlet series and complex analysis on polydiscs The Dirichlet-Bohr radius Manuel Maestre The Dirichlet-Bohr radius

Content Dirichlet series Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Dirichlet series 1 D = � n a n n s with coefficients a n ∈ C and variable s ∈ C Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Dirichlet series 1 D = � n a n n s with coefficients a n ∈ C and variable s ∈ C Convergence of Dirichlet series ✻ ✲ Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Dirichlet series 1 D = � n a n n s with coefficients a n ∈ C and variable s ∈ C Convergence of Dirichlet series ✻ ✲ conv. ✲ σ c Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Dirichlet series 1 D = � n a n n s with coefficients a n ∈ C and variable s ∈ C Convergence of Dirichlet series ✻ ✲ conv. ✲ abs. conv. ✲ σ c σ a Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Dirichlet series 1 D = � n a n n s with coefficients a n ∈ C and variable s ∈ C Convergence of Dirichlet series ✻ ✲ conv. ✲ unif. conv. ✲ abs. conv. ✲ σ c σ u σ a Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Dirichlet series 1 D = � n a n n s with coefficients a n ∈ C and variable s ∈ C Convergence of Dirichlet series ✻ holom. ✲ σ c σ u σ a Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Dirichlet series 1 D = � n a n n s with coefficients a n ∈ C and variable s ∈ C Convergence of Dirichlet series ✻ holom. ✲ σ c σ b σ u σ a Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Dirichlet series 1 D = � n a n n s with coefficients a n ∈ C and variable s ∈ C Convergence of Dirichlet series ✻ holom. & bdd. ✲ σ c σ b σ u σ a Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Bohr’s fundamental theorem σ u ( D ) = σ b ( D ) Convergence of Dirichlet series ✻ holom. & bdd. ✲ σ c σ u σ a � σ b Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Convergence of Dirichlet series ✻ ✲ unif. conv. ✲ abs. conv. ✲ σ u σ a Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Convergence of Dirichlet series ✻ ✲ unif. conv. ✲ abs. conv. ✲ σ u σ a Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Convergence of Dirichlet series ✻ ✲ unif. conv. ✲ abs. conv. ✛ ✲ ? ? ? ✲ σ u σ a Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Convergence of Dirichlet series ✻ ✲ unif. conv. ✲ abs. conv. ✛ ✲ ? ? ? ✲ σ u σ a Definition � 1 � S := sup σ a ( D ) − σ u ( D ) : D = � n a n n s Dirichlet series Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Convergence of Dirichlet series ✻ ✲ unif. conv. ✲ abs. conv. ✛ ✲ ? ? ? ✲ σ u σ a Bohr’s absolute convergence problem S = ? ? ? Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series Convergence of Dirichlet series ✻ ✲ unif. conv. ✲ abs. conv. ✛ ✲ ? ? ? ✲ σ u σ a Bohnenblust-Hille Theorem (1931 Annals of Math.) S = 1 2 Manuel Maestre The Dirichlet-Bohr radius

Vector valued Dirichlet series Theorem Let X be a complex Banach space, and let � ∞ a n n s be a Dirichlet series in X . i.e. n = 1 a n belongs to X for all n .Then S ( X ) = sup { σ a − σ u } � an ns Manuel Maestre The Dirichlet-Bohr radius

Vector valued Dirichlet series Theorem Let X be a complex Banach space, and let � ∞ a n n s be a Dirichlet series in X . i.e. n = 1 a n belongs to X for all n .Then S ( X ) = sup { σ a − σ u } � an ns Theorem. A. Defant, D. García, M. M. D. Pérez (Math. Annalen 2008) For every Banach space X � 1 1 � S ( X ) = inf = 1 − p ′ | Y has cotype p Cot ( X ) . Manuel Maestre The Dirichlet-Bohr radius

Vector valued Dirichlet series Theorem Let X be a complex Banach space, and let � ∞ a n n s be a Dirichlet series in X . i.e. n = 1 a n belongs to X for all n .Then S ( X ) = sup { σ a − σ u } � an ns Theorem. A. Defant, D. García, M. M. D. Pérez (Math. Annalen 2008) For every Banach space X � 1 1 � S ( X ) = inf = 1 − p ′ | Y has cotype p Cot ( X ) . Definition X has cotype p ( p ∈ [ 2 , + ∞ ] ) if there exists a constant K � 0 such that n n � � 1 � 1 � x k � p ) ε k ( ω ) x k � 2 d ω ) ( p � K ( 2 , � Ω k = 1 k = 1 Cot ( X ) := inf { 2 � p � ∞ | X has cotype p } Manuel Maestre The Dirichlet-Bohr radius

Vector valued Dirichlet series Recall � 2 if 1 ≤ p ≤ 2 Cot ( ℓ p ) = p if 2 ≤ p ≤ ∞ , Manuel Maestre The Dirichlet-Bohr radius

Vector valued Dirichlet series Recall � 2 if 1 ≤ p ≤ 2 Cot ( ℓ p ) = p if 2 ≤ p ≤ ∞ , Corollary  1 2 , 1 ≤ p ≤ 2     S ( ℓ p ) =    1 − 1 2 � p � ∞  p ,   In particular, S ( ℓ ∞ ) = 1 . Manuel Maestre The Dirichlet-Bohr radius

Vector valued Dirichlet series Recall � 2 if 1 ≤ p ≤ 2 Cot ( ℓ p ) = p if 2 ≤ p ≤ ∞ , Corollary  1 2 , 1 ≤ p ≤ 2     S ( ℓ p ) =    1 − 1 2 � p � ∞  p ,   In particular, S ( ℓ ∞ ) = 1 . Corollary For every t ∈ [ 1 2 , 1 ] there is a Banach space X for which t = S ( X ) . Manuel Maestre The Dirichlet-Bohr radius

Content Dirichlet series and complex analysis on polydiscs Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series and complex analysis on polydiscs p = the sequence of prime numbers: p 1 < p 2 < p 3 < . . . p α = p α 1 where α = ( α 1 , . . . , α n , 0 , . . . ) ∈ N ( N ) 1 × · · · × p α n n 0 Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series and complex analysis on polydiscs A one to one correspondence (The Borh transform): Dirichlet series D Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series and complex analysis on polydiscs A one to one correspondence (The Borh transform): Dirichlet series formal power series D − − − − − − − − − − − − − − − → P Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series and complex analysis on polydiscs A one to one correspondence (The Borh transform): Dirichlet series formal power series D − − − − − − − − − − − − − − − → P a n = a p α = c α 1 α c α z α � n a n − − − − − − − − − → � n s Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series and complex analysis on polydiscs A one to one correspondence (The Borh transform): Dirichlet series formal power series D − − − − − − − − − − − − − − − → P a n = a p α = c α 1 α c α z α � n a n − − − − − − − − − → � n s � H ∞ Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series and complex analysis on polydiscs A one to one correspondence (The Borh transform): Dirichlet series formal power series D − − − − − − − − − − − − − − − → P a n = a p α = c α 1 α c α z α � n a n − − − − − − − − − → � n s � H ∞ − − − − − − − − − − − − − − − → Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series and complex analysis on polydiscs A one to one correspondence (The Borh transform): Dirichlet series formal power series D − − − − − − − − − − − − − − − → P a n = a p α = c α 1 α c α z α � n a n − − − − − − − − − → � n s � H ∞ ? ? ? − − − − − − − − − − − − − − − → Manuel Maestre The Dirichlet-Bohr radius

Dirichlet series and complex analysis on polydiscs A one to one correspondence (The Borh transform): Dirichlet series formal power series D − − − − − − − − − − − − − − − → P a n = a p α = c α 1 α c α z α � n a n − − − − − − − − − → � n s � H ∞ − − − − − − − − − − − − − − − → Definition H ∞ := the set of all those Dirichlet series D which converge on [ Re > 0 ] and it is bounded on this halfplane. Manuel Maestre The Dirichlet-Bohr radius