[PPT] - Fourier analysis of Discrete Dirac n torus Nelson Faustino PowerPoint Presentation

SLIDE 1

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Fourier analysis of Discrete Dirac

perators on the n−torus Rn/2π

h Zn Nelson Faustino

Center of Mathematics, Computation and Cognition, UFABC nelson.faustino@ufabc.edu.br

19th Annual Workshop on Applications and Generalizations of Complex Analysis, March 23-24, 2018

1 / 22

SLIDE 2

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

1

New developments on discrete Dirac operators Discrete Dirac Operators The toroidal approach

2

Klein-Gordon type equations The model problem Wave type propagators

3

Playing around some remarkable connections A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

2 / 22

SLIDE 3

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The aim of this talk

Present an abridged version of the following preprints:

Faustino, N. (January 2018). Relativistic Wave Equations on the lattice: an operational perspective, arXiv:1801.09340. Faustino, N. (February 2018). A note on the discrete Cauchy-Kovaleskaya extension, arXiv:1802.08605.

Figure: I hope that the organizers have ordered donuts to stimulate the research discussions on the coffee break (A wishful thinking).

3 / 22

SLIDE 4

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The aim of this talk

Present an abridged version of the following preprints:

Faustino, N. (January 2018). Relativistic Wave Equations on the lattice: an operational perspective, arXiv:1801.09340. Faustino, N. (February 2018). A note on the discrete Cauchy-Kovaleskaya extension, arXiv:1802.08605.

Figure: I hope that the organizers have ordered donuts to stimulate the research discussions on the coffee break (A wishful thinking).

3 / 22

SLIDE 5

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Discrete Dirac Operators

Multivector formulation

Finite Difference Dirac Operators of Forward/Backward Type: D+

h = n

j=1

ej∂+j

h and D− h = n

j=1

ej∂−j

h ,

where ∂±j

h

correspond to the forward/backward finite difference operators

n the lattice hZn, and e1, e2, . . . , en corresponds to the basis of the

Clifford algebra of signature (0, n). Multivector operators acting on Cℓn,n ∼ = End(Cℓ0,n): ∂+

h = D+ h ∧ and ∂− h = D− h •, where:

∂+

h = D+ h ∧ = n

j=1

ej ∧ ∂+j

h stands the multivector counterpart of the

exterior derivative d. ∂−

h = D− h • = n

j=1

ej • ∂−j

h

stands the multivector counterpart of the co-differential form δ = ⋆−1d⋆.

4 / 22

SLIDE 6

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Discrete Dirac Operators

Multivector formulation

Finite Difference Dirac Operators of Forward/Backward Type: D+

h = n

j=1

ej∂+j

h and D− h = n

j=1

ej∂−j

h ,

where ∂±j

h

correspond to the forward/backward finite difference operators

n the lattice hZn, and e1, e2, . . . , en corresponds to the basis of the

Clifford algebra of signature (0, n). Multivector operators acting on Cℓn,n ∼ = End(Cℓ0,n): ∂+

h = D+ h ∧ and ∂− h = D− h •, where:

∂+

h = D+ h ∧ = n

j=1

ej ∧ ∂+j

h stands the multivector counterpart of the

exterior derivative d. ∂−

h = D− h • = n

j=1

ej • ∂−j

h

stands the multivector counterpart of the co-differential form δ = ⋆−1d⋆.

4 / 22

SLIDE 7

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Star-Laplacian factorization

There are several possibilities still.

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

2

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

3

A geometric calculus factorization: ∆h equals to

D+

h ∧ + D− h •

2 = −

D+

h ∧ − D− h •

2. Figure: The star laplacian in

hZ3. This construction is closely related to the one obtained in my joint paper with U. K¨ ahler and F. Sommen Discrete Dirac operators in Clifford analysis (Advances in Applied Clifford Algebras 17 (3), 451-467).

5 / 22

SLIDE 8

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Star-Laplacian factorization

There are several possibilities still.

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

2

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

3

A geometric calculus factorization: ∆h equals to

D+

h ∧ + D− h •

2 = −

D+

h ∧ − D− h •

2. Figure: The star laplacian in

hZ3. This construction is closely related to the one obtained in my joint paper with U. K¨ ahler and F. Sommen Discrete Dirac operators in Clifford analysis (Advances in Applied Clifford Algebras 17 (3), 451-467).

5 / 22

SLIDE 9

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Star-Laplacian factorization

There are several possibilities still.

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

2

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

3

A geometric calculus factorization: ∆h equals to

D+

h ∧ + D− h •

2 = −

D+

h ∧ − D− h •

2. Figure: The star laplacian in

hZ3. This construction is closely related to the one obtained in my joint paper with U. K¨ ahler and F. Sommen Discrete Dirac operators in Clifford analysis (Advances in Applied Clifford Algebras 17 (3), 451-467).

5 / 22

SLIDE 10

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Star-Laplacian factorization

There are several possibilities still.

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

2

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

3

A geometric calculus factorization: ∆h equals to

D+

h ∧ + D− h •

2 = −

D+

h ∧ − D− h •

2. Figure: The star laplacian in

hZ3. This construction is closely related to the one obtained in my joint paper with U. K¨ ahler and F. Sommen Discrete Dirac operators in Clifford analysis (Advances in Applied Clifford Algebras 17 (3), 451-467).

5 / 22

SLIDE 11

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Star-Laplacian factorization

There are several possibilities still.

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

2

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

3

A geometric calculus factorization: ∆h equals to

D+

h ∧ + D− h •

2 = −

D+

h ∧ − D− h •

2. Figure: The star laplacian in

hZ3. This construction is closely related to the one obtained in my joint paper with U. K¨ ahler and F. Sommen Discrete Dirac operators in Clifford analysis (Advances in Applied Clifford Algebras 17 (3), 451-467).

5 / 22

SLIDE 12

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Star-Laplacian factorization

There are several possibilities still.

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

2

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

3

A geometric calculus factorization: ∆h equals to

D+

h ∧ + D− h •

2 = −

D+

h ∧ − D− h •

2. Figure: The star laplacian in

hZ3. This construction is closely related to the one obtained in my joint paper with U. K¨ ahler and F. Sommen Discrete Dirac operators in Clifford analysis (Advances in Applied Clifford Algebras 17 (3), 451-467).

5 / 22

SLIDE 13

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Dirac-K¨ ahler operator on the lattice

Multivector Formulation

Finite Difference Discretization of the Dirac-K¨ ahler operator: DεΨ(x, t) =

n

j=1

ej Ψ(x + εej, t) − Ψ(x − εej, t) 2ε + +

n

j=1

en+j 2Ψ(x, t) − Ψ(x + εej, t) − Ψ(x − εej, t) 2ε

1

Pseudo-scalar representation: γ =

n

j=1

en+jej stands for the pseudo-scalar of Cℓn,n. Notice that γ2 = +1.

2

Factorization Property: (Dh − mγ)2 = −∆h + m2. For further details see my paper, entitled Solutions for the Klein-Gordon and Dirac equations on the lattice based on Chebyshev polynomials (Complex Anal. Oper. Theory, 10 : 379-399 (2016)).

6 / 22

SLIDE 14

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Dirac-K¨ ahler operator on the lattice

Multivector Formulation

Finite Difference Discretization of the Dirac-K¨ ahler operator: DεΨ(x, t) =

n

j=1

ej Ψ(x + εej, t) − Ψ(x − εej, t) 2ε + +

n

j=1

en+j 2Ψ(x, t) − Ψ(x + εej, t) − Ψ(x − εej, t) 2ε

1

Pseudo-scalar representation: γ =

n

j=1

en+jej stands for the pseudo-scalar of Cℓn,n. Notice that γ2 = +1.

2

Factorization Property: (Dh − mγ)2 = −∆h + m2. For further details see my paper, entitled Solutions for the Klein-Gordon and Dirac equations on the lattice based on Chebyshev polynomials (Complex Anal. Oper. Theory, 10 : 379-399 (2016)).

6 / 22

SLIDE 15

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

New classes of finite difference Dirac operators

Towards a fractional regularization

1

Fractional lattice: Rn

h,α := (1 − α)hZn ⊕ αhZn,

with h > 0 and 0 < α < 1 2.

2

Fractional regularization of the Dirac-K¨ ahler operator: Dh,α := (1 − α)D(1−α)h − αD†

αh,

where D(1−α)h denotes the Dirac-K¨ ahler discretization on the lattice (1 − α)hZn, and D†

αh the formal conjugation of Dαh on the lattice

αhZn.

3

Asymptotic formula on the lattice hZn: lim

α→0 Dh,α = Dh.

4

Asymptotic formula on the lattice h

2Zn:

lim

α→ 1

2

Dh,α = D+

h/2 + D− h/2

2 .

7 / 22

SLIDE 16

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

New classes of finite difference Dirac operators

Towards a fractional regularization

1

Fractional lattice: Rn

h,α := (1 − α)hZn ⊕ αhZn,

with h > 0 and 0 < α < 1 2.

2

Fractional regularization of the Dirac-K¨ ahler operator: Dh,α := (1 − α)D(1−α)h − αD†

αh,

where D(1−α)h denotes the Dirac-K¨ ahler discretization on the lattice (1 − α)hZn, and D†

αh the formal conjugation of Dαh on the lattice

αhZn.

3

Asymptotic formula on the lattice hZn: lim

α→0 Dh,α = Dh.

4

Asymptotic formula on the lattice h

2Zn:

lim

α→ 1

2

Dh,α = D+

h/2 + D− h/2

2 .

7 / 22

SLIDE 17

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

New classes of finite difference Dirac operators

Towards a fractional regularization

1

Fractional lattice: Rn

h,α := (1 − α)hZn ⊕ αhZn,

with h > 0 and 0 < α < 1 2.

2

Fractional regularization of the Dirac-K¨ ahler operator: Dh,α := (1 − α)D(1−α)h − αD†

αh,

where D(1−α)h denotes the Dirac-K¨ ahler discretization on the lattice (1 − α)hZn, and D†

αh the formal conjugation of Dαh on the lattice

αhZn.

3

Asymptotic formula on the lattice hZn: lim

α→0 Dh,α = Dh.

4

Asymptotic formula on the lattice h

2Zn:

lim

α→ 1

2

Dh,α = D+

h/2 + D− h/2

2 .

7 / 22

SLIDE 18

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

New classes of finite difference Dirac operators

Towards a fractional regularization

1

Fractional lattice: Rn

h,α := (1 − α)hZn ⊕ αhZn,

with h > 0 and 0 < α < 1 2.

2

Fractional regularization of the Dirac-K¨ ahler operator: Dh,α := (1 − α)D(1−α)h − αD†

αh,

where D(1−α)h denotes the Dirac-K¨ ahler discretization on the lattice (1 − α)hZn, and D†

αh the formal conjugation of Dαh on the lattice

αhZn.

3

Asymptotic formula on the lattice hZn: lim

α→0 Dh,α = Dh.

4

Asymptotic formula on the lattice h

2Zn:

lim

α→ 1

2

Dh,α = D+

h/2 + D− h/2

2 .

7 / 22

SLIDE 19

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The discrete Fourier analysis toolbox

Discrete Fourier Transform on the lattice Rn

h,α

(Fh,αg)(ξ, t) =          hn (2π)

n 2

x∈Rn

h,α

g(x, t)eix·ξ for ξ ∈ Qh for ξ ∈ Rn \ Qh . Here Qh =

− π

h , π h

n stands for the n−dimensional Brioullin zone representation of the n−torus Rn/ 2π

h Zn.

Isometric Isomorphism: Fα,h : ℓ2(Rn

h,α; C ⊗ Cℓn,n) → L2(Qh; C ⊗ Cℓn,n) is an isometry.

Mapping property at the level of distributions: Fh,α : S′(Rn

h,α; C ⊗ Cℓn,n) → C∞(Qh; C ⊗ Cℓn,n).

Discrete convolution formula: Fh,α [f(·, t) ⋆h,α Φ] = (Fh,αf(·, t)) (Fh,αΦ).

8 / 22

SLIDE 20

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The discrete Fourier analysis toolbox

Discrete Fourier Transform on the lattice Rn

h,α

(Fh,αg)(ξ, t) =          hn (2π)

n 2

x∈Rn

h,α

g(x, t)eix·ξ for ξ ∈ Qh for ξ ∈ Rn \ Qh . Here Qh =

− π

h , π h

n stands for the n−dimensional Brioullin zone representation of the n−torus Rn/ 2π

h Zn.

Isometric Isomorphism: Fα,h : ℓ2(Rn

h,α; C ⊗ Cℓn,n) → L2(Qh; C ⊗ Cℓn,n) is an isometry.

Mapping property at the level of distributions: Fh,α : S′(Rn

h,α; C ⊗ Cℓn,n) → C∞(Qh; C ⊗ Cℓn,n).

Discrete convolution formula: Fh,α [f(·, t) ⋆h,α Φ] = (Fh,αf(·, t)) (Fh,αΦ).

8 / 22

SLIDE 21

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The discrete Fourier analysis toolbox

Discrete Fourier Transform on the lattice Rn

h,α

(Fh,αg)(ξ, t) =          hn (2π)

n 2

x∈Rn

h,α

g(x, t)eix·ξ for ξ ∈ Qh for ξ ∈ Rn \ Qh . Here Qh =

− π

h , π h

n stands for the n−dimensional Brioullin zone representation of the n−torus Rn/ 2π

h Zn.

Isometric Isomorphism: Fα,h : ℓ2(Rn

h,α; C ⊗ Cℓn,n) → L2(Qh; C ⊗ Cℓn,n) is an isometry.

Mapping property at the level of distributions: Fh,α : S′(Rn

h,α; C ⊗ Cℓn,n) → C∞(Qh; C ⊗ Cℓn,n).

Discrete convolution formula: Fh,α [f(·, t) ⋆h,α Φ] = (Fh,αf(·, t)) (Fh,αΦ).

8 / 22

SLIDE 22

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The discrete Fourier analysis toolbox

Discrete Fourier Transform on the lattice Rn

h,α

(Fh,αg)(ξ, t) =          hn (2π)

n 2

x∈Rn

h,α

g(x, t)eix·ξ for ξ ∈ Qh for ξ ∈ Rn \ Qh . Here Qh =

− π

h , π h

n stands for the n−dimensional Brioullin zone representation of the n−torus Rn/ 2π

h Zn.

Isometric Isomorphism: Fα,h : ℓ2(Rn

h,α; C ⊗ Cℓn,n) → L2(Qh; C ⊗ Cℓn,n) is an isometry.

Mapping property at the level of distributions: Fh,α : S′(Rn

h,α; C ⊗ Cℓn,n) → C∞(Qh; C ⊗ Cℓn,n).

Discrete convolution formula: Fh,α [f(·, t) ⋆h,α Φ] = (Fh,αf(·, t)) (Fh,αΦ).

8 / 22

SLIDE 23

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Fourier multipliers

Fourier multilplier of Fh,α ◦ (−∆h) ◦ F −1

h,α

dh(ξ)2 = 4 h2

j=1

sin2 hξj 2

=
j=1
1 − e−ihξj

h eihθj

2

Fourier multilplier of Fh,α ◦ Dh,α ◦ F −1

h,α [choice θj = (1 − α)ξj]

zh,α(ξ) =

n

j=1

−iej sin((1 − α)hξj) + sin(αhξj) h + +

n

j=1

en+j cos(αhξj) − cos((1 − α)hξj) h satisfies the factorization property zh,α(ξ)2 = dh(ξ)2.

9 / 22

SLIDE 24

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Fourier multipliers

Fourier multilplier of Fh,α ◦ (−∆h) ◦ F −1

h,α

dh(ξ)2 = 4 h2

j=1

sin2 hξj 2

=
j=1
1 − e−ihξj

h eihθj

2

Fourier multilplier of Fh,α ◦ Dh,α ◦ F −1

h,α [choice θj = (1 − α)ξj]

zh,α(ξ) =

n

j=1

−iej sin((1 − α)hξj) + sin(αhξj) h + +

n

j=1

en+j cos(αhξj) − cos((1 − α)hξj) h satisfies the factorization property zh,α(ξ)2 = dh(ξ)2.

9 / 22

SLIDE 25

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The model problem

Second Order Evolution Problem Determine Ψ on Rn

h,α × T s.t

           L2

t Ψ(x, t) = ∆hΨ(x, t) − m2Ψ(x, t)

, (x, t) ∈ Rn

h,α × T

Ψ(x, 0) = Φ0(x) , x ∈ Rn

h,α

[LtΨ(x, t)]t=0 = Φ1(x) , x ∈ Rn

h,α

(1) associated to the discrete Laplacian ∆h on hZn, and a mass term m.

1

Differential-difference evolution problem Lt = ∂t and T = [0, ∞).

2

Difference-difference evolution problem LtΨ(x, t) = Ψ

x, t + τ

2

− Ψ
x, t − τ

2

τ

and T = kτ

2

: k = 0, 1, 2, . . .

.

3

General procedure Lt is a shift-invariant operator with respect to the translation semigroup {exp (τ∂t)}t≥0 (τ > 0) and T ⊆ [0, ∞).

10 / 22

SLIDE 26

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The model problem

Second Order Evolution Problem Determine Ψ on Rn

h,α × T s.t

           L2

t Ψ(x, t) = ∆hΨ(x, t) − m2Ψ(x, t)

, (x, t) ∈ Rn

h,α × T

Ψ(x, 0) = Φ0(x) , x ∈ Rn

h,α

[LtΨ(x, t)]t=0 = Φ1(x) , x ∈ Rn

h,α

(1) associated to the discrete Laplacian ∆h on hZn, and a mass term m.

1

Differential-difference evolution problem Lt = ∂t and T = [0, ∞).

2

Difference-difference evolution problem LtΨ(x, t) = Ψ

x, t + τ

2

− Ψ
x, t − τ

2

τ

and T = kτ

2

: k = 0, 1, 2, . . .

.

3

General procedure Lt is a shift-invariant operator with respect to the translation semigroup {exp (τ∂t)}t≥0 (τ > 0) and T ⊆ [0, ∞).

10 / 22

SLIDE 27

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The model problem

Second Order Evolution Problem Determine Ψ on Rn

h,α × T s.t

           L2

t Ψ(x, t) = ∆hΨ(x, t) − m2Ψ(x, t)

, (x, t) ∈ Rn

h,α × T

Ψ(x, 0) = Φ0(x) , x ∈ Rn

h,α

[LtΨ(x, t)]t=0 = Φ1(x) , x ∈ Rn

h,α

(1) associated to the discrete Laplacian ∆h on hZn, and a mass term m.

1

Differential-difference evolution problem Lt = ∂t and T = [0, ∞).

2

Difference-difference evolution problem LtΨ(x, t) = Ψ

x, t + τ

2

− Ψ
x, t − τ

2

τ

and T = kτ

2

: k = 0, 1, 2, . . .

.

3

General procedure Lt is a shift-invariant operator with respect to the translation semigroup {exp (τ∂t)}t≥0 (τ > 0) and T ⊆ [0, ∞).

10 / 22

SLIDE 28

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The model problem

Second Order Evolution Problem Determine Ψ on Rn

h,α × T s.t

           L2

t Ψ(x, t) = ∆hΨ(x, t) − m2Ψ(x, t)

, (x, t) ∈ Rn

h,α × T

Ψ(x, 0) = Φ0(x) , x ∈ Rn

h,α

[LtΨ(x, t)]t=0 = Φ1(x) , x ∈ Rn

h,α

(1) associated to the discrete Laplacian ∆h on hZn, and a mass term m.

1

Differential-difference evolution problem Lt = ∂t and T = [0, ∞).

2

Difference-difference evolution problem LtΨ(x, t) = Ψ

x, t + τ

2

− Ψ
x, t − τ

2

τ

and T = kτ

2

: k = 0, 1, 2, . . .

.

3

General procedure Lt is a shift-invariant operator with respect to the translation semigroup {exp (τ∂t)}t≥0 (τ > 0) and T ⊆ [0, ∞).

10 / 22

SLIDE 29

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The model problem

Formulation on the momentum space Qh × T Determine Ψ on Rn

h,α × T s.t

           L2

t [Fh,αΨ(ξ, t)] = −

dh(ξ)2 + m2

Fh,αΨ(ξ, t) , (ξ, t) ∈ Qh × T Fh,αΨ(ξ, 0) = Fh,αΦ0(ξ) , ξ ∈ Qh [LtFh,αΨ(ξ, t)]t=0 = Fh,αΦ1(ξ) , ξ ∈ Qh , (2) Here and elsewhere −(dh(ξ)2 + m2) denotes the Fourier multiplier of Fh,α ◦ (∆h − m2) ◦ F −1

h,α.

Umbral calculus formulation: Here and elsewhere we assume that Lt = L(∂t) admits the formal series expansion Lt =

∞

k=1

bk (∂t)k k! , with bk = [(Lt)ktk]t=0.

11 / 22

SLIDE 30

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The model problem

Formulation on the momentum space Qh × T Determine Ψ on Rn

h,α × T s.t

           L2

t [Fh,αΨ(ξ, t)] = −

dh(ξ)2 + m2

Fh,αΨ(ξ, t) , (ξ, t) ∈ Qh × T Fh,αΨ(ξ, 0) = Fh,αΦ0(ξ) , ξ ∈ Qh [LtFh,αΨ(ξ, t)]t=0 = Fh,αΦ1(ξ) , ξ ∈ Qh , (2) Here and elsewhere −(dh(ξ)2 + m2) denotes the Fourier multiplier of Fh,α ◦ (∆h − m2) ◦ F −1

h,α.

Umbral calculus formulation: Here and elsewhere we assume that Lt = L(∂t) admits the formal series expansion Lt =

∞

k=1

bk (∂t)k k! , with bk = [(Lt)ktk]t=0.

11 / 22

SLIDE 31

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Wave type propagators

Discrete convolution formula representation cosh

t L−1 √

∆h − m2 Φ(x) =

y∈Rn

h,α

hnΦ(y)K0(x − y, t) sinh

t L−1 √

∆h − m2 √ ∆h − m2 Φ(x) =

y∈Rn

h,α

hnΦ(x)K1(x − y, t), with K0(x, t) = 1 (2π)

n 2

Qh

cosh

t L−1

i

dh(ξ)2 + m2
e−ix·ξ dξ

K1(x, t) = 1 (2π)

n 2

Qh

sinh

t L−1

i

dh(ξ)2 + m2
i
dh(ξ)2 + m2

e−ix·ξ dξ. (3)

12 / 22

SLIDE 32

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Main Result

Theorem Let Φ0 and Φ1 be two Clifford-valued functions membership in S(Rn

h,α; C ⊗ Cℓn,n), and K0, K1 be the kernel functions defined viz (3).

Then we have the following: (i) The function Fh,αΨ(ξ, t) = cosh

tL−1

i

dh(ξ)2 + m2
Fh,αΦ0(ξ)+

+ sinh

tL−1

i

dh(ξ)2 + m2
i
dh(ξ)2 + m2

Fh,αΦ1(ξ) (4) solves the time-evolution problem (2).

13 / 22

SLIDE 33

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Main Result

Theorem Let Φ0 and Φ1 be two Clifford-valued functions membership in S(Rn

h,α; C ⊗ Cℓn,n), and K0, K1 be the kernel functions defined viz (3).

Then we have the following: (i) The function Fh,αΨ(ξ, t) = cosh

tL−1

i

dh(ξ)2 + m2
Fh,αΦ0(ξ)+

+ sinh

tL−1

i

dh(ξ)2 + m2
i
dh(ξ)2 + m2

Fh,αΦ1(ξ) (4) solves the time-evolution problem (2).

13 / 22

SLIDE 34

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Main Result

Continuation

Theorem (ii) The ansatz Ψ(x, t) = cosh

tL−1 √

∆h − m2 Φ0(x)+ + sinh

tL−1 √

∆h − m2 √ ∆h − m2 Φ1(x) (5) solves the discretized Klein-Gordon equation (1).

14 / 22

SLIDE 35

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Main Result

Continuation

Theorem (ii) The ansatz Ψ(x, t) = cosh

tL−1 √

∆h − m2 Φ0(x)+ + sinh

tL−1 √

∆h − m2 √ ∆h − m2 Φ1(x) (5) solves the discretized Klein-Gordon equation (1).

14 / 22

SLIDE 36

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

An Example

Central difference operator Ltψ(x, t) := ψ

x, t + τ

2

− ψ
x, t − τ

2

τ

= 2 τ sinh τ 2∂t

=ℓ(∂t )

ψ(x, t). Remark: ℓ−1(s) = 2

τ sinh−1 τ 2 s

On the Fourier domain:

Fh,αΨ(ξ, t) = cos 2t τ sin−1 τ 2

dh(ξ)2 + m2
Fh,αΦ0(ξ) +

+ sin 2t τ sin−1 τ 2

dh(ξ)2 + m2
dh(ξ)2 + m2

Fh,αΦ1(ξ) solves the second-order time-evolution problem (2) on the momentum space Qh × Z≥0 (remember statement (i))

15 / 22

SLIDE 37

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

An Example

The Chebyshev connection Fh,αΨ(ξ, t) = T 2t

τ

1 − τ 2

4 (dh(ξ)2 + m2)

Fh,αΦ0(ξ)+

+ τ 2U 2t

τ −1

1 − τ 2

4 (dh(ξ)2 + m2)

Fh,αΦ1(ξ),

Remark: The connection with the Chebyshev polynomials Tk(z) & Uk−1(z): is a direct consequence of the property sin−1(z) = cos−1(1 − z2) (0 ≤ z ≤ 1).

16 / 22

SLIDE 38

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Difference-Difference Klein-Gordon problem

The Laplace Transform Tecnhique

Formal solution on the Fourier domain Fh,αΨ(ξ, t) = T 2t

τ

1 − τ 2

4 (dh(ξ)2 + m2)

Fh,αΦ0(ξ)+

+ τ 2U 2t

τ −1

1 − τ 2

4 (dh(ξ)2 + m2)

Fh,αΦ1(ξ),

Fourier+Cauchy principal value representation lead to the following singular integral representation Fh,αΨ(ξ, t) = τ 4π

2π

τ

− 2π

τ

−i sin ωτ

2

Fh,αΦ0(ξ) + τ

2 Fh,αΦ1(ξ)

cos ωτ

2

−
1 − τ2

4 (dh(ξ)2 + m2)

e−iωtdω.

17 / 22

SLIDE 39

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Difference-Difference Klein-Gordon problem

The Laplace Transform Tecnhique

Formal solution on the Fourier domain Fh,αΨ(ξ, t) = T 2t

τ

1 − τ 2

4 (dh(ξ)2 + m2)

Fh,αΦ0(ξ)+

+ τ 2U 2t

τ −1

1 − τ 2

4 (dh(ξ)2 + m2)

Fh,αΦ1(ξ),

Fourier+Cauchy principal value representation lead to the following singular integral representation Fh,αΨ(ξ, t) = τ 4π

2π

τ

− 2π

τ

−i sin ωτ

2

Fh,αΦ0(ξ) + τ

2 Fh,αΦ1(ξ)

cos ωτ

2

−
1 − τ2

4 (dh(ξ)2 + m2)

e−iωtdω.

17 / 22

SLIDE 40

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Difference-Difference Klein-Gordon problem

The Fourier Transform Tecnhique

Hypersingular integral representation Fh,αΨ(ξ, t) = τ 4π

2π

τ

− 2π

τ

−i sin ωτ

2

Fh,αΦ0(ξ) + τ

2 Fh,αΦ1(ξ)

cos ωτ

2

−
1 − τ2

4 (dh(ξ)2 + m2)

e−iωtdω. Fourier inversion formula: The solution Ψ(x, t) of the Klein-Gordon problem of difference-difference type is equal to τ (2π)

n 2 +2

Qh
π

τ

− π

τ

−i sin ωτ

2

Fh,αΦ0(ξ) + τ

2 Fh,αΦ1(ξ)

cos ωτ

2

−
1 − τ2

4 (dh(ξ)2 + m2)

e−i(ωt+x·ξ)dωdξ. A toroidal Fourier transform in disguise: Here we recall that from the isomorphism Qh ×

− 2π

τ , 2π τ

∼ =

Rn/ 2π

h Zn

×

R/ 4π

τ Z

, the

resulting integral representation formula may be interpreted as a space-time toroidal Fourier transform.

18 / 22

SLIDE 41

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Difference-Difference Klein-Gordon problem

The Fourier Transform Tecnhique

Hypersingular integral representation Fh,αΨ(ξ, t) = τ 4π

2π

τ

− 2π

τ

−i sin ωτ

2

Fh,αΦ0(ξ) + τ

2 Fh,αΦ1(ξ)

cos ωτ

2

−
1 − τ2

4 (dh(ξ)2 + m2)

e−iωtdω. Fourier inversion formula: The solution Ψ(x, t) of the Klein-Gordon problem of difference-difference type is equal to τ (2π)

n 2 +2

Qh
π

τ

− π

τ

−i sin ωτ

2

Fh,αΦ0(ξ) + τ

2 Fh,αΦ1(ξ)

cos ωτ

2

−
1 − τ2

4 (dh(ξ)2 + m2)

e−i(ωt+x·ξ)dωdξ. A toroidal Fourier transform in disguise: Here we recall that from the isomorphism Qh ×

− 2π

τ , 2π τ

∼ =

Rn/ 2π

h Zn

×

R/ 4π

τ Z

, the

resulting integral representation formula may be interpreted as a space-time toroidal Fourier transform.

18 / 22

SLIDE 42

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Difference-Difference Klein-Gordon problem

The Fourier Transform Tecnhique

Hypersingular integral representation Fh,αΨ(ξ, t) = τ 4π

2π

τ

− 2π

τ

−i sin ωτ

2

Fh,αΦ0(ξ) + τ

2 Fh,αΦ1(ξ)

cos ωτ

2

−
1 − τ2

4 (dh(ξ)2 + m2)

e−iωtdω. Fourier inversion formula: The solution Ψ(x, t) of the Klein-Gordon problem of difference-difference type is equal to τ (2π)

n 2 +2

Qh
π

τ

− π

τ

−i sin ωτ

2

Fh,αΦ0(ξ) + τ

2 Fh,αΦ1(ξ)

cos ωτ

2

−
1 − τ2

4 (dh(ξ)2 + m2)

e−i(ωt+x·ξ)dωdξ. A toroidal Fourier transform in disguise: Here we recall that from the isomorphism Qh ×

− 2π

τ , 2π τ

∼ =

Rn/ 2π

h Zn

×

R/ 4π

τ Z

, the

resulting integral representation formula may be interpreted as a space-time toroidal Fourier transform.

18 / 22

SLIDE 43

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Difference-Difference Klein Gordon

The Fourier-Laplace technique

Starting from the Laplace transform identity ∞ epλ2pβ−1Eα,β (spα) dp = λ−2β 1 − sλ−2α , ℜ(λ2) > |s|

1 α & ℜ(β) > 0

involving the generalized Mittag-Leffler functions Eα,β(z) we realize that 1 cos ωτ

2

−
1 − τ2

4 (dh(ξ)2 + m2)

= = − ∞ e− pτ2

4

dh(ξ)2 E 1

2 , 1 2

cos

ωτ

2

√p

√p

e

p

1− τ2

4 m2

dp

so that Fh,αΨ(ξ, t) = − τ (2π)

n 2 +2

2π

τ

− 2π

τ

∞ e− pτ2

4

dh(ξ)2 E 1

2 , 1 2

cos

ωτ

2

√p

√p

× ×

−i sin

ωτ 2

Fh,αΦ0(ξ) + τ

2Fh,αΦ1(ξ)

e

p

1− τ2

4 m2

e−iωtdpdω.

19 / 22

SLIDE 44

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Difference-Difference Klein Gordon

The Fourier-Laplace technique

A Fourier-Laplace inversion formula Ψ(x, t) = ∞ exp pτ 2 4 ∆h

[Φ(x, t; p)]dp,

with Φ(x, t; p) = − τ 4π

2π

τ

− 2π

τ

−i sin

ωτ 2

Φ0(x) + τ

2Φ1(x)

×

× E 1

2 , 1 2

cos

ωτ

2

√p

√p

e

p

1− τ2

4 m2

e−iωtdω.

Remark: The action exp

pτ2

4 ∆h

[Φ(x, t; p)] corresponds to a discrete

convolution formula. The associated discrete heat kernel K

x, pτ2

4

may

be recovered as a product of the modified Bessel functions of the first kind.

20 / 22

SLIDE 45

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

The discrete Cauchy-Kovaleskaya approach explained

The Cauchy problem that we propose to study here is of the form              e0 Ψ(x, t + τ) − Ψ(x, t − τ) 2τ + e2n+1 2Ψ(x, t) − Ψ(x, t + τ) − Ψ(x, t − τ) 2τ = −DhΨ(x, t), for (x, t) ∈ hZn × τZ≥0 Ψ(x, 0) = Φ0(x), for x ∈ hZn How to rid the ’Sommen-Weyl’ symmetries? From the basic identities e2

0 = −1 and e0e2n+1 = −e2n+1e0, there holds

Ψ(x, t + τ) − Ψ(x, t − τ) =

2τe0Dh + τ 2e2n+1e0∆h
Ψ(x, t). Moreover,

from the formal identity Ψ(x, t + τ) − Ψ(x, t − τ) = 2 sinh(τ∂t)Ψ(x, t) there holds that Ψ(x, t) =

∞

k=0

Gk(t; −τ, 2τ) k!

2τe0Dh + τ 2e2n+1e0∆h

k Φ0(x) may be written in terms of Gould polynomials Gk(t; −τ, 2τ) (and whence, in terms of Chebyshev...)

21 / 22

SLIDE 46

Fourier analysis

f Discrete Dirac
perators on the

n−torus Nelson Faustino New developments on discrete Dirac

perators

Discrete Dirac Operators The toroidal approach

Klein-Gordon type equations

The model problem Wave type propagators

Playing around some remarkable connections

A space-time Fourier inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension

Thank you for your attention!

Figure: The Colossus of Rhodes

You are invited to present your recent research in 14-th Symposium Clifford Analysis and Applications. Our Symposium is part of the International Conference of Numerical Analysis and Applied Mathematics (ICNAAM) 2018 in Rhodes.

22 / 22