SLIDE 1 The closed span of an exponential system in Lp spaces on simple closed rectifiable curves in the complex plane and P´
- lya singularity results for
Taylor-Dirichlet series
Elias Zikkos Cyprus Ministry of Education and Culture New Developments in Complex Analysis and Function Theory Crete 2018, July 2
Short version
SLIDE 2 The Polya Theorem for Exponential Dirichlet Series
Short version
SLIDE 3 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers,
Short version
SLIDE 4 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞.
Short version
SLIDE 5 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞. consider the class of Dirichlet series of the form
∞
cneλnz, lim sup
n→∞
log |cn| λn = ξ ∈ R.
Short version
SLIDE 6 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞. consider the class of Dirichlet series of the form
∞
cneλnz, lim sup
n→∞
log |cn| λn = ξ ∈ R. The series is analytic in the left half-plane ℜz < −ξ, it converges uniformly on compact subsets, and it diverges for all z such ℜz > −ξ.
Short version
SLIDE 7 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞. consider the class of Dirichlet series of the form
∞
cneλnz, lim sup
n→∞
log |cn| λn = ξ ∈ R. The series is analytic in the left half-plane ℜz < −ξ, it converges uniformly on compact subsets, and it diverges for all z such ℜz > −ξ. The line ℜz = −ξ is called the Abscissa of Convergence (pointwise and absolute).
Short version
SLIDE 8 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞. consider the class of Dirichlet series of the form
∞
cneλnz, lim sup
n→∞
log |cn| λn = ξ ∈ R. The series is analytic in the left half-plane ℜz < −ξ, it converges uniformly on compact subsets, and it diverges for all z such ℜz > −ξ. The line ℜz = −ξ is called the Abscissa of Convergence (pointwise and absolute). POLYA:
Short version
SLIDE 9 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞. consider the class of Dirichlet series of the form
∞
cneλnz, lim sup
n→∞
log |cn| λn = ξ ∈ R. The series is analytic in the left half-plane ℜz < −ξ, it converges uniformly on compact subsets, and it diverges for all z such ℜz > −ξ. The line ℜz = −ξ is called the Abscissa of Convergence (pointwise and absolute). POLYA: the series has at least One singularity
Short version
SLIDE 10 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞. consider the class of Dirichlet series of the form
∞
cneλnz, lim sup
n→∞
log |cn| λn = ξ ∈ R. The series is analytic in the left half-plane ℜz < −ξ, it converges uniformly on compact subsets, and it diverges for all z such ℜz > −ξ. The line ℜz = −ξ is called the Abscissa of Convergence (pointwise and absolute). POLYA: the series has at least One singularity in every
- pen interval whose length Exceeds 2πd and lies on the abscissa of
convergence.
Short version
SLIDE 11 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞. consider the class of Dirichlet series of the form
∞
cneλnz, lim sup
n→∞
log |cn| λn = ξ ∈ R. The series is analytic in the left half-plane ℜz < −ξ, it converges uniformly on compact subsets, and it diverges for all z such ℜz > −ξ. The line ℜz = −ξ is called the Abscissa of Convergence (pointwise and absolute). POLYA: the series has at least One singularity in every
- pen interval whose length Exceeds 2πd and lies on the abscissa of
convergence. Example (trivial): ez 1 − ez
Short version
SLIDE 12 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞. consider the class of Dirichlet series of the form
∞
cneλnz, lim sup
n→∞
log |cn| λn = ξ ∈ R. The series is analytic in the left half-plane ℜz < −ξ, it converges uniformly on compact subsets, and it diverges for all z such ℜz > −ξ. The line ℜz = −ξ is called the Abscissa of Convergence (pointwise and absolute). POLYA: the series has at least One singularity in every
- pen interval whose length Exceeds 2πd and lies on the abscissa of
convergence. Example (trivial): ez 1 − ez =
∞
enz, ℜz < 0,
Short version
SLIDE 13 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞. consider the class of Dirichlet series of the form
∞
cneλnz, lim sup
n→∞
log |cn| λn = ξ ∈ R. The series is analytic in the left half-plane ℜz < −ξ, it converges uniformly on compact subsets, and it diverges for all z such ℜz > −ξ. The line ℜz = −ξ is called the Abscissa of Convergence (pointwise and absolute). POLYA: the series has at least One singularity in every
- pen interval whose length Exceeds 2πd and lies on the abscissa of
convergence. Example (trivial): ez 1 − ez =
∞
enz, ℜz < 0, Density = 1.
Short version
SLIDE 14 The Polya Theorem for Exponential Dirichlet Series
Given a strictly increasing sequence Λ = {λn}∞
n=1, of positive real
numbers, uniformly separated and having Density d, λn+1 − λn > c > 0, lim
n→∞
n λn = d < ∞. consider the class of Dirichlet series of the form
∞
cneλnz, lim sup
n→∞
log |cn| λn = ξ ∈ R. The series is analytic in the left half-plane ℜz < −ξ, it converges uniformly on compact subsets, and it diverges for all z such ℜz > −ξ. The line ℜz = −ξ is called the Abscissa of Convergence (pointwise and absolute). POLYA: the series has at least One singularity in every
- pen interval whose length Exceeds 2πd and lies on the abscissa of
convergence. Example (trivial): ez 1 − ez =
∞
enz, ℜz < 0, Density = 1. Singularities at the points 2kπi, k ∈ Z
Short version
SLIDE 15 First Goal: Generalizing The Polya Theorem
Short version
SLIDE 16 First Goal: Generalizing The Polya Theorem
We consider Taylor-Dirichlet series
∞
µn−1
cn,kzk
associated to a multiplicity sequence Λ = {λn, µn}∞
n=1
{λn, µn}∞
n=1 := {λ1, λ1, . . . , λ1
, λ2, λ2, . . . , λ2
, . . . , λk, λk, . . . , λk
, . . . }
Short version
SLIDE 17 First Goal: Generalizing The Polya Theorem
We consider Taylor-Dirichlet series
∞
µn−1
cn,kzk
associated to a multiplicity sequence Λ = {λn, µn}∞
n=1
{λn, µn}∞
n=1 := {λ1, λ1, . . . , λ1
, λ2, λ2, . . . , λ2
, . . . , λk, λk, . . . , λk
, . . . } {λn}∞
n=1 is a strictly increasing sequence of positive real numbers
diverging to infinity,
Short version
SLIDE 18 First Goal: Generalizing The Polya Theorem
We consider Taylor-Dirichlet series
∞
µn−1
cn,kzk
associated to a multiplicity sequence Λ = {λn, µn}∞
n=1
{λn, µn}∞
n=1 := {λ1, λ1, . . . , λ1
, λ2, λ2, . . . , λ2
, . . . , λk, λk, . . . , λk
, . . . } {λn}∞
n=1 is a strictly increasing sequence of positive real numbers
diverging to infinity, AND {µn}∞
n=1 is a sequence of positive integers,
Not Necessarily Bounded.
Short version
SLIDE 19 First Goal: Generalizing The Polya Theorem
We consider Taylor-Dirichlet series
∞
µn−1
cn,kzk
associated to a multiplicity sequence Λ = {λn, µn}∞
n=1
{λn, µn}∞
n=1 := {λ1, λ1, . . . , λ1
, λ2, λ2, . . . , λ2
, . . . , λk, λk, . . . , λk
, . . . } {λn}∞
n=1 is a strictly increasing sequence of positive real numbers
diverging to infinity, AND {µn}∞
n=1 is a sequence of positive integers,
Not Necessarily Bounded. We impose two conditions:
Short version
SLIDE 20 First Goal: Generalizing The Polya Theorem
We consider Taylor-Dirichlet series
∞
µn−1
cn,kzk
associated to a multiplicity sequence Λ = {λn, µn}∞
n=1
{λn, µn}∞
n=1 := {λ1, λ1, . . . , λ1
, λ2, λ2, . . . , λ2
, . . . , λk, λk, . . . , λk
, . . . } {λn}∞
n=1 is a strictly increasing sequence of positive real numbers
diverging to infinity, AND {µn}∞
n=1 is a sequence of positive integers,
Not Necessarily Bounded. We impose two conditions: (A) Λ has Density d counting multiplicities
Short version
SLIDE 21 First Goal: Generalizing The Polya Theorem
We consider Taylor-Dirichlet series
∞
µn−1
cn,kzk
associated to a multiplicity sequence Λ = {λn, µn}∞
n=1
{λn, µn}∞
n=1 := {λ1, λ1, . . . , λ1
, λ2, λ2, . . . , λ2
, . . . , λk, λk, . . . , λk
, . . . } {λn}∞
n=1 is a strictly increasing sequence of positive real numbers
diverging to infinity, AND {µn}∞
n=1 is a sequence of positive integers,
Not Necessarily Bounded. We impose two conditions: (A) Λ has Density d counting multiplicities lim
t→∞
nΛ(t) t = d < ∞, nΛ(t) =
µn
Short version
SLIDE 22 First Goal: Generalizing The Polya Theorem
We consider Taylor-Dirichlet series
∞
µn−1
cn,kzk
associated to a multiplicity sequence Λ = {λn, µn}∞
n=1
{λn, µn}∞
n=1 := {λ1, λ1, . . . , λ1
, λ2, λ2, . . . , λ2
, . . . , λk, λk, . . . , λk
, . . . } {λn}∞
n=1 is a strictly increasing sequence of positive real numbers
diverging to infinity, AND {µn}∞
n=1 is a sequence of positive integers,
Not Necessarily Bounded. We impose two conditions: (A) Λ has Density d counting multiplicities lim
t→∞
nΛ(t) t = d < ∞, nΛ(t) =
µn (if µn = 1 for all n ∈ N then n/λn → d)
Short version
SLIDE 23 First Goal: Generalizing The Polya Theorem
We consider Taylor-Dirichlet series
∞
µn−1
cn,kzk
associated to a multiplicity sequence Λ = {λn, µn}∞
n=1
{λn, µn}∞
n=1 := {λ1, λ1, . . . , λ1
, λ2, λ2, . . . , λ2
, . . . , λk, λk, . . . , λk
, . . . } {λn}∞
n=1 is a strictly increasing sequence of positive real numbers
diverging to infinity, AND {µn}∞
n=1 is a sequence of positive integers,
Not Necessarily Bounded. We impose two conditions: (A) Λ has Density d counting multiplicities lim
t→∞
nΛ(t) t = d < ∞, nΛ(t) =
µn (if µn = 1 for all n ∈ N then n/λn → d) (B) λn+1 − λn > c > 0, (Uniformly Separated).
Short version
SLIDE 24 Region of Convergence, Taylor-Dirichlet series:
Short version
SLIDE 25 Region of Convergence, Taylor-Dirichlet series:
Assuming (A) and (B) then Λ = {λn, µn}∞
n=1 satisfies
lim
n→∞
log n λn = 0 lim
n→∞
µn λn = 0.
Short version
SLIDE 26 Region of Convergence, Taylor-Dirichlet series:
Assuming (A) and (B) then Λ = {λn, µn}∞
n=1 satisfies
lim
n→∞
log n λn = 0 lim
n→∞
µn λn = 0. Valiron (1929) :
Short version
SLIDE 27 Region of Convergence, Taylor-Dirichlet series:
Assuming (A) and (B) then Λ = {λn, µn}∞
n=1 satisfies
lim
n→∞
log n λn = 0 lim
n→∞
µn λn = 0. Valiron (1929) : consider the series g(z) =
∞
µn−1
cn,kzk
Cn = max{|cn,k| : k = 0, 1, 2, . . . , µn−1} lim sup
n→∞
log Cn λn = ξ ∈ R, P−ξ := {z : ℜz < −ξ}.
Short version
SLIDE 28 Region of Convergence, Taylor-Dirichlet series:
Assuming (A) and (B) then Λ = {λn, µn}∞
n=1 satisfies
lim
n→∞
log n λn = 0 lim
n→∞
µn λn = 0. Valiron (1929) : consider the series g(z) =
∞
µn−1
cn,kzk
Cn = max{|cn,k| : k = 0, 1, 2, . . . , µn−1} lim sup
n→∞
log Cn λn = ξ ∈ R, P−ξ := {z : ℜz < −ξ}. Then g(z) is an analytic function in the left half-plane P−ξ converging uniformly on compact subsets.
Short version
SLIDE 29 Region of Convergence, Taylor-Dirichlet series:
Assuming (A) and (B) then Λ = {λn, µn}∞
n=1 satisfies
lim
n→∞
log n λn = 0 lim
n→∞
µn λn = 0. Valiron (1929) : consider the series g(z) =
∞
µn−1
cn,kzk
Cn = max{|cn,k| : k = 0, 1, 2, . . . , µn−1} lim sup
n→∞
log Cn λn = ξ ∈ R, P−ξ := {z : ℜz < −ξ}. Then g(z) is an analytic function in the left half-plane P−ξ converging uniformly on compact subsets. We call the line ℜz = −ξ the abscissa of convergence for g(z).
Short version
SLIDE 30 Region of Convergence, Taylor-Dirichlet series:
Assuming (A) and (B) then Λ = {λn, µn}∞
n=1 satisfies
lim
n→∞
log n λn = 0 lim
n→∞
µn λn = 0. Valiron (1929) : consider the series g(z) =
∞
µn−1
cn,kzk
Cn = max{|cn,k| : k = 0, 1, 2, . . . , µn−1} lim sup
n→∞
log Cn λn = ξ ∈ R, P−ξ := {z : ℜz < −ξ}. Then g(z) is an analytic function in the left half-plane P−ξ converging uniformly on compact subsets. We call the line ℜz = −ξ the abscissa of convergence for g(z). Question :
Short version
SLIDE 31 Region of Convergence, Taylor-Dirichlet series:
Assuming (A) and (B) then Λ = {λn, µn}∞
n=1 satisfies
lim
n→∞
log n λn = 0 lim
n→∞
µn λn = 0. Valiron (1929) : consider the series g(z) =
∞
µn−1
cn,kzk
Cn = max{|cn,k| : k = 0, 1, 2, . . . , µn−1} lim sup
n→∞
log Cn λn = ξ ∈ R, P−ξ := {z : ℜz < −ξ}. Then g(z) is an analytic function in the left half-plane P−ξ converging uniformly on compact subsets. We call the line ℜz = −ξ the abscissa of convergence for g(z). Question : is it True that in every interval having length greater than 2πd on the line ℜz = −ξ, the series has at least One singularity?
Short version
SLIDE 32 Positive Answers to the Singularity Question
Short version
SLIDE 33 Positive Answers to the Singularity Question
Suppose that Λ = {λn, µn} satisfies (A) Λ has Density d : lim
t→∞
t = d < ∞, (B) λn+1 − λn > c > 0, (Uniformly Separated).
Short version
SLIDE 34 Positive Answers to the Singularity Question
Suppose that Λ = {λn, µn} satisfies (A) Λ has Density d : lim
t→∞
t = d < ∞, (B) λn+1 − λn > c > 0, (Uniformly Separated).
◮ Zikkos (2005 Complex Variables):
Short version
SLIDE 35 Positive Answers to the Singularity Question
Suppose that Λ = {λn, µn} satisfies (A) Λ has Density d : lim
t→∞
t = d < ∞, (B) λn+1 − λn > 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.
Short version
SLIDE 36 Positive Answers to the Singularity Question
Suppose that Λ = {λn, µn} satisfies (A) Λ has Density d : lim
t→∞
t = d < ∞, (B) λn+1 − λn > 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. ):
Short version
SLIDE 37 Positive Answers to the Singularity Question
Suppose that Λ = {λn, µn} satisfies (A) Λ has Density d : lim
t→∞
t = d < ∞, (B) λn+1 − λn > 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 SΛ is Equal to 0,
Short version
SLIDE 38 Positive Answers to the Singularity Question
Suppose that Λ = {λn, µn} satisfies (A) Λ has Density d : lim
t→∞
t = d < ∞, (B) λn+1 − λn > 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 SΛ is Equal to 0, then the answer is YES.
Short version
SLIDE 39 Another Positive Answer
◮ Zikkos (2018):
Short version
SLIDE 40 Another Positive Answer
◮ Zikkos (2018):
If Λ belongs to the class U(d, 0), then the Krivosheev characteristic SΛ = 0,
Short version
SLIDE 41 Another Positive Answer
◮ Zikkos (2018):
If Λ belongs to the class U(d, 0), then the Krivosheev characteristic SΛ = 0, hence the answer is YES.
Short version
SLIDE 42 Another Positive Answer
◮ Zikkos (2018):
If Λ belongs to the class U(d, 0), then the Krivosheev characteristic SΛ = 0, hence the answer is YES. Examples in U(d, 0) :
Short version
SLIDE 43 Another Positive Answer
◮ Zikkos (2018):
If Λ belongs to the class U(d, 0), then the Krivosheev characteristic SΛ = 0, hence the answer is YES. Examples in U(d, 0) : (1) If (A) and (B) hold and µn = O(1), then Λ ∈ U(d, 0).
Short version
SLIDE 44 Another Positive Answer
◮ Zikkos (2018):
If Λ belongs to the class U(d, 0), then the Krivosheev characteristic SΛ = 0, hence the answer is YES. Examples in U(d, 0) : (1) If (A) and (B) hold and µn = O(1), then Λ ∈ U(d, 0). (2) Let {pn} be the prime numbers
Short version
SLIDE 45 Another Positive Answer
◮ Zikkos (2018):
If Λ belongs to the class U(d, 0), then the Krivosheev characteristic SΛ = 0, hence the answer is YES. Examples in U(d, 0) : (1) If (A) and (B) hold and µn = O(1), then Λ ∈ U(d, 0). (2) Let {pn} be the prime numbers and let µn = pn+1 − pn.
Short version
SLIDE 46 Another Positive Answer
◮ Zikkos (2018):
If Λ belongs to the class U(d, 0), then the Krivosheev characteristic SΛ = 0, hence the answer is YES. Examples in U(d, 0) : (1) If (A) and (B) hold and µn = O(1), then Λ ∈ U(d, 0). (2) Let {pn} be the prime numbers and let µn = pn+1 − pn. Then Λ = {pn, µn} belongs to the class U(1, 0).
Short version
SLIDE 47 Another Positive Answer
◮ Zikkos (2018):
If Λ belongs to the class U(d, 0), then the Krivosheev characteristic SΛ = 0, hence the answer is YES. Examples in U(d, 0) : (1) If (A) and (B) hold and µn = O(1), then Λ ∈ U(d, 0). (2) Let {pn} be the prime numbers and let µn = pn+1 − pn. Then Λ = {pn, µn} belongs to the class U(1, 0).
Theorem
The Taylor-Dirichlet series g(z) =
∞
µn−1
zk
cn,k ∈ C defines an analytic function in the half-plane {z : ℜz < 0}
Short version
SLIDE 48 Another Positive Answer
◮ Zikkos (2018):
If Λ belongs to the class U(d, 0), then the Krivosheev characteristic SΛ = 0, hence the answer is YES. Examples in U(d, 0) : (1) If (A) and (B) hold and µn = O(1), then Λ ∈ U(d, 0). (2) Let {pn} be the prime numbers and let µn = pn+1 − pn. Then Λ = {pn, µn} belongs to the class U(1, 0).
Theorem
The Taylor-Dirichlet series g(z) =
∞
µn−1
zk
cn,k ∈ C defines an analytic function in the half-plane {z : ℜz < 0} and it has at least One singularity
Short version
SLIDE 49 Another Positive Answer
◮ Zikkos (2018):
If Λ belongs to the class U(d, 0), then the Krivosheev characteristic SΛ = 0, hence the answer is YES. Examples in U(d, 0) : (1) If (A) and (B) hold and µn = O(1), then Λ ∈ U(d, 0). (2) Let {pn} be the prime numbers and let µn = pn+1 − pn. Then Λ = {pn, µn} belongs to the class U(1, 0).
Theorem
The Taylor-Dirichlet series g(z) =
∞
µn−1
zk
cn,k ∈ C defines an analytic function in the half-plane {z : ℜz < 0} and it has at least One singularity in every open interval of length exceeding 2π and lying on the Imaginary axis.
Short version
SLIDE 50 A Negative Answer
Short version
SLIDE 51 A Negative Answer
Zikkos ( Ufa Math J.):
Short version
SLIDE 52 A Negative Answer
Zikkos ( Ufa Math J.): for every d ≥ 0,
Short version
SLIDE 53 A Negative Answer
Zikkos ( Ufa Math J.): for every d ≥ 0, there exists a multiplicity sequence Λ = {λn, µn} with µn → ∞, such that
Short version
SLIDE 54 A Negative Answer
Zikkos ( Ufa Math J.): for every d ≥ 0, there exists a multiplicity sequence Λ = {λn, µn} with µn → ∞, such that (A) Λ has Density d : lim
t→∞
t = d < ∞, (B) λn+1 − λn > c > 0, (Uniformly Separated).
Short version
SLIDE 55 A Negative Answer
Zikkos ( Ufa Math J.): for every d ≥ 0, there exists a multiplicity sequence Λ = {λn, µn} with µn → ∞, such that (A) Λ has Density d : lim
t→∞
t = d < ∞, (B) λn+1 − λn > c > 0, (Uniformly Separated). (C) SΛ < 0
Short version
SLIDE 56 A Negative Answer
Zikkos ( Ufa Math J.): for every d ≥ 0, there exists a multiplicity sequence Λ = {λn, µn} with µn → ∞, such that (A) Λ has Density d : lim
t→∞
t = d < ∞, (B) λn+1 − λn > c > 0, (Uniformly Separated). (C) SΛ < 0 and hence (Krivosheeva 2012 St. Petersburg Math. J.):
Short version
SLIDE 57 A Negative Answer
Zikkos ( Ufa Math J.): for every d ≥ 0, there exists a multiplicity sequence Λ = {λn, µn} with µn → ∞, such that (A) Λ has Density d : lim
t→∞
t = d < ∞, (B) λn+1 − λn > c > 0, (Uniformly Separated). (C) SΛ < 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.
Short version
SLIDE 58 The class U(d, 0)
Short version
SLIDE 59 The class U(d, 0)
Zikkos (2005 Complex Variables, 2010 CMFT) :
Short version
SLIDE 60 The class U(d, 0)
Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence {an} of positive real numbers, having density d with uniformly separated terms n/an → d, an+1 − an > c > 0.
Short version
SLIDE 61 The class U(d, 0)
Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence {an} of positive real numbers, having density d with uniformly separated terms n/an → d, an+1 − an > c > 0. Choose two positive numbers α < 1, δ < c.
Short version
SLIDE 62 The class U(d, 0)
Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence {an} of positive real numbers, having density d with uniformly separated terms n/an → d, an+1 − an > c > 0. Choose two positive numbers α < 1, δ < c. For each term an consider the closed disk B(an, |an|α) = {z : |z − an| ≤ aα
n }.
Short version
SLIDE 63 The class U(d, 0)
Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence {an} of positive real numbers, having density d with uniformly separated terms n/an → d, an+1 − an > c > 0. Choose two positive numbers α < 1, δ < c. For each term an consider the closed disk B(an, |an|α) = {z : |z − an| ≤ aα
n }.
Choose a point in B(an, |an|α) ∩ R, call it bn, in an almost arbitrary way,
Short version
SLIDE 64 The class U(d, 0)
Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence {an} of positive real numbers, having density d with uniformly separated terms n/an → d, an+1 − an > c > 0. Choose two positive numbers α < 1, δ < c. For each term an consider the closed disk B(an, |an|α) = {z : |z − an| ≤ aα
n }.
Choose a point in B(an, |an|α) ∩ R, call it bn, in an almost arbitrary way, such that for all n = m either (I) bm = bn
Short version
SLIDE 65 The class U(d, 0)
Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence {an} of positive real numbers, having density d with uniformly separated terms n/an → d, an+1 − an > c > 0. Choose two positive numbers α < 1, δ < c. For each term an consider the closed disk B(an, |an|α) = {z : |z − an| ≤ aα
n }.
Choose a point in B(an, |an|α) ∩ R, call it bn, in an almost arbitrary way, such that for all n = m either (I) bm = bn
(II) |bm − bn| ≥ δ.
Short version
SLIDE 66 The class U(d, 0)
Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence {an} of positive real numbers, having density d with uniformly separated terms n/an → d, an+1 − an > c > 0. Choose two positive numbers α < 1, δ < c. For each term an consider the closed disk B(an, |an|α) = {z : |z − an| ≤ aα
n }.
Choose a point in B(an, |an|α) ∩ R, call it bn, in an almost arbitrary way, such that for all n = m either (I) bm = bn
(II) |bm − bn| ≥ δ. Rename {bn} into Λ = {λn, µn}.
Short version
SLIDE 67 The class U(d, 0)
Zikkos (2005 Complex Variables, 2010 CMFT) : Consider a strictly increasing sequence {an} of positive real numbers, having density d with uniformly separated terms n/an → d, an+1 − an > c > 0. Choose two positive numbers α < 1, δ < c. For each term an consider the closed disk B(an, |an|α) = {z : |z − an| ≤ aα
n }.
Choose a point in B(an, |an|α) ∩ R, call it bn, in an almost arbitrary way, such that for all n = m either (I) bm = bn
(II) |bm − bn| ≥ δ. Rename {bn} into Λ = {λn, µn}. Then we say that Λ ∈ U(d, 0).
Short version
SLIDE 68 The Class U(d, 0)
Short version
SLIDE 69
The Class U(d, 0)
an
SLIDE 70 The Class U(d, 0)
an R = aα
n
SLIDE 71 The Class U(d, 0)
an R = aα
n
bn
SLIDE 72 The Class U(d, 0)
an R = aα
n
bn an+1
SLIDE 73 The Class U(d, 0)
an R = aα
n
bn an+1 R = aα
n+1
SLIDE 74 The Class U(d, 0)
an R = aα
n
bn an+1 R = aα
n+1
bn+1
SLIDE 75 The Class U(d, 0)
an R = aα
n
bn an+1 R = aα
n+1
bn+1 an+2
SLIDE 76 The Class U(d, 0)
an R = aα
n
bn an+1 R = aα
n+1
bn+1 an+2 R = aα
n+2
SLIDE 77 The Class U(d, 0)
an R = aα
n
bn an+1 R = aα
n+1
bn+1 an+2 R = aα
n+2
= bn+2
SLIDE 78 The Class U(d, 0)
an R = aα
n
bn an+1 R = aα
n+1
bn+1 an+2 R = aα
n+2
= bn+2 an+3
SLIDE 79 The Class U(d, 0)
an R = aα
n
bn an+1 R = aα
n+1
bn+1 an+2 R = aα
n+2
= bn+2 an+3 R = aα
n+3
SLIDE 80 The Class U(d, 0)
an R = aα
n
bn an+1 R = aα
n+1
bn+1 an+2 R = aα
n+2
= bn+2 an+3 R = aα
n+3
= bn+3
Short version
SLIDE 81 Singularities of Taylor-Dirichlet series
Short version
SLIDE 82 Singularities of Taylor-Dirichlet series
Theorem A
Let the multiplicity-sequence Λ = {λn, µn}∞
n=1 belong to the class U(d, 0)
for some d > 0, and consider the Taylor-Dirichlet series g(z) =
∞
µn−1
cn,kzk
cn,k ∈ C lim sup
n→∞
log Cn λn = ξ ∈ R, where Cn = max{|cn,k| : k = 0, 1, . . . , µn−1}.
Short version
SLIDE 83 Singularities of Taylor-Dirichlet series
Theorem A
Let the multiplicity-sequence Λ = {λn, µn}∞
n=1 belong to the class U(d, 0)
for some d > 0, and consider the Taylor-Dirichlet series g(z) =
∞
µn−1
cn,kzk
cn,k ∈ C lim sup
n→∞
log Cn λn = ξ ∈ R, where Cn = max{|cn,k| : k = 0, 1, . . . , µn−1}. Then g(z) defines an analytic function in the half-plane {z : ℜz < −ξ} and it has at least One singularity in every open interval of length exceeding 2πd and lying on the line ℜz = −ξ.
Short version
SLIDE 84 Second Goal
Short version
SLIDE 85 Second Goal
Given Λ = {λn, µn}∞
n=1 in U(d, 0)
Short version
SLIDE 86 Second Goal
Given Λ = {λn, µn}∞
n=1 in U(d, 0)
Characterize the closed span of the exponential system EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}
Short version
SLIDE 87 Second Goal
Given Λ = {λn, µn}∞
n=1 in U(d, 0)
Characterize the closed span of the exponential system EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1} in Lp(l) spaces where l is a simple closed rectifiable curve in C, and Gl is the domain bounded by the curve.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Short version
SLIDE 88 Second Goal
Given Λ = {λn, µn}∞
n=1 in U(d, 0)
Characterize the closed span of the exponential system EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1} in Lp(l) spaces where l is a simple closed rectifiable curve in C, and Gl is the domain bounded by the curve.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
If f is in the closed span of EΛ in Lp(l),
Short version
SLIDE 89 Second Goal
Given Λ = {λn, µn}∞
n=1 in U(d, 0)
Characterize the closed span of the exponential system EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1} in Lp(l) spaces where l is a simple closed rectifiable curve in C, and Gl is the domain bounded by the curve.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
If f is in the closed span of EΛ in Lp(l), then f is in the Lp closure of polynomials,
Short version
SLIDE 90 Second Goal
Given Λ = {λn, µn}∞
n=1 in U(d, 0)
Characterize the closed span of the exponential system EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1} in Lp(l) spaces where l is a simple closed rectifiable curve in C, and Gl is the domain bounded by the curve.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
If f is in the closed span of EΛ in Lp(l), then f is in the Lp closure of polynomials, hence f ∈ E p(Gl).
Short version
SLIDE 91 Curve l is surrounded by a rectangle whose height is less than 2πd
Short version
SLIDE 92 Curve l is surrounded by a rectangle whose height is less than 2πd
Height < 2πd
Theorem B
Suppose the Domain Gl bounded by the curve l is a Smirnov domain.
Short version
SLIDE 93 Curve l is surrounded by a rectangle whose height is less than 2πd
Height < 2πd
Theorem B
Suppose the Domain Gl bounded by the curve l is a Smirnov domain. Suppose also that Λ = {λn, µn} has Density d.
Short version
SLIDE 94 Curve l is surrounded by a rectangle whose height is less than 2πd
Height < 2πd
Theorem B
Suppose the Domain Gl bounded by the curve l is a Smirnov domain. Suppose also that Λ = {λn, µn} has Density d. Then the closed span of the exponential system EΛ in the space Lp(l) for p ≥ 1
Short version
SLIDE 95 Curve l is surrounded by a rectangle whose height is less than 2πd
Height < 2πd
Theorem B
Suppose the Domain Gl bounded by the curve l is a Smirnov domain. Suppose also that Λ = {λn, µn} has Density d. Then the closed span of the exponential system EΛ in the space Lp(l) for p ≥ 1 Coincides with the Smirnov space E p(Gl).
Short version
SLIDE 96 Proof
It is enough to show that E p(Gl) is a subspace of the closed span of the exponential system EΛ in Lp(l).
Short version
SLIDE 97 Proof
It is enough to show that E p(Gl) is a subspace of the closed span of the exponential system EΛ in Lp(l). Since Gl is a Smirnov domain we have to show that the Lp closure of polynomials is a subspace of the closed span of the exponential system EΛ in Lp(l).
Short version
SLIDE 98 Proof
It is enough to show that E p(Gl) is a subspace of the closed span of the exponential system EΛ in Lp(l). Since Gl is a Smirnov domain we have to show that the Lp closure of polynomials is a subspace of the closed span of the exponential system EΛ in Lp(l). Let H(K) be the space of functions analytic in the rectangle K with the topology of uniform convergence on compact subsets.
Short version
SLIDE 99 Proof
It is enough to show that E p(Gl) is a subspace of the closed span of the exponential system EΛ in Lp(l). Since Gl is a Smirnov domain we have to show that the Lp closure of polynomials is a subspace of the closed span of the exponential system EΛ in Lp(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):
Short version
SLIDE 100 Proof
It is enough to show that E p(Gl) is a subspace of the closed span of the exponential system EΛ in Lp(l). Since Gl is a Smirnov domain we have to show that the Lp closure of polynomials is a subspace of the closed span of the exponential system EΛ in Lp(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,
Short version
SLIDE 101 Proof
It is enough to show that E p(Gl) is a subspace of the closed span of the exponential system EΛ in Lp(l). Since Gl is a Smirnov domain we have to show that the Lp closure of polynomials is a subspace of the closed span of the exponential system EΛ in Lp(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, AND the height of the rectangle is less than 2πd,
Short version
SLIDE 102 Proof
It is enough to show that E p(Gl) is a subspace of the closed span of the exponential system EΛ in Lp(l). Since Gl is a Smirnov domain we have to show that the Lp closure of polynomials is a subspace of the closed span of the exponential system EΛ in Lp(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, AND the height of the rectangle is less than 2πd, then the system EΛ is Complete in H(K).
Short version
SLIDE 103 Proof
It is enough to show that E p(Gl) is a subspace of the closed span of the exponential system EΛ in Lp(l). Since Gl is a Smirnov domain we have to show that the Lp closure of polynomials is a subspace of the closed span of the exponential system EΛ in Lp(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, AND the height of the rectangle is less than 2πd, then the system EΛ is Complete in H(K). Hence polynomials are approximated uniformly on the curve l by exponential polynomials.
Short version
SLIDE 104 The curve l is Surrounding a rectangle whose height is 2πd
Short version
SLIDE 105 The curve l is Surrounding a rectangle whose height is 2πd
Height of rectangle ≥ 2πd
Short version
SLIDE 106 The curve l is Surrounding a rectangle whose height is 2πd
Height of rectangle ≥ 2πd
Theorem C
Suppose that Λ = {λn, µn} has Density d.
Short version
SLIDE 107 The curve l is Surrounding a rectangle whose height is 2πd
Height of rectangle ≥ 2πd
Theorem C
Suppose that Λ = {λn, µn} has Density d. Then the closed span of the exponential system EΛ in the space Lp(l) for p ≥ 1 is a Proper subspace
- f the Smirnov space E p(Gl).
- E. Zikkos
Short version
SLIDE 108 The curve l is Surrounding a rectangle whose height is 2πd
Height of rectangle ≥ 2πd
Theorem C
Suppose that Λ = {λn, µn} has Density d. Then the closed span of the exponential system EΛ in the space Lp(l) for p ≥ 1 is a Proper subspace
- f the Smirnov space E p(Gl).
For any λ / ∈ {λn}, the function eλz does not belong to the closed span of the system.
Short version
SLIDE 109 The curve l is Surrounding a rectangle whose height is 2πd
Height of rectangle ≥ 2πd
Theorem C
Suppose that Λ = {λn, µn} has Density d. Then the closed span of the exponential system EΛ in the space Lp(l) for p ≥ 1 is a Proper subspace
- f the Smirnov space E p(Gl).
For any λ / ∈ {λn}, the function eλz does not belong to the closed span of the system. Question:
Short version
SLIDE 110 The curve l is Surrounding a rectangle whose height is 2πd
Height of rectangle ≥ 2πd
Theorem C
Suppose that Λ = {λn, µn} has Density d. Then the closed span of the exponential system EΛ in the space Lp(l) for p ≥ 1 is a Proper subspace
- f the Smirnov space E p(Gl).
For any λ / ∈ {λn}, the function eλz does not belong to the closed span of the system. Question: Can we characterize the closed span of the exponential system EΛ in the space Lp(l) for p ≥ 1?
Short version
SLIDE 111 The curve l is Surrounding a rectangle whose height is 2πd
Height of rectangle ≥ 2πd
Theorem C
Suppose that Λ = {λn, µn} has Density d. Then the closed span of the exponential system EΛ in the space Lp(l) for p ≥ 1 is a Proper subspace
- f the Smirnov space E p(Gl).
For any λ / ∈ {λn}, the function eλz does not belong to the closed span of the system. Question: Can we characterize the closed span of the exponential system EΛ in the space Lp(l) for p ≥ 1? We give an answer when Λ ∈ U(d, 0).
Short version
SLIDE 112 Characterizing the closed span of EΛ
Short version
SLIDE 113 Characterizing the closed span of EΛ
Let Λ belong to the class U(d, 0). Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}.
Short version
SLIDE 114
Characterizing the closed span of EΛ
Let Λ belong to the class U(d, 0). Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}. Curve ld, Domain Gld length > 2πd
SLIDE 115
Characterizing the closed span of EΛ
Let Λ belong to the class U(d, 0). Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}. Curve ld, Domain Gld length > 2πd Sld the set of all such line segments
SLIDE 116
Characterizing the closed span of EΛ
Let Λ belong to the class U(d, 0). Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}. Curve ld, Domain Gld length > 2πd Sld the set of all such line segments qld := sup{ℜz : ∀ z ∈ Sld}
SLIDE 117
Characterizing the closed span of EΛ
Let Λ belong to the class U(d, 0). Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}. Curve ld, Domain Gld length > 2πd Sld the set of all such line segments qld := sup{ℜz : ∀ z ∈ Sld}
SLIDE 118
Characterizing the closed span of EΛ
Let Λ belong to the class U(d, 0). Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}. Curve ld, Domain Gld length > 2πd Sld the set of all such line segments qld := sup{ℜz : ∀ z ∈ Sld} If f ∈ span(EΛ) in Lp(ld),
SLIDE 119 Characterizing the closed span of EΛ
Let Λ belong to the class U(d, 0). Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}. Curve ld, Domain Gld length > 2πd Sld the set of all such line segments qld := sup{ℜz : ∀ z ∈ Sld} If f ∈ span(EΛ) in Lp(ld), f extends analytically in ℜz < qld as a Taylor-Dirichlet series
Short version
SLIDE 120 The closed span of EΛ in Lp(ld)
Short version
SLIDE 121 The closed span of EΛ in Lp(ld)
Theorem D
Let Λ = {λn, µn}∞
n=1 ∈ U(d, 0) and consider an ld curve and its qld
constant.
◮ Then every function f belonging to the closed span of EΛ in Lp(ld)
for p ≥ 1, not only extends analytically in the domain Gld and belongs to the Smirnov space E p(Gld).
◮ But it is also extended analytically in the half-plane
Hqld := {z : ℜz < qld}, admitting a unique Taylor-Dirichlet series representation of the form g(z) =
∞
µn−1
cn,kzk
cn,k ∈ C, ∀ z ∈ Hqld with the series converging uniformly on compact subsets of Hqld .
Short version
SLIDE 122 Crucial Tool: Distances in Lp(ld)
Short version
SLIDE 123 Crucial Tool: Distances in Lp(ld)
Suppose that Λ = {λn, µn}∞
n=1 belongs to the class U(d, 0) and consider
an ld curve and its qld constant. Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}.
Short version
SLIDE 124 Crucial Tool: Distances in Lp(ld)
Suppose that Λ = {λn, µn}∞
n=1 belongs to the class U(d, 0) and consider
an ld curve and its qld constant. Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}. Let pn,k(z) := zkeλnz And EΛn,k := EΛ \ {pn,k}.
Short version
SLIDE 125 Crucial Tool: Distances in Lp(ld)
Suppose that Λ = {λn, µn}∞
n=1 belongs to the class U(d, 0) and consider
an ld curve and its qld constant. Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}. Let pn,k(z) := zkeλnz And EΛn,k := EΛ \ {pn,k}. Define the Distance between pn,k and the closed span of EΛn,k in Lp(ld) Dp,n,k := inf
g∈span(EΛn,k ) ||pn,k − g||Lp(ld)
Short version
SLIDE 126 Crucial Tool: Distances in Lp(ld)
Suppose that Λ = {λn, µn}∞
n=1 belongs to the class U(d, 0) and consider
an ld curve and its qld constant. Let EΛ = {zkeλnz : n ∈ N, k = 0, 1, . . . , µn − 1}. Let pn,k(z) := zkeλnz And EΛn,k := EΛ \ {pn,k}. Define the Distance between pn,k and the closed span of EΛn,k in Lp(ld) Dp,n,k := inf
g∈span(EΛn,k ) ||pn,k − g||Lp(ld)
Theorem E
For every ǫ > 0 there is a constant uǫ > 0, independent of p ≥ 1, n ∈ N and k = 0, 1, . . . , µn − 1, but depending on Λ the curve ld, so that Dp,n,k ≥ uǫe(qld −ǫ)λn.
Short version
SLIDE 127 A Biorthogonal sequence to EΛ in E 2(Gld) and a solution to a Moment Problem
Theorem F
◮ Let Λ = {λn, µn}∞ n=1 belong to the class U(d, 0) and consider an ld
curve and its qld constant.
Short version
SLIDE 128 A Biorthogonal sequence to EΛ in E 2(Gld) and a solution to a Moment Problem
Theorem F
◮ Let Λ = {λn, µn}∞ n=1 belong to the class U(d, 0) and consider an ld
curve and its qld constant. Then there exists a family of functions {rn,k ∈ E 2(Gld) : n ∈ N, k = 0, 1, . . . , µn − 1} such that this family is the Unique Biorthogonal sequence to the system EΛ in E 2(Gld), belonging to span(EΛ) in E 2(Gld).
Short version
SLIDE 129 A Biorthogonal sequence to EΛ in E 2(Gld) and a solution to a Moment Problem
Theorem F
◮ Let Λ = {λn, µn}∞ n=1 belong to the class U(d, 0) and consider an ld
curve and its qld constant. Then there exists a family of functions {rn,k ∈ E 2(Gld) : n ∈ N, k = 0, 1, . . . , µn − 1} such that this family is the Unique Biorthogonal sequence to the system EΛ in E 2(Gld), belonging to span(EΛ) in E 2(Gld).
◮ Moreover, for every ǫ > 0 there is a constant mǫ > 0, independent
- f n and k, but depending on Λ and the curve ld, so that
||rn,k||E 2(Gld ) ≤ mǫe(−qld +ǫ)λn, ∀ n ∈ N, k = 0, 1, . . . , µn − 1.
Short version
SLIDE 130 ◮ Let {dn,k : n ∈ N, k = 0, 1, . . . , µn − 1} be a doubly-indexed
sequence of complex numbers such that lim sup
n→∞
log An λn < qld where An = max{|dn,k| : k = 0, 1, . . . , µn−1}.
Short version
SLIDE 131 ◮ Let {dn,k : n ∈ N, k = 0, 1, . . . , µn − 1} be a doubly-indexed
sequence of complex numbers such that lim sup
n→∞
log An λn < qld where An = max{|dn,k| : k = 0, 1, . . . , µn−1}. Then the function f (z) :=
∞
µn−1
dn,krn,k(z)
Short version
SLIDE 132 ◮ Let {dn,k : n ∈ N, k = 0, 1, . . . , µn − 1} be a doubly-indexed
sequence of complex numbers such that lim sup
n→∞
log An λn < qld where An = max{|dn,k| : k = 0, 1, . . . , µn−1}. Then the function f (z) :=
∞
µn−1
dn,krn,k(z)
- belongs to E 2(Gld) and it is a solution to the moment problem
- ld
zkeλnzf (z) |dz| = dn,k ∀ n ∈ N and k = 0, 1, 2, . . . µn − 1.
Short version
SLIDE 133
- A. S. Krivosheev, A fundamental principle for invariant subspaces in
convex domains, Izv. Ross. Acad. Nauk Ser. Mat. 68 no. 2 (2004), 71-136, English transl., Izv. Math. 68 no. 2 (2004), 291-353.
- O. A. Krivosheeva, Singular points of the sum of a series of
exponential monomials on the boundary of the convergence domain, Algebra i Analiz 23 no. 2 (2011), 162-205; English transl., St. Petersburg Math. J. 23 no. 2 (2012), 321-350.
- O. A. Krivosheeva, A. S. Krivosheev, Singular Points of the Sum of a
Dirichlet Series on the Convergence Line, Funktsional. Anal. i
- Prilozhen. 49, no. 2 (2015), 54-69; English transl., Funct. Anal.
- Appl. 49 no. 2 (2015), 122-134.
G, Polya, On converse gap theorems. Trans. Amer. Math. Soc. 52, (1942). 65-71.
- M. G. Valiron, Sur les solutions des ´
equations diff´ erentielles lin´ eaires d’ordre infini et a coefficients constants, Ann. Ecole Norm. (3) 46 (1929), 25-53.
Short version
SLIDE 134
- E. Zikkos, On a theorem of Norman Levinson and a variation of the
Fabry Gap theorem, Complex Var. and Ell. Eqns. 50 no. 4 (2005), 229-255.
- E. Zikkos, Analytic continuation of Taylor-Dirichlet series and
non-vanishing solutions of a differential equation of infinite order, CMFT 10 no. 1 (2010), 367-398.
- E. Zikkos, A Taylor-Dirichlet series with no singularities on its
abscissa of convergence.
Short version
SLIDE 135 THANK YOU VERY MUCH!!! ΣΑΣ ΕYΧΑΡΙΣΤΩ ΠΑΡ Α ΠΟΛY !!!
Short version
