Recent results in discrete Clifford analysis

Transcription

1 Recent results in discrete Clifford analysis Hilde De Ridder joint work with F. Sommen Department of Mathematical Analysis, Ghent University Mathematical Optics, Image Modelling and Algorithms / 30

2 Overview Introduction to discrete Clifford analysis Some early results Recent results: connection between continuous setting and the discrete grid 1 / 30

3 Discrete Clifford analysis complex analysis dim = 2 Clifford analysis dim 2 discrete complex analysis dim = 2 discrete Clifford analysis dim 2 2 / 30

4 By no means a comprehensive list... R. P. Isaacs, A finite difference function theory, Univ. Nac. Tucumán. Revista A. 2, 1941, p J. Ferrand, Fonctions préharmoniques et functions préholomorphes, Bull. Sci. Math., 68, 1944, p R.P. Isaacs, Monodiffric functions, Construction and applications of conformal maps: Proceedings of a symposium, 18, 1952, U. S. Government Printing Office, Washington D.C., p R.J. Duffin, Basic properties of discrete analytic functions, Duke Math. J., 23, 1956, p A. Dimakis, F. Müller-Hoissen, Discrete Differential Calculus, Graphs, Topologies and Gauge Theory, J. Math. Phys., 35, 1994, p A. Hommel, Fundamentallösungen partieller Differenzenoperatoren und die Lösung diskreter Randwertprobleme mit Hilfe von Differenzenpotentialen, 1998, PhD-thesis, Bauhaus Universität Weimar. K. Gürlebeck and A. Hommel, On finite difference potentials and their applications in a discrete function theory, Math. Meth. Appl. Sci., 25, 2002, p / 30

5 I. Kanamori, N. Kawamoto, Dirac-Kaehler Fermion from Clifford Product with Noncommutative Differential Form on a Lattice, Int. J. Mod. Phys., A19, 2004, p A.I. Bobenko, C. Mercat, Y. Suris, Linear and nonlinear theories of discrete analytic functions. Integrable structures abd isomonodromic Green s function, J. Reine Angew. Math., 583, 2005, p N. Faustino, K. Gürlebeck, A. Hommel, U. Kähler, Difference Potentials for the Navier-Stokes equations in unbounded domains, J. Diff. Eq. & Appl., 12 (6), 2006, p N. Faustino, U. Kähler, F. Sommen, Discrete Dirac operators in Clifford Analysis, Adv. Appl. Cliff. Alg., 17 (3), 2007, p F. Brackx, H. De Schepper, F. Sommen, L. Van de Voorde, Discrete Clifford Analysis: A Germ of Function Theory, Hypercomplex Analysis, eds. I. Sabadini and M. Shapiro and F. Sommen, 2009, Birkhäuser, p H. De Schepper, F. Sommen, L. Van de Voorde, A basic framework for discrete Clifford analysis, Experiment. Math., 18 (4), 2009, p N. Faustino, Discrete Clifford analysis, PhD-thesis, Universidade de Aveiro, Aveiro, / 30

6 Goal Working over a grid Z m h (often we choose h = 1) Define a discrete Dirac operator as a refinement of the discrete Laplacian Construction of a discrete function theory, both as a counterpart of the theory of holomorphic functions in the complex plane as of its higher dimensional version(s) Connect the classical properties in Clifford analysis with the discrete structure of Z m 5 / 30

7 Framework Z m h = {(x 1h,..., x m h) x 1,..., x m Z} Orthonormal basis e 1,..., e m One-sided forward/backward differences, j = 1,..., m: + j f (x) = f (x + he j) f (x), h j f (x) = f (x) f (x he j) h Discrete Laplacian w.r.t. the graph: star Laplacian [f ] (x) = For now: h = 1 m ( ) f (x + hej ) + f (x he j ) h 2 2m h 2 f (x) j=1 6 / 30

8 Defining a discrete Dirac operator ± j decompose the star Laplacian : m m [f ] (x) = + j j [f ] (x) = j + j j=1 j=1 [f ] (x) First consider the Clifford algebra R m,0 of order (m, 0): e 2 i = 1, e i e j + e j e i = 2 δ ij Decompose the basis vectors of R m,0 as e j = e + j + e j where e ± j are forward /backward basis vectors that also carry an orientation. 7 / 30

9 Free algebra over {e + j, e j } satisfying 3 assumptions 1. The forward and the backward basis vector in each particular cartesian direction add up to the traditional basis vector in that direction e + j + e j = e j, j = 1,..., m 2. Dimensional equivalence: all cartesian directions play the same role in the metric ( rotational invariance ) 3. The positive and negative orientations of any cartesian direction play an equivalent role ( reflection invariance ) We arrive at: ( e + j ) 2 = ( e j ) 2 = 0 e + j e j + e j e + j = 1 { } e ± j, e ± k = 0, j k 8 / 30

10 Discrete Dirac operator Discrete Dirac operator = m ( e + j + j + e j j ) j=1 Denote the co-ordinate differences j = e + j + j + e j j Refinement of harmonicity: 2 = : A discrete function f is discrete monogenic if f = 0 9 / 30

11 Operator calculus Continuous setting Discrete setting x = m e j xj = j=1 m j=1 ( e + j + j + e j j ) = m j=1 j m x = e j x j ξ = m j=1 j=1 { x, x} = 2E + m E ξ j 10 / 30

12 Discrete vector variable and Euler operator = m j=1 ( e + j + j + e j j ) The difference operators j behave as lowering operators since they lower the degree of a polynomial by one. We will introduce raising operators ξ j, raising the degree of polynomials by one, by assuming ξj is vector-valued: ξ j = e + j X j + e j X + j with X ± j scalar operators m Denote ξ = j=1 ξ j, then there exists an operator E such that ξ + ξ = 2 E + m E ξ ξ E = ξ E E = 11 / 30

13 By assuming that j and ξ j are vector-valued and by imposing the intertwining relations ξ + ξ = 2 E + m E ξ ξ E = ξ E E = we find the skew-weyl relations: j ξ j ξ j j = 1 { j, ξ k } = { j, k } = {ξ j, ξ k } = 0, j k At the level of the operators X ± j, ± j : + j X + j X j j = 1 j X j X + j + j = 1 12 / 30

14 Discrete homogeneous polynomials A polynomial P k is discrete homogeneous of degree k if E P k = k P k where E = m j=1 ξ j j Since E ξ j = ξ j (E + 1), we find that ξj k [1] are (basic) discrete homogeneous polynomials of degree k ξ 2k+1 j [1] = x j k ( x 2 j s 2 h 2) ( ) e + j + e j s=1 ( ξj 2k [1] = xj 2 + k x j h ( e + j e j e j e + j )) k 1 (xj 2 s 2 h 2 ) s=1 13 / 30

15 Basic discrete homogeneous polynomials of degree k: ξ 2k+1 j [1] = x j k ( x 2 j s 2 h 2) ( ) e + j + e j s=1 ( ξj 2k [1] = xj 2 + k x j h ( e + j e j e j e + j )) k 1 (xj 2 s 2 h 2 ) s=1 Note that for fixed x j h Z h : ξ k j [1](x j h) = 0, for all k 2 x j + 1 Because of skew-weyl rule j ξ j ξ j j = 1, we have that j ξ k j [1] = k ξ k 1 j [1] ξ j is an operator acting on discrete functions, it is not the multiplication with x j. H. De Schepper, F. Sommen, L. Van de Voorde, A basic framework for discrete Clifford analysis, Exp. Math. 18 (4), 2009, p H. De Ridder, H. De Schepper, U. Kähler, F. Sommen, Discrete function theory based on skew Weyl relations, Proc. Amer. Math. Soc. 138, 2010, p / 30

16 Some early results: building an operator theory 15 / 30

17 Function-theory We now consider discrete functions f, defined on Z m : f = A f A e A, A = {a 1,..., a k }, 1 a 1 <... < a k m and k e A = e as, e as { e + s, e s, e + s e s, e s e + } s. s=1 H: space of discrete harmonic functions: null-functions of M: space of discrete monogenic functions: null-functions of H k and M k : spaces of discrete k-homogeneous harmonic, resp. monogenic, functions. 16 / 30

18 Function-theory centered around monogenic functions Lemma (Fischer decomposition) Let P k be the space of discrete homogeneous polynomials of degree k, then k P k = ξ s M k s s=0 Lemma (Taylor series decomposition) Let f be a discrete function defined on Z m, then the Taylor series of f is well-defined f (x) = k=0 1 k! m... l 1 =1 m ξ l1... ξ lk [1](x) ( lk... l1 f ) (0) l k =1 Furthermore, if f is discrete monogenic, then f can be developed into a convergent series of discrete monogenics. 17 / 30

19 Example: delta-function Consider the (1D) discrete delta-function δ 0 which take the value 1 in the origin and value 0 everywhere else. Apply Fischer decomposition: δ 0 (x 1 ) = l=0 ( 1) l l!l! ξ 2l 1 [1] + l=0 ( 1) l+1 (l + 1)! l! ξ2l+1 1 [1] ( e + 1 ) e 1 If we now replace our expressions for ξ1 k [1](x) in this formula, for x 1 R we see that δ 0 (x 1 ) becomes sin(π x 1 ) π x 1 H. De Ridder, H. De Schepper, F. Sommen, Taylor series expansion in discrete Clifford analysis, Compl. Anal. Op. Theory 8 (2), 2014, p / 30

20 Theorem (Sampling theorem for lowpass functions, 1948, Shannon) Let f (t) contain no frequencies higher than W. Then f (t) = n= sin (π (2W t n)) ( n ) X n, with X n = f π (2W t n) 2W 19 / 30

21 Theorem (Sampling theorem for lowpass functions, 1948, Shannon) Let f (t) contain no frequencies higher than W. Then f (t) = n= sin (π (2W t n)) ( n ) X n, with X n = f π (2W t n) 2W The theorem states that if a function contains no frequencies higher than a given W cycles per second, then it is completely determined by giving its ordinates at a series of sampling points spaced 1 2W seconds apart. 19 / 30

22 Theorem (Sampling theorem for lowpass functions, 1948, Shannon) Let f (t) contain no frequencies higher than W. Then f (t) = n= sin (π (2W t n)) ( n ) X n, with X n = f π (2W t n) 2W The theorem states that if a function contains no frequencies higher than a given W cycles per second, then it is completely determined by giving its ordinates at a series of sampling points spaced 1 2W seconds apart. Sampling a continuous function f thus is nothing else then restricting f to the grid, decomposing it into discrete delta functions and subsequently expanding it again to the whole of R: f (x) = n= f (n) sin(π(x n)) π(x n) 19 / 30

23 Construction of monogenic functions By means of Cauchy-Kovalevskaya extension: if f is defined on Z m 1, then there exists a unique monogenic extension of f to Z m : Definition The CK extension CK [f ] of a discrete function f (x 2,..., x m ) is the discrete monogenic function CK [f ] (x 1, x 2,..., x m ) = k=0 ξ k 1 [1](x 1) k! f k (x 2,..., x m ) (1) where f 0 = f and f k+1 = ( 1) k+1 m j=2 j f k. H. De Ridder, H. De Schepper, F. Sommen, The Cauchy-Kovalevskaya Extension Theorem in Discrete Clifford Analysis, Comm. Pure Appl. Math 10(4), pp / 30

24 Construction of monogenic functions Or by means of Fueter s theorem: Theorem Let m 2 be even and f (ξ 1, ξ 2 ) be a discrete monogenic function in ξ 1,ξ 2, i.e. ( ) f (ξ 1, ξ 2 ) = 0. Denote = m j=2 j and ξ = m j=2 ξ j. Replacing every ξ 2 by ξ and letting the discrete Laplace operator act m 2 1 times on f (ξ 1, ξ ) gives a discrete monogenic function, i.e. ( 1 + ) m 2 1 f (ξ 1, ξ ) = 0 H. De Ridder, F. Sommen, Fueters theorem in discrete Clifford analysis, Math. Meth. Appl. Sciences, 39 (7), p , / 30

25 Discrete Heat equation Consider a discrete version of the Heat equation: ( t ) u(x, t) = 0, x Z m, t R +, i.e. the case where space is discrete and time continuous. Then t k G t = H(t) k! 2k δ 0 = H(t) exp ( t 2) δ 0 k=0 with H(t) the continuous Heaviside, is a fundamental solution of the discrete Heat equation, i.e. it satisfies ( t ) G t (x, t) = δ(t) δ 0 in distributional sense. It consists of continuous distributions in t combined with discrete distributions in x. 22 / 30

26 The continuous Heat polynomials p β (x, t) are polynomial solutions to the Heat equation ( t ) u(x, t) = 0, x R, t > 0, with initial condition p β (x, 0) = x β. For β Z, we get the following solutions: p n (x, t) = n! n 2 k=0 t k (n 2k)! k! x n 2k Discrete: the discrete Heat polynomials p n (x, t) p n (x, t) = n 2 l=0 t l l! n! (n 2l)! ξn 2l [1]. are polynomial solutions of the discrete Heat equation ( t ) u(x, t) = 0, x Z, t > 0, with initial condition p n (x, 0) = ξ n [1]. F. Baaske, S. Bernstein, H. De Ridder, F. Sommen, On solutions of a discretized heat equation in discrete Clifford analysis, J. Diff. Eq. Appl. 20 (2), pp / 30

27 Rotations of the discrete grid How to connect the discrete grid Z m with rotations over all possible angles θ? Because the function-theory is centered around the concept of monogenic functions, we want rotation operators to commute with the Dirac operator, as to not disrupt the monogenicity. 24 / 30

28 In classical harmonic analysis, we consider the infinitesimal rotations L a,b = x a xb x b xa which leave the Laplacian invariant. In Euclidean Clifford analysis, we consider dr(e a,b ) = x a xb x b xa 1 2 e a e b which are symmetries of the Dirac operator. 25 / 30

29 Definition The discrete angular momentum operators are given by L a,a = 0 and Ω a,b = R b R a (ξ a b + ξ b a ), a b. The operators Ω a,b commute with, ξ 2 and E. Definition The discrete operators dr(e a,b ) are given by dr(e a,a ) = 0 and ( dr(e a,b ) = R b R a ξ a b + ξ b a 1 ), a b. 2 The operators dr(e a,b ) commute with, ξ and E. Here, R b is an operator that commutes with ξ j and j and anticommutes with ξ k and k, k j. It acts on 1 as multiplication by e b. 26 / 30

30 Example: rotation of δ (2,0) in dimension 2 exp (θ Ω a,b ) resp. exp (θ dr(e a,b )) defines the rotation in the a, b-plane over (continuous) angle θ [0, 2π[. H. De Ridder, T. Raeymaekers, F. Sommen. Rotations in Discrete Clifford Analysis, Applied Mathematics and Computation 285, 2016, p / 30

31 Discrete translations Consider (h = 1): T j f (x) = f (x e j ) This translation does not work well with the ξ j, j -formalism What about continuous translations? What do we want? Some symmetry of the discrete Dirac operator that translates discrete functions in the e j -direction. The obvious choice ± j does not work: ( ) exp ± j xj 2 = xj 2 + (2 x j ± 1) (2) = (x j + 1) 2 ± 1. Combining ± j into the co-ordinate difference j does not give a symmetry of the discrete Dirac operator: [ j, ] = 2 k j j k. 28 / 30

32 Definition Let R j be again a discrete operator, commuting with ξ j and j, anti-commuting with ξ k and k (k j) and R j [1] = e j. Then the translation T j of a discrete function f in the e j -direction is T j [f ] = exp (R j j ) f. Example Basic homogeneous polynomials: T j ξ k j [1] = k s=0 ( ) k ξj s [1] (e j ) s. s This is not equal to (ξ j + e j ) k j [1] since ξ j e j e j ξ j. 29 / 30

33 Properties Translations T j and T k mutually commute. Continuous translations: let m j=1 a j e j, with a j R then m exp a j R j j f j=1 The image of a delta function /distribution is then a linear combination of delta functions /distributions. Translating ξ 2 [1] over a = m j=1 a j e j, a j R, gives us an appropriate definition for m ξ, a = ξ j R j a j j=1 This will serve as inspiration for the introduction of discrete plane waves. 30 / 30