Taylor Monomialsin Time Scales
Pauline Ballesteros, Juan Batista, Samantha Bell
Department of MathematicsKansas State UniversityManhattan, KS 66506
Mathematics REUJuly 24, 2012
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Time Scales
Definition
A Time Scale T is an arbitrary, non-empty closed subset of R.
Examples: R, N, [0, 1], and [−5, 0] ∪ [3, 4] ∪ [8]
Q, (0, 1), and (3, 8] are not time scales.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Time Scales
Definition
A Time Scale T is an arbitrary, non-empty closed subset of R.
Examples: R, N, [0, 1], and [−5, 0] ∪ [3, 4] ∪ [8]
Q, (0, 1), and (3, 8] are not time scales.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Time Scales
Definition
A Time Scale T is an arbitrary, non-empty closed subset of R.
Examples: R, N, [0, 1], and [−5, 0] ∪ [3, 4] ∪ [8]
Q, (0, 1), and (3, 8] are not time scales.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Operators
Definition
The forward jump operatorσ : T→ T is defined by
σ(t) := inf {s ∈ T : s > t}.
Definition
The backward jump operatorρ : T→ T is defined by
ρ(t) := sup{s ∈ T : s < t}.
left scattered ρ(t) < t
right scattered σ(t) > t
left dense ρ(t) = t
right dense σ(t) = t
isolated ρ(t) < t < σ(t)
ρ2(t) ρ(t) t σ(t) σ2(t)
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Operators
Definition
The forward jump operatorσ : T→ T is defined by
σ(t) := inf {s ∈ T : s > t}.
Definition
The backward jump operatorρ : T→ T is defined by
ρ(t) := sup{s ∈ T : s < t}.
left scattered ρ(t) < t
right scattered σ(t) > t
left dense ρ(t) = t
right dense σ(t) = t
isolated ρ(t) < t < σ(t)
ρ2(t) ρ(t) t σ(t) σ2(t)
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Operators
Definition
The forward jump operatorσ : T→ T is defined by
σ(t) := inf {s ∈ T : s > t}.
Definition
The backward jump operatorρ : T→ T is defined by
ρ(t) := sup{s ∈ T : s < t}.
left scattered ρ(t) < t
right scattered σ(t) > t
left dense ρ(t) = t
right dense σ(t) = t
isolated ρ(t) < t < σ(t)
ρ2(t) ρ(t) t σ(t) σ2(t)
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Delta Derivatives
Definition
Let f : T→ R and t ∈ T. Then f ∆(t) the real number, if it exists,such that the following holds: ∀ε > 0 ∃δε > 0 such that if|s − t| < δε and s ∈ T then
|f (σ(t))− f (s)− f ∆(t)(σ(t)− s)| ≤ ε|σ(t)− s|.
Theorem
Assume f : T→ R is a continuous function and fix t ∈ Tκ.If t is right dense, then
f ∆(t) = lims→t
f (t)− f (s)
t − s= f ′(t).
If t is right scattered, then
f ∆(t) =f (σ(t))− f (t)
µ(t).
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Delta Derivatives
Definition
Let f : T→ R and t ∈ T. Then f ∆(t) the real number, if it exists,such that the following holds: ∀ε > 0 ∃δε > 0 such that if|s − t| < δε and s ∈ T then
|f (σ(t))− f (s)− f ∆(t)(σ(t)− s)| ≤ ε|σ(t)− s|.
Theorem
Assume f : T→ R is a continuous function and fix t ∈ Tκ.If t is right dense, then
f ∆(t) = lims→t
f (t)− f (s)
t − s= f ′(t).
If t is right scattered, then
f ∆(t) =f (σ(t))− f (t)
µ(t).
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Nabla Derivatives
Definition
Let f : T→ R and t ∈ T. Then f ∇(t) is the real number, if itexists, such that the following holds:∀ε > 0 ∃δε > 0 such that if|s − t| < δε and s ∈ T then
|f (ρ(t))− f (s)− f ∇(t)(ρ(t)− s)| ≤ ε|ρ(t)− s|.
Theorem
Assume f : T→ R is a continuous function and fix t ∈ Tκ.If t is left dense, then
f ∇(t) = lims→t
f (t)− f (s)
t − s= f ′(t).
If t is left scattered, then
f ∇(t) =f (t)− f (ρ(t))
ν(t).
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Nabla Derivatives
Definition
Let f : T→ R and t ∈ T. Then f ∇(t) is the real number, if itexists, such that the following holds:∀ε > 0 ∃δε > 0 such that if|s − t| < δε and s ∈ T then
|f (ρ(t))− f (s)− f ∇(t)(ρ(t)− s)| ≤ ε|ρ(t)− s|.
Theorem
Assume f : T→ R is a continuous function and fix t ∈ Tκ.If t is left dense, then
f ∇(t) = lims→t
f (t)− f (s)
t − s= f ′(t).
If t is left scattered, then
f ∇(t) =f (t)− f (ρ(t))
ν(t).
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Existance of Pre-Antiderivatives
A function f : T→ R is:
regulated if right hand limits are finite at right dense pointsand left hand limits are finite at left dense points.
right dense continuous if it is regulated and continuous atright dense points (denoted as Crd(T)).
pre-differentiable on the region D if Tκ\D is countable and Dcontains no right scattered elements.
Theorem
Let f be regulated. Then there exists a pre-differentiable F withregion of differentiation D such that
F∆(t) = f (t) ∀t ∈ D.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Existance of Pre-Antiderivatives
A function f : T→ R is:
regulated if right hand limits are finite at right dense pointsand left hand limits are finite at left dense points.
right dense continuous if it is regulated and continuous atright dense points (denoted as Crd(T)).
pre-differentiable on the region D if Tκ\D is countable and Dcontains no right scattered elements.
Theorem
Let f be regulated. Then there exists a pre-differentiable F withregion of differentiation D such that
F∆(t) = f (t) ∀t ∈ D.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Existance of Pre-Antiderivatives
A function f : T→ R is:
regulated if right hand limits are finite at right dense pointsand left hand limits are finite at left dense points.
right dense continuous if it is regulated and continuous atright dense points (denoted as Crd(T)).
pre-differentiable on the region D if Tκ\D is countable and Dcontains no right scattered elements.
Theorem
Let f be regulated. Then there exists a pre-differentiable F withregion of differentiation D such that
F∆(t) = f (t) ∀t ∈ D.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Existance of Pre-Antiderivatives
A function f : T→ R is:
regulated if right hand limits are finite at right dense pointsand left hand limits are finite at left dense points.
right dense continuous if it is regulated and continuous atright dense points (denoted as Crd(T)).
pre-differentiable on the region D if Tκ\D is countable and Dcontains no right scattered elements.
Theorem
Let f be regulated. Then there exists a pre-differentiable F withregion of differentiation D such that
F∆(t) = f (t) ∀t ∈ D.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Existance of Pre-Antiderivatives
A function f : T→ R is:
regulated if right hand limits are finite at right dense pointsand left hand limits are finite at left dense points.
right dense continuous if it is regulated and continuous atright dense points (denoted as Crd(T)).
pre-differentiable on the region D if Tκ\D is countable and Dcontains no right scattered elements.
Theorem
Let f be regulated. Then there exists a pre-differentiable F withregion of differentiation D such that
F∆(t) = f (t) ∀t ∈ D.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Existance of Antiderivatives
Definition
Let f : T→ R be a regulated function and F be apre-differentiable function of f . Define the indefinite integral of fby ∫
f (t)∆t = F (t) + C .
Define the Cauchy integral by∫ s
rf (t)∆t = F (s)− F (r). ∀r , s ∈ T
Theorem
Every rd-continuous function has an antiderivative. If t0 ∈ T, then
F (t) :=∫ ttof (τ)∆τ for t ∈ T is an antiderivative of f .
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Existance of Antiderivatives
Definition
Let f : T→ R be a regulated function and F be apre-differentiable function of f . Define the indefinite integral of fby ∫
f (t)∆t = F (t) + C .
Define the Cauchy integral by∫ s
rf (t)∆t = F (s)− F (r). ∀r , s ∈ T
Theorem
Every rd-continuous function has an antiderivative. If t0 ∈ T, then
F (t) :=∫ ttof (τ)∆τ for t ∈ T is an antiderivative of f .
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Delta Integration
Theorem
If f ∈ Crd and t ∈ Tκ, then∫ σ(t)
tf (τ)∆τ = µ(t)f (t).
Corollary
If [a, b] consists of only isolated points, then
∫ b
af (t)∆t =
ρ(b)∑t=a
µ(t)f (t) : a < b
0 : a = b
−ρ(a)∑t=b
µ(t)f (t) : a > b.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Delta Integration
Theorem
If f ∈ Crd and t ∈ Tκ, then∫ σ(t)
tf (τ)∆τ = µ(t)f (t).
Corollary
If [a, b] consists of only isolated points, then
∫ b
af (t)∆t =
ρ(b)∑t=a
µ(t)f (t) : a < b
0 : a = b
−ρ(a)∑t=b
µ(t)f (t) : a > b.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Nabla Integration
Theorem
If f ∈ Cld and t ∈ Tκ, then∫ t
ρ(t)f (τ)∇τ = ν(t)f (t).
Corollary
If [a, b] consists of only isolated points, then
∫ b
af (t)∇t =
b∑t=σ(a)
ν(t)f (t) : a < b
0 : a = b
−a∑
t=σ(b)
ν(t)f (t) : a > b.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Nabla Integration
Theorem
If f ∈ Cld and t ∈ Tκ, then∫ t
ρ(t)f (τ)∇τ = ν(t)f (t).
Corollary
If [a, b] consists of only isolated points, then
∫ b
af (t)∇t =
b∑t=σ(a)
ν(t)f (t) : a < b
0 : a = b
−a∑
t=σ(b)
ν(t)f (t) : a > b.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Integrals
Proof.
We have ∫ tρ(t) f (τ)∇τ = F (t)− F (ρ(t))
= ν(t)F∇(t)= ν(t)f (t).
For a < b, ∫ ba f (τ)∇τ =
ρ(b)∑t=a
∫ σ(t)
tf (τ)∇τ
=
ρ(b)∑t=a
ν(σ(t))f (σ(t))
=b∑
t=σ(a)
ν(t)f (t).
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Integrals
Proof.
We have ∫ tρ(t) f (τ)∇τ = F (t)− F (ρ(t))
= ν(t)F∇(t)= ν(t)f (t).
For a < b, ∫ ba f (τ)∇τ =
ρ(b)∑t=a
∫ σ(t)
tf (τ)∇τ
=
ρ(b)∑t=a
ν(σ(t))f (σ(t))
=b∑
t=σ(a)
ν(t)f (t).
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Taylor Formula
Taylor’s Formula on R:
Theorem
Assume f ∈ Cn+1(R). Fix s. Then
f (t) =n∑
k=0
f (k)′(s)hk(t, s) +
∫ t
shn+1(t, τ)f (n+1)′(τ)dτ
where
h0(t, s) = 1,
hk+1(t, s) =∫ ts hk(τ, s)dτ, k ≥ 0.
When T = Rhk(t, s) = (t−s)k
k!
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Taylor Formula
Taylor’s Formula on R:
Theorem
Assume f ∈ Cn+1(R). Fix s. Then
f (t) =n∑
k=0
f (k)′(s)hk(t, s) +
∫ t
shn+1(t, τ)f (n+1)′(τ)dτ
where
h0(t, s) = 1,
hk+1(t, s) =∫ ts hk(τ, s)dτ, k ≥ 0.
When T = Rhk(t, s) = (t−s)k
k!
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Taylor Formula
Taylor’s Formula on R:
Theorem
Assume f ∈ Cn+1(R). Fix s. Then
f (t) =n∑
k=0
f (k)′(s)hk(t, s) +
∫ t
shn+1(t, τ)f (n+1)′(τ)dτ
where
h0(t, s) = 1,
hk+1(t, s) =∫ ts hk(τ, s)dτ, k ≥ 0.
When T = Rhk(t, s) = (t−s)k
k!
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Taylor Formula
Taylor’s Formula on R:
Theorem
Assume f ∈ Cn+1(R). Fix s. Then
f (t) =n∑
k=0
f (k)′(s)hk(t, s) +
∫ t
shn+1(t, τ)f (n+1)′(τ)dτ
where
h0(t, s) = 1,
hk+1(t, s) =∫ ts hk(τ, s)dτ, k ≥ 0.
When T = Rhk(t, s) = (t−s)k
k!
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Taylor Formula cont.
Taylor’s Formula on T for ∆ derivatives:
Theorem
Assume f ∈ Cn+1rd (T). Fix s ∈ T. Then
f (t) =n∑
k=0
f ∆k(s)hk∆(t, s) +
∫ t
sh(n+1)∆(t, σ(τ))f ∆n+1
(τ)∆τ
where
h0∆(t, s) = 1,
h(k+1)∆(t, s) =∫ ts hk∆(τ, s)∆τ, k ≥ 0.
hk∇(t, s) defined similarly.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Taylor Formula cont.
Taylor’s Formula on T for ∆ derivatives:
Theorem
Assume f ∈ Cn+1rd (T). Fix s ∈ T. Then
f (t) =n∑
k=0
f ∆k(s)hk∆(t, s) +
∫ t
sh(n+1)∆(t, σ(τ))f ∆n+1
(τ)∆τ
where
h0∆(t, s) = 1,
h(k+1)∆(t, s) =∫ ts hk∆(τ, s)∆τ, k ≥ 0.
hk∇(t, s) defined similarly.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Taylor Formula cont.
Taylor’s Formula on T for ∆ derivatives:
Theorem
Assume f ∈ Cn+1rd (T). Fix s ∈ T. Then
f (t) =n∑
k=0
f ∆k(s)hk∆(t, s) +
∫ t
sh(n+1)∆(t, σ(τ))f ∆n+1
(τ)∆τ
where
h0∆(t, s) = 1,
h(k+1)∆(t, s) =∫ ts hk∆(τ, s)∆τ, k ≥ 0.
hk∇(t, s) defined similarly.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Taylor Formula cont.
Taylor’s Formula on T for ∆ derivatives:
Theorem
Assume f ∈ Cn+1rd (T). Fix s ∈ T. Then
f (t) =n∑
k=0
f ∆k(s)hk∆(t, s) +
∫ t
sh(n+1)∆(t, σ(τ))f ∆n+1
(τ)∆τ
where
h0∆(t, s) = 1,
h(k+1)∆(t, s) =∫ ts hk∆(τ, s)∆τ, k ≥ 0.
hk∇(t, s) defined similarly.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
T = hZ = {0,±h,±2h,±3h, ...}
Example ([1])
For T = hZ and t > s,
hk∆(t, s) =
∏(k−1)i=0 (t − ih − s)
k!
Compute hk∆ for small k .
Identify a pattern.
Verify by Taylor’s Formula.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
T = hZ = {0,±h,±2h,±3h, ...}
Example ([1])
For T = hZ and t > s,
hk∆(t, s) =
∏(k−1)i=0 (t − ih − s)
k!
Compute hk∆ for small k .
Identify a pattern.
Verify by Taylor’s Formula.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
How to Find h2∇(t, s)
h1∇(t, s) = t − s
h2∇(t, s) =
∫ t
sh1∇(τ, s)∇τ
=t∑
σ(s)
h1∇(τ, s)ν(τ)
=t∑
σ(s)
(τ − s) · 1
=k∑
τ=1
τ
=k(k + 1)
2
=(t − s)(t − s + 1)
2
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
hk∇(t, s)
Example (B3)
For T = hZ and t > s,
hk∇(t, s) =
∏k−1i=0 (t + ih − s)
k!
Recall:
hk∆(t, s) =
∏(k−1)i=0 (t − ih − s)
k!
Differ by sign on ih
hk∆(t, s) is 0 for large k
hk∇(t, s) 6= 0 for large k
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
hk∇(t, s)
Example (B3)
For T = hZ and t > s,
hk∇(t, s) =
∏k−1i=0 (t + ih − s)
k!
Recall:
hk∆(t, s) =
∏(k−1)i=0 (t − ih − s)
k!
Differ by sign on ih
hk∆(t, s) is 0 for large k
hk∇(t, s) 6= 0 for large k
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
hk∇(t, s)
Example (B3)
For T = hZ and t > s,
hk∇(t, s) =
∏k−1i=0 (t + ih − s)
k!
Recall:
hk∆(t, s) =
∏(k−1)i=0 (t − ih − s)
k!
Differ by sign on ih
hk∆(t, s) is 0 for large k
hk∇(t, s) 6= 0 for large k
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
T = qZ
Example ([1])
For T = qZ and t > s,
hk∆(t, s) =k−1∏m=0
t − qmsm∑j=0
qj
Example (B3)
hk∇(t, s) =k−1∏m=0
tqm − sm∑j=0
qj
Similar to hZ.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
When t < s...
Define gk∆(t, s), gk∇(t, s) to be Taylor monomials when t < s.
Example (B3)
In hZ,gk∆(t, s) =
gk∇(t, s) =(−1)k
∏ki=0(s − t − ih)
(k − 1)!
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Comparisons
T = hZ
hk∆(t, s) =∏(k−1)
i=0 (t−ih−s)k! hk∇(t, s) =
∏k−1i=0 (t+ih−s)
k!
gk∆(t, s) =(−1)k
∏k−1i=0 (s+ih−t)k! gk∇(t, s) =
(−1)k∏k
i=0(s−t−ih)(k−1)!
T = qZ
hk∆(t, s) =∏k−1
m=0t−qmsm∑j=0
qjhk∇(t, s) =
∏k−1m=0
tqm−sm∑j=0
qj
gk∆(t, s) =∏k−1
m=0 s−qmtm∑j=0
qjgk∇(t, s) =
∏k−1m=0 sq
m−km∑j=0
qj
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Main Theorem
Theorem
Let T be an isolated or dense time scale, s ∈ Tκ, t ∈ Tκ, witht > s. Then
hk∆(t, s) = (−1)kgk∇(s, t)
if
(hk∆(t, s))(µ(t)− ν(s)) = µ(t)hk∆(t, ρ(s))− ν(s)hk∆(σ(t), s)
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof of Main Theorem
Proof
We prove this by induction
Fix s, t ∈ T, with s < t.Base Case: h0∆(t, s) ≡ 1 ≡ g0∇(s, t) ∀t, s ∈ TAssume k th step: hk∆(t, s) = (−1)kgk∇(s, t)Need: hk+1∆(t, s) = (−1)k+1gk+1∇(s, t)
Show this by existence/uniqueness of solutions to differentialequations:
Note that hk+1∆(s, s) = (−1)k+1gk+1∇(s, s) = 0Consider the following differential equation:
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof of Main Theorem
Proof
We prove this by induction
Fix s, t ∈ T, with s < t.Base Case: h0∆(t, s) ≡ 1 ≡ g0∇(s, t) ∀t, s ∈ TAssume k th step: hk∆(t, s) = (−1)kgk∇(s, t)Need: hk+1∆(t, s) = (−1)k+1gk+1∇(s, t)
Show this by existence/uniqueness of solutions to differentialequations:
Note that hk+1∆(s, s) = (−1)k+1gk+1∇(s, s) = 0Consider the following differential equation:
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof of Main Theorem
Proof
We prove this by induction
Fix s, t ∈ T, with s < t.
Base Case: h0∆(t, s) ≡ 1 ≡ g0∇(s, t) ∀t, s ∈ TAssume k th step: hk∆(t, s) = (−1)kgk∇(s, t)Need: hk+1∆(t, s) = (−1)k+1gk+1∇(s, t)
Show this by existence/uniqueness of solutions to differentialequations:
Note that hk+1∆(s, s) = (−1)k+1gk+1∇(s, s) = 0Consider the following differential equation:
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof of Main Theorem
Proof
We prove this by induction
Fix s, t ∈ T, with s < t.Base Case: h0∆(t, s) ≡ 1 ≡ g0∇(s, t) ∀t, s ∈ T
Assume k th step: hk∆(t, s) = (−1)kgk∇(s, t)Need: hk+1∆(t, s) = (−1)k+1gk+1∇(s, t)
Show this by existence/uniqueness of solutions to differentialequations:
Note that hk+1∆(s, s) = (−1)k+1gk+1∇(s, s) = 0Consider the following differential equation:
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof of Main Theorem
Proof
We prove this by induction
Fix s, t ∈ T, with s < t.Base Case: h0∆(t, s) ≡ 1 ≡ g0∇(s, t) ∀t, s ∈ TAssume k th step: hk∆(t, s) = (−1)kgk∇(s, t)
Need: hk+1∆(t, s) = (−1)k+1gk+1∇(s, t)
Show this by existence/uniqueness of solutions to differentialequations:
Note that hk+1∆(s, s) = (−1)k+1gk+1∇(s, s) = 0Consider the following differential equation:
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof of Main Theorem
Proof
We prove this by induction
Fix s, t ∈ T, with s < t.Base Case: h0∆(t, s) ≡ 1 ≡ g0∇(s, t) ∀t, s ∈ TAssume k th step: hk∆(t, s) = (−1)kgk∇(s, t)Need: hk+1∆(t, s) = (−1)k+1gk+1∇(s, t)
Show this by existence/uniqueness of solutions to differentialequations:
Note that hk+1∆(s, s) = (−1)k+1gk+1∇(s, s) = 0Consider the following differential equation:
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof of Main Theorem
Proof
We prove this by induction
Fix s, t ∈ T, with s < t.Base Case: h0∆(t, s) ≡ 1 ≡ g0∇(s, t) ∀t, s ∈ TAssume k th step: hk∆(t, s) = (−1)kgk∇(s, t)Need: hk+1∆(t, s) = (−1)k+1gk+1∇(s, t)
Show this by existence/uniqueness of solutions to differentialequations:
Note that hk+1∆(s, s) = (−1)k+1gk+1∇(s, s) = 0Consider the following differential equation:
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof of Main Theorem
Proof
We prove this by induction
Fix s, t ∈ T, with s < t.Base Case: h0∆(t, s) ≡ 1 ≡ g0∇(s, t) ∀t, s ∈ TAssume k th step: hk∆(t, s) = (−1)kgk∇(s, t)Need: hk+1∆(t, s) = (−1)k+1gk+1∇(s, t)
Show this by existence/uniqueness of solutions to differentialequations:
Note that hk+1∆(s, s) = (−1)k+1gk+1∇(s, s) = 0
Consider the following differential equation:
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof of Main Theorem
Proof
We prove this by induction
Fix s, t ∈ T, with s < t.Base Case: h0∆(t, s) ≡ 1 ≡ g0∇(s, t) ∀t, s ∈ TAssume k th step: hk∆(t, s) = (−1)kgk∇(s, t)Need: hk+1∆(t, s) = (−1)k+1gk+1∇(s, t)
Show this by existence/uniqueness of solutions to differentialequations:
Note that hk+1∆(s, s) = (−1)k+1gk+1∇(s, s) = 0Consider the following differential equation:
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof Continued
Proof.
Computing the derivative gives
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
⇔ hk+1∆(t, s)− hk+1∆(t, ρ(s))
ν(s)= (−1)k+1gk∇(s, t)
Inserting the inductive hypothesis and inserting the definitionof hk∆(t, s):
hk+1δ(t, s)− hk+1δ(t, ρ(s))
ν(s)= −(hk∆(σ(t), s)− hk∆(t, s))
µ(t)
Algebraic rearrangement gives
(hk∆(t, s))(µ(t)− ν(s)) = µ(t)hk∆(t, ρ(s))− ν(s)hk∆(σ(t), s)
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof Continued
Proof.
Computing the derivative gives
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
⇔ hk+1∆(t, s)− hk+1∆(t, ρ(s))
ν(s)= (−1)k+1gk∇(s, t)
Inserting the inductive hypothesis and inserting the definitionof hk∆(t, s):
hk+1δ(t, s)− hk+1δ(t, ρ(s))
ν(s)= −(hk∆(σ(t), s)− hk∆(t, s))
µ(t)
Algebraic rearrangement gives
(hk∆(t, s))(µ(t)− ν(s)) = µ(t)hk∆(t, ρ(s))− ν(s)hk∆(σ(t), s)
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof Continued
Proof.
Computing the derivative gives
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
⇔ hk+1∆(t, s)− hk+1∆(t, ρ(s))
ν(s)= (−1)k+1gk∇(s, t)
Inserting the inductive hypothesis and inserting the definitionof hk∆(t, s):
hk+1δ(t, s)− hk+1δ(t, ρ(s))
ν(s)= −(hk∆(σ(t), s)− hk∆(t, s))
µ(t)
Algebraic rearrangement gives
(hk∆(t, s))(µ(t)− ν(s)) = µ(t)hk∆(t, ρ(s))− ν(s)hk∆(σ(t), s)
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Proof Continued
Proof.
Computing the derivative gives
(hk+1∆(t, s))∇s = ((−1)k+1gk+1∇(s, t))∇s
⇔ hk+1∆(t, s)− hk+1∆(t, ρ(s))
ν(s)= (−1)k+1gk∇(s, t)
Inserting the inductive hypothesis and inserting the definitionof hk∆(t, s):
hk+1δ(t, s)− hk+1δ(t, ρ(s))
ν(s)= −(hk∆(σ(t), s)− hk∆(t, s))
µ(t)
Algebraic rearrangement gives
(hk∆(t, s))(µ(t)− ν(s)) = µ(t)hk∆(t, ρ(s))− ν(s)hk∆(σ(t), s)
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Future Research
Taylor Monomials on non-isolated Time Scales.
Fourier and Laplacian Analysis on Time Scales.
Multivariate Taylor Expansions.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Future Research
Taylor Monomials on non-isolated Time Scales.
Fourier and Laplacian Analysis on Time Scales.
Multivariate Taylor Expansions.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Future Research
Taylor Monomials on non-isolated Time Scales.
Fourier and Laplacian Analysis on Time Scales.
Multivariate Taylor Expansions.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Future Research
Taylor Monomials on non-isolated Time Scales.
Fourier and Laplacian Analysis on Time Scales.
Multivariate Taylor Expansions.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
References
Reagan J. Higgins and Allan Peterson. Cauchy Functions andTaylor’s Formula for Time Scales. Department ofMathematics, University of Nebraska-Lincoln.
M. Bohner and A. Peterson, Dynamic Equations On TimeScales: An Introduction With Applications, Birkhauser,Boston, 2001.
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
Special Thanks To...
National Science Foundation
Dr. Nathan Pennington
P. Ballesteros, J. Batista, S. Bell Taylor Monomials
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