A Survey of Recent Results on the Hardy Space of Dirichlet Series Gregory Zitelli University of Tennessee, Knoxville September 2013 Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Notation D is the open unit disc. T is the unit circle. C ρ is the right half plane with real part > ρ . C + = C 0 . Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Hardy Spaces We begin with the standard definition of the Hardy-Hilbert space on D , a Hilbert space of holomorphic functions on D with a square summable power series. � ∞ ∞ � a n z n : | a n | 2 < ∞ � � H 2 ( D ) = f ( z ) = n =0 n =0 where the inner product is given by � ∞ ∞ � ∞ � a n z n , � b n z n � � f, g � H 2 ( D ) = = a n b n n =0 n =0 n =0 H 2 ( D ) Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Hardy Spaces This formulation of the Hardy-Hilbert space H 2 ( D ) is useful because it emphasizes the canonical equivalence of H 2 ( D ) and ℓ 2 ( N ) , namely ∞ a n z n ∼ ( a 0 , a 1 , a 2 , . . . ) � f ( z ) = n =0 Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Hardy Spaces Interestingly, the Hardy-Hilbert space norm is equivalent to a growth condition on the radial boundary values of its functions, so that if f ( z ) = � ∞ n =0 a n z n , � 2 π ∞ | f ( re it ) | 2 dt | a n | 2 = sup � f � 2 � H 2 ( D ) = 2 π 0 ≤ r< 1 0 n =0 Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Hardy Spaces Hardy-Hilbert space functions living on D have nontangential boundary values almost everywhere on T , allowing us to extend functions f ∈ H 2 ( D ) to functions ˜ f ∈ H 2 ( T ) ⊆ L 2 ( T ) , where � 2 π � 2 π | f ( re it ) | 2 dt f ( e it ) | 2 dt � f � 2 | ˜ 2 π = � ˜ f � 2 H 2 ( D ) = sup 2 π = H 2 ( T ) 0 ≤ r< 1 0 0 and H 2 ( T ) is the subspace of L 2 ( T ) whose elements have only nonnegative Fourier coefficients. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Hardy Spaces For general 1 < p < ∞ , we can form the Hardy space H p similarly, with � 2 π � 2 π | f ( re it ) | p dt f ( e it ) | p dt � f � p f � p | ˜ 2 π = � ˜ H p ( D ) = sup 2 π = H p ( T ) 0 ≤ r< 1 0 0 = H p ( T ) ⊆ L p ( T ) . We treat H p as both H p ( D ) and Here H p ( D ) ∼ H p ( T ) interchangeably. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Hardy Spaces The Hardy spaces H p can be thought of both as the holomorphic functions on D which satisfy a growth condition on the boundary, and the nontangential boundary functions which live inside of the L p space on that boundary. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Hardy Spaces There are three important properties posessed by the Hardy space H 2 as a Hilbert space which we would like to contrast: Reproducing kernels k λ Zero sets { z n } Multiplier algebra M ( H 2 ) Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Reproducing Kernels for the space H 2 ( D ) Point evaluations are bounded linear functionals on H 2 ( D ) , and can therefore be expressed as inner products with appropriate reproducing kernels. n =0 a n z n ∈ H 2 ( D ) , then the reproducing If λ ∈ D and f ( z ) = � ∞ n =0 λ n z n is such that kernel k λ ( z ) = � ∞ ∞ ∞ a n λ n = � � a n ( λ n ) = � f, k λ � H 2 ( D ) f ( λ ) = n =0 n =0 � 2 < ∞ , so that k λ ∈ H 2 ( D ) . Note that � ∞ � � λ n � n =0 Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Zero Sets of H 2 ( D ) Given a sequence of points { z n } ⊆ D , there is a nontrivial function f ∈ H 2 ( D ) which vanishes at each z n if and only if { z n } satisfies the Blaschke condition ∞ � (1 − | z n | ) < ∞ n =1 Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Multiplier Algebra of H 2 ( D ) The multipliers M ( H 2 ( D )) are precisely H ∞ ( D ) , the bounded holomorphic functions on the disc. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Hardy Spaces in Half Planes There is similarly a Hardy space for the half plane C + using the growth condition on the imaginary line � ∞ � � H p ( C + ) = | f ( σ + it ) | p dt < ∞ f ∈ Hol( C + ) : sup σ> 0 −∞ One can also define spaces H p ( C ρ ) for arbitrary ρ . Like the Hardy space H 2 ( D ) , H 2 ( C + ) has well understood reproducing kernels, zero sets, and a multiplier algebra. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Reproducing Kernels for the space H 2 ( C + ) Point evaluations are bounded linear functionals on H 2 ( C + ) , and can therefore be expressed as inner products with appropriate reproducing kernels. If λ ∈ C + and f ∈ H 2 ( C + ) , then the reproducing kernel 1 k λ ( z ) = z + λ is such that f ( λ ) = � f, k λ � H 2 ( D ) 2 � ∞ � � 1 dt < ∞ , so that k λ ∈ H 2 ( D ) . Note that sup x> 0 � � −∞ x + it + λ � � Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Zero Sets of H 2 ( C + ) Given a sequence of points { z n } ⊆ C + , there is a nontrivial function f ∈ H 2 ( C + ) which vanishes at each z n if and only if { z n } satisfies the following condition ∞ x n � 1 + | z n | 2 < ∞ n =1 where z n = x n + iy n . If the sequence { z n } is bounded, then we recover a Blaschke-type condition ∞ � x n < ∞ n =1 Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Multiplier Algebra of H 2 ( C + ) The multipliers M ( H 2 ( C + )) are precisely H ∞ ( C + ) , the bounded holomorphic functions on the right half plane. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Dirichlet Series A Dirichlet series is a series of the form f ( s ) = � ∞ a n n s . We write n =1 s = σ + it , and let Ω ρ denote the half plane with real part > ρ . Unlike power series, the “radius” of convergence and absolute convergence may be different. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Dirichlet Series For a particular f ( s ) = � ∞ a n n s , we write n =1 � ∞ � a n � σ c ( f ) = inf R ( s ) : n s converges n =1 � ∞ � a n � σ b ( f ) = inf ρ : n s converges to a bounded function in Ω ρ n =1 � ∞ � a n � σ u ( f ) = inf ρ : n s converges uniformly in Ω ρ n =1 � ∞ � a n � σ a ( f ) = inf R ( s ) : n s converges absolutely n =1 σ c ≤ σ b = σ u ≤ σ a ≤ σ c + 1 Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Riesz-Basis √ The Riesz-Fischer theorem states that ϕ ( x ) = 2 sin( πx ) can be dilated to form a complete orthonormal basis � √ √ � 2 sin( πx ) , 2 sin( π 2 x ) , . . . = { ϕ ( x ) , ϕ (2 x ) , . . . } for L 2 (0 , 1) . Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Riesz-Basis A natural extension of the theorem would be to ask which functions ϕ can take the place of sin so that { ϕ ( nx ) } n ≥ 1 forms an orthonormal basis for L 2 (0 , 1) under an equivalent norm. Such a sequence is called a Riesz basis. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Riesz-Basis The characterization of Riesz-type sets which are complete in L 2 (0 , 1) was characterized by Beurling in 1945, by transforming the expression ∞ √ � ϕ ( x ) = a n 2 sin( nπx ) n =1 into ∞ a n � Sϕ ( s ) = n s n =1 and analyzing properties of the analytic Sϕ . In 1995, Hedenmalm, Lindqvist, and Seip solved the Reisz-basis problem completely by exploiting a Hilbert space of analytic functions of this form. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Riesz-Basis Theorem (Hedenmalm, Lindqvist, Seip) The system { ϕ ( nx ) } n ≥ 1 is a Reisz basis for L 2 (0 , 1) if and only if Sϕ and 1 /Sϕ are in the multiplier algebra M ( H 2 ) . Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Hardy Space of Dirichlet Series The proof used the Hardy space of Dirichlet series (or Hardy-Dirichlet space), � ∞ ∞ � a n H 2 = | a n | 2 < ∞ � � f ( s ) = n s : n =1 n =1 along with the characterization of the multipliers M ( H 2 ) of the space. The paper also further established Bohr’s work on the connection between the Hardy-Dirichlet space H 2 and the Hardy space of the infinite polycircle H 2 ( T ∞ ) . Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
Hardy Space of Dirichlet Series The work by Hedehmalm, Lindqvist, and Seip inspired an investigation of the space H 2 and various related spaces over the next 15 years. Contributors in analysis include Aleman, Andersson, Bayart, McCarthy, Olsen, Saskman. Topics included Multipliers Reproducing kernels Zero sets for H 2 and related H p spaces Boundary behavior (What happens on the line σ = 1 / 2 ? Can you look at behavior of the function on the line σ = 0 ?) Connections with the infinite polycircle H p ( T ∞ ) Carleson measures Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville
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