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

Fourier analysis of Discrete Dirac operators on the Fourier analysis of Discrete Dirac n − torus Nelson Faustino operators on the n − torus R n / 2 π h Z n New developments on discrete Dirac operators Discrete Dirac Operators Nelson Faustino The toroidal approach Klein-Gordon type equations Center of Mathematics, Computation and Cognition, UFABC The model problem Wave type nelson.faustino@ufabc.edu.br propagators Playing around some remarkable 19th Annual Workshop on Applications and Generalizations of connections A space-time Fourier Complex Analysis, March 23-24, 2018 inversion formula The discrete heat semigroup connection The Cauchy-Kovaleskaya extension 1 / 22

Fourier analysis of Discrete Dirac operators on the n − torus New developments on discrete Dirac operators 1 Nelson Faustino Discrete Dirac Operators New The toroidal approach developments on discrete Dirac operators Discrete Dirac Operators Klein-Gordon type equations 2 The toroidal approach The model problem Klein-Gordon Wave type propagators type equations The model problem Wave type propagators Playing around some remarkable connections 3 Playing around some remarkable A space-time Fourier inversion formula connections A space-time Fourier The discrete heat semigroup connection inversion formula The discrete heat The Cauchy-Kovaleskaya extension semigroup connection The Cauchy-Kovaleskaya extension 2 / 22

The aim of this talk Present an abridged version of the following preprints: Fourier analysis Faustino, N. (January 2018) . Relativistic Wave Equations on of Discrete Dirac operators on the the lattice: an operational perspective , arXiv:1801.09340. n − torus Nelson Faustino Faustino, N. (February 2018) . A note on the discrete New Cauchy-Kovaleskaya extension , arXiv:1802.08605. developments on discrete Dirac operators 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 Figure: I hope that the organizers have ordered donuts to stimulate the Cauchy-Kovaleskaya extension research discussions on the coffee break ( A wishful thinking ). 3 / 22

The aim of this talk Present an abridged version of the following preprints: Fourier analysis Faustino, N. (January 2018) . Relativistic Wave Equations on of Discrete Dirac operators on the the lattice: an operational perspective , arXiv:1801.09340. n − torus Nelson Faustino Faustino, N. (February 2018) . A note on the discrete New Cauchy-Kovaleskaya extension , arXiv:1802.08605. developments on discrete Dirac operators 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 Figure: I hope that the organizers have ordered donuts to stimulate the Cauchy-Kovaleskaya extension research discussions on the coffee break ( A wishful thinking ). 3 / 22

Discrete Dirac Operators Multivector formulation Finite Difference Dirac Operators of Forward/Backward Type: Fourier analysis of Discrete Dirac operators on the n n n − torus � � D + e j ∂ + j h and D − e j ∂ − j h = h = h , Nelson Faustino j = 1 j = 1 New developments on where ∂ ± j discrete Dirac correspond to the forward/backward finite difference operators h operators on the lattice h Z n , and e 1 , e 2 , . . . , e n corresponds to the basis of the Discrete Dirac Operators Clifford algebra of signature ( 0 , n ) . The toroidal approach Klein-Gordon Multivector operators acting on C ℓ n , n ∼ type equations = End ( C ℓ 0 , n ) : The model problem ∂ + h = D + h ∧ and ∂ − h = D − h • , where: Wave type propagators n Playing around � ∂ + h = D + e j ∧ ∂ + j h ∧ = h stands the multivector counterpart of the some remarkable connections j = 1 A space-time Fourier inversion formula exterior derivative d . The discrete heat semigroup n connection � ∂ − h = D − e j • ∂ − j h • = stands the multivector counterpart of the The h Cauchy-Kovaleskaya extension j = 1 co-differential form δ = ⋆ − 1 d ⋆ . 4 / 22

Discrete Dirac Operators Multivector formulation Finite Difference Dirac Operators of Forward/Backward Type: Fourier analysis of Discrete Dirac operators on the n n n − torus � � D + e j ∂ + j h and D − e j ∂ − j h = h = h , Nelson Faustino j = 1 j = 1 New developments on where ∂ ± j discrete Dirac correspond to the forward/backward finite difference operators h operators on the lattice h Z n , and e 1 , e 2 , . . . , e n corresponds to the basis of the Discrete Dirac Operators Clifford algebra of signature ( 0 , n ) . The toroidal approach Klein-Gordon Multivector operators acting on C ℓ n , n ∼ type equations = End ( C ℓ 0 , n ) : The model problem ∂ + h = D + h ∧ and ∂ − h = D − h • , where: Wave type propagators n Playing around � ∂ + h = D + e j ∧ ∂ + j h ∧ = h stands the multivector counterpart of the some remarkable connections j = 1 A space-time Fourier inversion formula exterior derivative d . The discrete heat semigroup n connection � ∂ − h = D − e j • ∂ − j h • = stands the multivector counterpart of the The h Cauchy-Kovaleskaya extension j = 1 co-differential form δ = ⋆ − 1 d ⋆ . 4 / 22

Star-Laplacian factorization There are several possibilities still. Fourier analysis of Discrete Dirac n f ( x + h e j ) + f ( x − h e j ) − 2 f ( x ) � operators on the Star Laplacian: ∆ h f ( x ) = n − torus h 2 Nelson Faustino j = 1 New Using forward and backward Dirac 1 developments on � � operators: ∆ h = − 1 D + h D − h + D − h D + discrete Dirac h operators 2 Discrete Dirac Using a central difference Dirac 2 Operators � � 2 The toroidal operator: ∆ h = − 1 D + h / 2 + D − approach h / 2 4 Klein-Gordon type equations A geometric calculus factorization: ∆ h 3 The model problem equals to Wave type propagators � 2 = − Playing around � � � 2 . Figure: The star laplacian in D + h ∧ + D − D + h ∧ − D − h • h • some remarkable connections h Z 3 . A space-time Fourier inversion formula The discrete heat This construction is closely related to the one obtained in my joint semigroup connection paper with U. K¨ ahler and F. Sommen Discrete Dirac operators in Clifford The Cauchy-Kovaleskaya analysis ( Advances in Applied Clifford Algebras 17 (3), 451-467 ). extension 5 / 22

Star-Laplacian factorization There are several possibilities still. Fourier analysis of Discrete Dirac n f ( x + h e j ) + f ( x − h e j ) − 2 f ( x ) � operators on the Star Laplacian: ∆ h f ( x ) = n − torus h 2 Nelson Faustino j = 1 New Using forward and backward Dirac 1 developments on � � operators: ∆ h = − 1 D + h D − h + D − h D + discrete Dirac h operators 2 Discrete Dirac Using a central difference Dirac 2 Operators � � 2 The toroidal operator: ∆ h = − 1 D + h / 2 + D − approach h / 2 4 Klein-Gordon type equations A geometric calculus factorization: ∆ h 3 The model problem equals to Wave type propagators � 2 = − Playing around � � � 2 . Figure: The star laplacian in D + h ∧ + D − D + h ∧ − D − h • h • some remarkable connections h Z 3 . A space-time Fourier inversion formula The discrete heat This construction is closely related to the one obtained in my joint semigroup connection paper with U. K¨ ahler and F. Sommen Discrete Dirac operators in Clifford The Cauchy-Kovaleskaya analysis ( Advances in Applied Clifford Algebras 17 (3), 451-467 ). extension 5 / 22

Star-Laplacian factorization There are several possibilities still. Fourier analysis of Discrete Dirac n f ( x + h e j ) + f ( x − h e j ) − 2 f ( x ) � operators on the Star Laplacian: ∆ h f ( x ) = n − torus h 2 Nelson Faustino j = 1 New Using forward and backward Dirac 1 developments on � � operators: ∆ h = − 1 D + h D − h + D − h D + discrete Dirac h operators 2 Discrete Dirac Using a central difference Dirac 2 Operators � � 2 The toroidal operator: ∆ h = − 1 D + h / 2 + D − approach h / 2 4 Klein-Gordon type equations A geometric calculus factorization: ∆ h 3 The model problem equals to Wave type propagators � 2 = − Playing around � � � 2 . Figure: The star laplacian in D + h ∧ + D − D + h ∧ − D − h • h • some remarkable connections h Z 3 . A space-time Fourier inversion formula The discrete heat This construction is closely related to the one obtained in my joint semigroup connection paper with U. K¨ ahler and F. Sommen Discrete Dirac operators in Clifford The Cauchy-Kovaleskaya analysis ( Advances in Applied Clifford Algebras 17 (3), 451-467 ). extension 5 / 22