spaces on simple closed rectifiable curves in the complex plane and - PowerPoint PPT Presentation

First Goal: Generalizing The Polya Theorem We consider Taylor-Dirichlet series � µ n − 1 � ∞ � � c n , k z k e λ n z n =1 k =0 associated to a multiplicity sequence Λ = { λ n , µ n } ∞ n =1 { λ n , µ n } ∞ n =1 := { λ 1 , λ 1 , . . . , λ 1 , λ 2 , λ 2 , . . . , λ 2 , . . . , λ k , λ k , . . . , λ k , . . . } � �� � � �� � � �� � µ 1 − times µ 2 − times µ k − times { λ n } ∞ n =1 is a strictly increasing sequence of positive real numbers AND { µ n } ∞ diverging to infinity, n =1 is a sequence of positive integers, Not Necessarily Bounded. E. Zikkos Short version

First Goal: Generalizing The Polya Theorem We consider Taylor-Dirichlet series � µ n − 1 � ∞ � � c n , k z k e λ n z n =1 k =0 associated to a multiplicity sequence Λ = { λ n , µ n } ∞ n =1 { λ n , µ n } ∞ n =1 := { λ 1 , λ 1 , . . . , λ 1 , λ 2 , λ 2 , . . . , λ 2 , . . . , λ k , λ k , . . . , λ k , . . . } � �� � � �� � � �� � µ 1 − times µ 2 − times µ k − times { λ n } ∞ n =1 is a strictly increasing sequence of positive real numbers AND { µ n } ∞ diverging to infinity, n =1 is a sequence of positive integers, Not Necessarily Bounded. We impose two conditions: E. Zikkos Short version

First Goal: Generalizing The Polya Theorem We consider Taylor-Dirichlet series � µ n − 1 � ∞ � � c n , k z k e λ n z n =1 k =0 associated to a multiplicity sequence Λ = { λ n , µ n } ∞ n =1 { λ n , µ n } ∞ n =1 := { λ 1 , λ 1 , . . . , λ 1 , λ 2 , λ 2 , . . . , λ 2 , . . . , λ k , λ k , . . . , λ k , . . . } � �� � � �� � � �� � µ 1 − times µ 2 − times µ k − times { λ n } ∞ n =1 is a strictly increasing sequence of positive real numbers AND { µ n } ∞ diverging to infinity, n =1 is a sequence of positive integers, Not Necessarily Bounded. We impose two conditions: (A) Λ has Density d counting multiplicities E. Zikkos Short version

First Goal: Generalizing The Polya Theorem We consider Taylor-Dirichlet series � µ n − 1 � ∞ � � c n , k z k e λ n z n =1 k =0 associated to a multiplicity sequence Λ = { λ n , µ n } ∞ n =1 { λ n , µ n } ∞ n =1 := { λ 1 , λ 1 , . . . , λ 1 , λ 2 , λ 2 , . . . , λ 2 , . . . , λ k , λ k , . . . , λ k , . . . } � �� � � �� � � �� � µ 1 − times µ 2 − times µ k − times { λ n } ∞ n =1 is a strictly increasing sequence of positive real numbers AND { µ n } ∞ diverging to infinity, n =1 is a sequence of positive integers, Not Necessarily Bounded. We impose two conditions: (A) Λ has Density d counting multiplicities n Λ ( t ) � lim = d < ∞ , n Λ ( t ) = µ n t t →∞ λ n ≤ t E. Zikkos Short version

First Goal: Generalizing The Polya Theorem We consider Taylor-Dirichlet series � µ n − 1 � ∞ � � c n , k z k e λ n z n =1 k =0 associated to a multiplicity sequence Λ = { λ n , µ n } ∞ n =1 { λ n , µ n } ∞ n =1 := { λ 1 , λ 1 , . . . , λ 1 , λ 2 , λ 2 , . . . , λ 2 , . . . , λ k , λ k , . . . , λ k , . . . } � �� � � �� � � �� � µ 1 − times µ 2 − times µ k − times { λ n } ∞ n =1 is a strictly increasing sequence of positive real numbers AND { µ n } ∞ diverging to infinity, n =1 is a sequence of positive integers, Not Necessarily Bounded. We impose two conditions: (A) Λ has Density d counting multiplicities n Λ ( t ) � lim = d < ∞ , n Λ ( t ) = µ n t t →∞ λ n ≤ t (if µ n = 1 for all n ∈ N then n /λ n → d ) E. Zikkos Short version

First Goal: Generalizing The Polya Theorem We consider Taylor-Dirichlet series � µ n − 1 � ∞ � � c n , k z k e λ n z n =1 k =0 associated to a multiplicity sequence Λ = { λ n , µ n } ∞ n =1 { λ n , µ n } ∞ n =1 := { λ 1 , λ 1 , . . . , λ 1 , λ 2 , λ 2 , . . . , λ 2 , . . . , λ k , λ k , . . . , λ k , . . . } � �� � � �� � � �� � µ 1 − times µ 2 − times µ k − times { λ n } ∞ n =1 is a strictly increasing sequence of positive real numbers AND { µ n } ∞ diverging to infinity, n =1 is a sequence of positive integers, Not Necessarily Bounded. We impose two conditions: (A) Λ has Density d counting multiplicities n Λ ( t ) � lim = d < ∞ , n Λ ( t ) = µ n t t →∞ λ n ≤ t (if µ n = 1 for all n ∈ N then n /λ n → d ) ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . E. Zikkos Short version

Region of Convergence, Taylor-Dirichlet series: E. Zikkos Short version

Region of Convergence, Taylor-Dirichlet series: Assuming (A) and (B) then Λ = { λ n , µ n } ∞ n =1 satisfies log n µ n lim = 0 lim = 0 . λ n λ n n →∞ n →∞ E. Zikkos Short version

Region of Convergence, Taylor-Dirichlet series: Assuming (A) and (B) then Λ = { λ n , µ n } ∞ n =1 satisfies log n µ n lim = 0 lim = 0 . λ n λ n n →∞ n →∞ Valiron (1929) : E. Zikkos Short version

Region of Convergence, Taylor-Dirichlet series: Assuming (A) and (B) then Λ = { λ n , µ n } ∞ n =1 satisfies log n µ n lim = 0 lim = 0 . λ n λ n n →∞ n →∞ Valiron (1929) : consider the series � µ n − 1 � ∞ � � c n , k z k e λ n z , g ( z ) = C n = max {| c n , k | : k = 0 , 1 , 2 , . . . , µ n − 1 } n =1 k =0 log C n lim sup = ξ ∈ R , P − ξ := { z : ℜ z < − ξ } . λ n n →∞ E. Zikkos Short version

Region of Convergence, Taylor-Dirichlet series: Assuming (A) and (B) then Λ = { λ n , µ n } ∞ n =1 satisfies log n µ n lim = 0 lim = 0 . λ n λ n n →∞ n →∞ Valiron (1929) : consider the series � µ n − 1 � ∞ � � c n , k z k e λ n z , g ( z ) = C n = max {| c n , k | : k = 0 , 1 , 2 , . . . , µ n − 1 } n =1 k =0 log C n lim sup = ξ ∈ R , P − ξ := { z : ℜ z < − ξ } . λ n n →∞ Then g ( z ) is an analytic function in the left half-plane P − ξ converging uniformly on compact subsets. E. Zikkos Short version

Region of Convergence, Taylor-Dirichlet series: Assuming (A) and (B) then Λ = { λ n , µ n } ∞ n =1 satisfies log n µ n lim = 0 lim = 0 . λ n λ n n →∞ n →∞ Valiron (1929) : consider the series � µ n − 1 � ∞ � � c n , k z k e λ n z , g ( z ) = C n = max {| c n , k | : k = 0 , 1 , 2 , . . . , µ n − 1 } n =1 k =0 log C n lim sup = ξ ∈ R , P − ξ := { z : ℜ z < − ξ } . λ n n →∞ Then g ( z ) is an analytic function in the left half-plane P − ξ converging uniformly on compact subsets. We call the line ℜ z = − ξ the abscissa of convergence for g ( z ). E. Zikkos Short version

Region of Convergence, Taylor-Dirichlet series: Assuming (A) and (B) then Λ = { λ n , µ n } ∞ n =1 satisfies log n µ n lim = 0 lim = 0 . λ n λ n n →∞ n →∞ Valiron (1929) : consider the series � µ n − 1 � ∞ � � c n , k z k e λ n z , g ( z ) = C n = max {| c n , k | : k = 0 , 1 , 2 , . . . , µ n − 1 } n =1 k =0 log C n lim sup = ξ ∈ R , P − ξ := { z : ℜ z < − ξ } . λ n n →∞ Then g ( z ) is an analytic function in the left half-plane P − ξ converging uniformly on compact subsets. We call the line ℜ z = − ξ the abscissa of convergence for g ( z ). Question : E. Zikkos Short version

Region of Convergence, Taylor-Dirichlet series: Assuming (A) and (B) then Λ = { λ n , µ n } ∞ n =1 satisfies log n µ n lim = 0 lim = 0 . λ n λ n n →∞ n →∞ Valiron (1929) : consider the series � µ n − 1 � ∞ � � c n , k z k e λ n z , g ( z ) = C n = max {| c n , k | : k = 0 , 1 , 2 , . . . , µ n − 1 } n =1 k =0 log C n lim sup = ξ ∈ R , P − ξ := { z : ℜ z < − ξ } . λ n n →∞ Then g ( z ) is an analytic function in the left half-plane P − ξ converging uniformly on compact subsets. We call the line ℜ z = − ξ the abscissa of convergence for g ( z ). Question : is it True that in every interval having length greater than 2 π d on the line ℜ z = − ξ , the series has at least One singularity? E. Zikkos Short version

Positive Answers to the Singularity Question E. Zikkos Short version

Positive Answers to the Singularity Question Suppose that Λ = { λ n , µ n } satisfies � λ n ≤ t µ n ( A ) Λ has Density d : lim = d < ∞ , t t →∞ ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . E. Zikkos Short version

Positive Answers to the Singularity Question Suppose that Λ = { λ n , µ n } satisfies � λ n ≤ t µ n ( A ) Λ has Density d : lim = d < ∞ , t t →∞ ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . ◮ Zikkos (2005 Complex Variables): E. Zikkos Short version

Positive Answers to the Singularity Question Suppose that Λ = { λ n , µ n } satisfies � λ n ≤ t µ n ( A ) Λ has Density d : lim = d < ∞ , t t →∞ ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . ◮ Zikkos (2005 Complex Variables): If Λ belongs to a certain class denoted by U ( d , 0), with a restriction on the coefficients, the answer is YES. E. Zikkos Short version

Positive Answers to the Singularity Question Suppose that Λ = { λ n , µ n } satisfies � λ n ≤ t µ n ( A ) Λ has Density d : lim = d < ∞ , t t →∞ ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . ◮ Zikkos (2005 Complex Variables): If Λ belongs to a certain class denoted by U ( d , 0), with a restriction on the coefficients, the answer is YES. ◮ O. A. Krivosheeva (2012 St. Petersburg Math. J. ): E. Zikkos Short version

Positive Answers to the Singularity Question Suppose that Λ = { λ n , µ n } satisfies � λ n ≤ t µ n ( A ) Λ has Density d : lim = d < ∞ , t t →∞ ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . ◮ Zikkos (2005 Complex Variables): If Λ belongs to a certain class denoted by U ( d , 0), with a restriction on the coefficients, the answer is YES. ◮ O. A. Krivosheeva (2012 St. Petersburg Math. J. ): If the Krivosheev characteristic S Λ is Equal to 0, E. Zikkos Short version

Positive Answers to the Singularity Question Suppose that Λ = { λ n , µ n } satisfies � λ n ≤ t µ n ( A ) Λ has Density d : lim = d < ∞ , t t →∞ ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . ◮ Zikkos (2005 Complex Variables): If Λ belongs to a certain class denoted by U ( d , 0), with a restriction on the coefficients, the answer is YES. ◮ O. A. Krivosheeva (2012 St. Petersburg Math. J. ): If the Krivosheev characteristic S Λ is Equal to 0, then the answer is YES. E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): If Λ belongs to the class U ( d , 0), then the Krivosheev characteristic S Λ = 0, E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): If Λ belongs to the class U ( d , 0), then the Krivosheev characteristic S Λ = 0, hence the answer is YES. E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): If Λ belongs to the class U ( d , 0), then the Krivosheev characteristic S Λ = 0, hence the answer is YES. Examples in U ( d , 0) : E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): If Λ belongs to the class U ( d , 0), then the Krivosheev characteristic S Λ = 0, hence the answer is YES. Examples in U ( d , 0) : (1) If (A) and (B) hold and µ n = O (1), then Λ ∈ U ( d , 0). E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): If Λ belongs to the class U ( d , 0), then the Krivosheev characteristic S Λ = 0, hence the answer is YES. Examples in U ( d , 0) : (1) If (A) and (B) hold and µ n = O (1), then Λ ∈ U ( d , 0). (2) Let { p n } be the prime numbers E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): If Λ belongs to the class U ( d , 0), then the Krivosheev characteristic S Λ = 0, hence the answer is YES. Examples in U ( d , 0) : (1) If (A) and (B) hold and µ n = O (1), then Λ ∈ U ( d , 0). (2) Let { p n } be the prime numbers and let µ n = p n +1 − p n . E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): If Λ belongs to the class U ( d , 0), then the Krivosheev characteristic S Λ = 0, hence the answer is YES. Examples in U ( d , 0) : (1) If (A) and (B) hold and µ n = O (1), then Λ ∈ U ( d , 0). (2) Let { p n } be the prime numbers and let µ n = p n +1 − p n . Then Λ = { p n , µ n } belongs to the class U (1 , 0). E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): If Λ belongs to the class U ( d , 0), then the Krivosheev characteristic S Λ = 0, hence the answer is YES. Examples in U ( d , 0) : (1) If (A) and (B) hold and µ n = O (1), then Λ ∈ U ( d , 0). (2) Let { p n } be the prime numbers and let µ n = p n +1 − p n . Then Λ = { p n , µ n } belongs to the class U (1 , 0). Theorem The Taylor-Dirichlet series � µ n − 1 � ∞ � � z k e p n z , g ( z ) = c n , k ∈ C n =1 k =0 defines an analytic function in the half-plane { z : ℜ z < 0 } E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): If Λ belongs to the class U ( d , 0), then the Krivosheev characteristic S Λ = 0, hence the answer is YES. Examples in U ( d , 0) : (1) If (A) and (B) hold and µ n = O (1), then Λ ∈ U ( d , 0). (2) Let { p n } be the prime numbers and let µ n = p n +1 − p n . Then Λ = { p n , µ n } belongs to the class U (1 , 0). Theorem The Taylor-Dirichlet series � µ n − 1 � ∞ � � z k e p n z , g ( z ) = c n , k ∈ C n =1 k =0 defines an analytic function in the half-plane { z : ℜ z < 0 } and it has at least One singularity E. Zikkos Short version

Another Positive Answer ◮ Zikkos (2018): If Λ belongs to the class U ( d , 0), then the Krivosheev characteristic S Λ = 0, hence the answer is YES. Examples in U ( d , 0) : (1) If (A) and (B) hold and µ n = O (1), then Λ ∈ U ( d , 0). (2) Let { p n } be the prime numbers and let µ n = p n +1 − p n . Then Λ = { p n , µ n } belongs to the class U (1 , 0). Theorem The Taylor-Dirichlet series � µ n − 1 � ∞ � � z k e p n z , g ( z ) = c n , k ∈ C n =1 k =0 defines an analytic function in the half-plane { z : ℜ z < 0 } and it has at least One singularity in every open interval of length exceeding 2 π and lying on the Imaginary axis. E. Zikkos Short version

A Negative Answer E. Zikkos Short version

A Negative Answer Zikkos ( Ufa Math J.): E. Zikkos Short version

A Negative Answer Zikkos ( Ufa Math J.): for every d ≥ 0, E. Zikkos Short version

A Negative Answer Zikkos ( Ufa Math J.): for every d ≥ 0, there exists a multiplicity sequence Λ = { λ n , µ n } with µ n → ∞ , such that E. Zikkos Short version

A Negative Answer Zikkos ( Ufa Math J.): for every d ≥ 0, there exists a multiplicity sequence Λ = { λ n , µ n } with µ n → ∞ , such that � λ n ≤ t µ n ( A ) Λ has Density d : lim = d < ∞ , t t →∞ ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . E. Zikkos Short version

A Negative Answer Zikkos ( Ufa Math J.): for every d ≥ 0, there exists a multiplicity sequence Λ = { λ n , µ n } with µ n → ∞ , such that � λ n ≤ t µ n ( A ) Λ has Density d : lim = d < ∞ , t t →∞ ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . ( C ) S Λ < 0 E. Zikkos Short version

A Negative Answer Zikkos ( Ufa Math J.): for every d ≥ 0, there exists a multiplicity sequence Λ = { λ n , µ n } with µ n → ∞ , such that � λ n ≤ t µ n ( A ) Λ has Density d : lim = d < ∞ , t t →∞ ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . ( C ) S Λ < 0 and hence (Krivosheeva 2012 St. Petersburg Math. J.): E. Zikkos Short version

A Negative Answer Zikkos ( Ufa Math J.): for every d ≥ 0, there exists a multiplicity sequence Λ = { λ n , µ n } with µ n → ∞ , such that � λ n ≤ t µ n ( A ) Λ has Density d : lim = d < ∞ , t t →∞ ( B ) λ n +1 − λ n > c > 0 , ( Uniformly Separated ) . ( C ) S Λ < 0 and hence (Krivosheeva 2012 St. Petersburg Math. J.): there exists a Taylor-Dirichlet series such that it Can be Continued Analytically across the abscissa of convergence. E. Zikkos Short version

The class U ( d , 0) E. Zikkos Short version

The class U ( d , 0) Zikkos (2005 Complex Variables, 2010 CMFT) : E. Zikkos Short version

The class U ( d , 0) Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence { a n } of positive real numbers, having density d with uniformly separated terms n / a n → d , a n +1 − a n > c > 0 . E. Zikkos Short version

The class U ( d , 0) Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence { a n } of positive real numbers, having density d with uniformly separated terms n / a n → d , a n +1 − a n > c > 0 . Choose two positive numbers α < 1 , δ < c . E. Zikkos Short version

The class U ( d , 0) Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence { a n } of positive real numbers, having density d with uniformly separated terms n / a n → d , a n +1 − a n > c > 0 . Choose two positive numbers α < 1 , δ < c . For each term a n consider the closed disk B ( a n , | a n | α ) = { z : | z − a n | ≤ a α n } . E. Zikkos Short version

The class U ( d , 0) Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence { a n } of positive real numbers, having density d with uniformly separated terms n / a n → d , a n +1 − a n > c > 0 . Choose two positive numbers α < 1 , δ < c . For each term a n consider the closed disk B ( a n , | a n | α ) = { z : | z − a n | ≤ a α n } . Choose a point in B ( a n , | a n | α ) ∩ R , call it b n , in an almost arbitrary way, E. Zikkos Short version

The class U ( d , 0) Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence { a n } of positive real numbers, having density d with uniformly separated terms n / a n → d , a n +1 − a n > c > 0 . Choose two positive numbers α < 1 , δ < c . For each term a n consider the closed disk B ( a n , | a n | α ) = { z : | z − a n | ≤ a α n } . Choose a point in B ( a n , | a n | α ) ∩ R , call it b n , in an almost arbitrary way, such that for all n � = m either ( I ) b m = b n E. Zikkos Short version

The class U ( d , 0) Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence { a n } of positive real numbers, having density d with uniformly separated terms n / a n → d , a n +1 − a n > c > 0 . Choose two positive numbers α < 1 , δ < c . For each term a n consider the closed disk B ( a n , | a n | α ) = { z : | z − a n | ≤ a α n } . Choose a point in B ( a n , | a n | α ) ∩ R , call it b n , in an almost arbitrary way, such that for all n � = m either ( I ) b m = b n or ( II ) | b m − b n | ≥ δ. E. Zikkos Short version

The class U ( d , 0) Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence { a n } of positive real numbers, having density d with uniformly separated terms n / a n → d , a n +1 − a n > c > 0 . Choose two positive numbers α < 1 , δ < c . For each term a n consider the closed disk B ( a n , | a n | α ) = { z : | z − a n | ≤ a α n } . Choose a point in B ( a n , | a n | α ) ∩ R , call it b n , in an almost arbitrary way, such that for all n � = m either ( I ) b m = b n or ( II ) | b m − b n | ≥ δ. Rename { b n } into Λ = { λ n , µ n } . E. Zikkos Short version

The class U ( d , 0) Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence { a n } of positive real numbers, having density d with uniformly separated terms n / a n → d , a n +1 − a n > c > 0 . Choose two positive numbers α < 1 , δ < c . For each term a n consider the closed disk B ( a n , | a n | α ) = { z : | z − a n | ≤ a α n } . Choose a point in B ( a n , | a n | α ) ∩ R , call it b n , in an almost arbitrary way, such that for all n � = m either ( I ) b m = b n or ( II ) | b m − b n | ≥ δ. Rename { b n } into Λ = { λ n , µ n } . Then we say that Λ ∈ U ( d , 0). E. Zikkos Short version

The Class U ( d , 0) E. Zikkos Short version

The Class U ( d , 0) a n

The Class U ( d , 0) a n R = a α n

The Class U ( d , 0) b n a n R = a α n

The Class U ( d , 0) b n a n a n +1 R = a α n

The Class U ( d , 0) b n a n a n +1 R = a α R = a α n n +1

The Class U ( d , 0) b n b n +1 a n a n +1 R = a α R = a α n n +1

The Class U ( d , 0) b n b n +1 a n a n +1 a n +2 R = a α R = a α n n +1

The Class U ( d , 0) b n b n +1 a n a n +1 a n +2 R = a α n +2 R = a α R = a α n n +1

The Class U ( d , 0) b n = b n +2 b n +1 a n a n +1 a n +2 R = a α n +2 R = a α R = a α n n +1

The Class U ( d , 0) b n = b n +2 b n +1 a n a n +1 a n +2 a n +3 R = a α n +2 R = a α R = a α n n +1

The Class U ( d , 0) b n = b n +2 b n +1 a n a n +1 a n +2 a n +3 R = a α n +3 R = a α n +2 R = a α R = a α n n +1

The Class U ( d , 0) b n = b n +2 = b n +3 b n +1 a n a n +1 a n +2 a n +3 R = a α n +3 R = a α n +2 R = a α R = a α n n +1 E. Zikkos Short version

Singularities of Taylor-Dirichlet series E. Zikkos Short version

Singularities of Taylor-Dirichlet series Theorem A Let the multiplicity-sequence Λ = { λ n , µ n } ∞ n =1 belong to the class U ( d , 0) for some d > 0 , and consider the Taylor-Dirichlet series � µ n − 1 � ∞ � � c n , k z k e λ n z , g ( z ) = c n , k ∈ C n =1 k =0 log C n lim sup = ξ ∈ R , where C n = max {| c n , k | : k = 0 , 1 , . . . , µ n − 1 } . λ n n →∞ E. Zikkos Short version

Singularities of Taylor-Dirichlet series Theorem A Let the multiplicity-sequence Λ = { λ n , µ n } ∞ n =1 belong to the class U ( d , 0) for some d > 0 , and consider the Taylor-Dirichlet series � µ n − 1 � ∞ � � c n , k z k e λ n z , g ( z ) = c n , k ∈ C n =1 k =0 log C n lim sup = ξ ∈ R , where C n = max {| c n , k | : k = 0 , 1 , . . . , µ n − 1 } . λ n n →∞ Then g ( z ) defines an analytic function in the half-plane { z : ℜ z < − ξ } and it has at least One singularity in every open interval of length exceeding 2 π d and lying on the line ℜ z = − ξ . E. Zikkos Short version

Second Goal E. Zikkos Short version

Second Goal Given Λ = { λ n , µ n } ∞ n =1 in U ( d , 0) E. Zikkos Short version

Second Goal Given Λ = { λ n , µ n } ∞ n =1 in U ( d , 0) Characterize the closed span of the exponential system E Λ = { z k e λ n z : n ∈ N , k = 0 , 1 , . . . , µ n − 1 } E. Zikkos Short version

Second Goal Given Λ = { λ n , µ n } ∞ n =1 in U ( d , 0) Characterize the closed span of the exponential system E Λ = { z k e λ n z : n ∈ N , k = 0 , 1 , . . . , µ n − 1 } in L p ( l ) spaces where l is a simple closed rectifiable curve in C , and G l is the domain bounded by the curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Zikkos Short version

Second Goal Given Λ = { λ n , µ n } ∞ n =1 in U ( d , 0) Characterize the closed span of the exponential system E Λ = { z k e λ n z : n ∈ N , k = 0 , 1 , . . . , µ n − 1 } in L p ( l ) spaces where l is a simple closed rectifiable curve in C , and G l is the domain bounded by the curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . If f is in the closed span of E Λ in L p ( l ), E. Zikkos Short version

Second Goal Given Λ = { λ n , µ n } ∞ n =1 in U ( d , 0) Characterize the closed span of the exponential system E Λ = { z k e λ n z : n ∈ N , k = 0 , 1 , . . . , µ n − 1 } in L p ( l ) spaces where l is a simple closed rectifiable curve in C , and G l is the domain bounded by the curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . If f is in the closed span of E Λ in L p ( l ), then f is in the L p closure of polynomials, E. Zikkos Short version

Second Goal Given Λ = { λ n , µ n } ∞ n =1 in U ( d , 0) Characterize the closed span of the exponential system E Λ = { z k e λ n z : n ∈ N , k = 0 , 1 , . . . , µ n − 1 } in L p ( l ) spaces where l is a simple closed rectifiable curve in C , and G l is the domain bounded by the curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . If f is in the closed span of E Λ in L p ( l ), then f is in the L p closure of polynomials, hence f ∈ E p ( G l ). E. Zikkos Short version

Curve l is surrounded by a rectangle whose height is less than 2 π d E. Zikkos Short version

Curve l is surrounded by a rectangle whose height is less than 2 π d Height < 2 π d Theorem B Suppose the Domain G l bounded by the curve l is a Smirnov domain. E. Zikkos Short version

Curve l is surrounded by a rectangle whose height is less than 2 π d Height < 2 π d Theorem B Suppose the Domain G l bounded by the curve l is a Smirnov domain. Suppose also that Λ = { λ n , µ n } has Density d. E. Zikkos Short version

Curve l is surrounded by a rectangle whose height is less than 2 π d Height < 2 π d Theorem B Suppose the Domain G l bounded by the curve l is a Smirnov domain. Suppose also that Λ = { λ n , µ n } has Density d. Then the closed span of the exponential system E Λ in the space L p ( l ) for p ≥ 1 E. Zikkos Short version

Curve l is surrounded by a rectangle whose height is less than 2 π d Height < 2 π d Theorem B Suppose the Domain G l bounded by the curve l is a Smirnov domain. Suppose also that Λ = { λ n , µ n } has Density d. Then the closed span of the exponential system E Λ in the space L p ( l ) for p ≥ 1 Coincides with the Smirnov space E p ( G l ) . E. Zikkos Short version

Proof It is enough to show that E p ( G l ) is a subspace of the closed span of the exponential system E Λ in L p ( l ). E. Zikkos Short version

Proof It is enough to show that E p ( G l ) is a subspace of the closed span of the exponential system E Λ in L p ( l ). Since G l is a Smirnov domain we have to show that the L p closure of polynomials is a subspace of the closed span of the exponential system E Λ in L p ( l ). E. Zikkos Short version

Proof It is enough to show that E p ( G l ) is a subspace of the closed span of the exponential system E Λ in L p ( l ). Since G l is a Smirnov domain we have to show that the L p closure of polynomials is a subspace of the closed span of the exponential system E Λ in L p ( l ). Let H ( K ) be the space of functions analytic in the rectangle K with the topology of uniform convergence on compact subsets. E. Zikkos Short version

Proof It is enough to show that E p ( G l ) is a subspace of the closed span of the exponential system E Λ in L p ( l ). Since G l is a Smirnov domain we have to show that the L p closure of polynomials is a subspace of the closed span of the exponential system E Λ in L p ( l ). Let H ( K ) be the space of functions analytic in the rectangle K with the topology of uniform convergence on compact subsets. ( B. Ya. Levin , A. F. Leont’ev): E. Zikkos Short version

Proof It is enough to show that E p ( G l ) is a subspace of the closed span of the exponential system E Λ in L p ( l ). Since G l is a Smirnov domain we have to show that the L p closure of polynomials is a subspace of the closed span of the exponential system E Λ in L p ( l ). Let H ( K ) be the space of functions analytic in the rectangle K with the topology of uniform convergence on compact subsets. ( B. Ya. Levin , A. F. Leont’ev): Since the density of Λ is d , E. Zikkos Short version