Wavelet Sets, Fractal Surfaces and Coxeter Groups

Transcription

1 Wavelet Sets, Fractal Surfaces and Coxeter Groups David Larson Peter Massopust Workshop on Harmonic Analysis and Fractal Geometry Louisiana State University February 24-26, 26 Harmonic Analysis and Fractal Geometry p. /24

2 Outline Wavelet Sets based on Dilation/Translation Structure Fractal Functions and Fractal Surfaces Foldable Figures and Coxeter Groups Multiresolution Structures for L 2 (R n ) on Foldable Figures Wavelet Sets based on Dilation/Reflection Structure Harmonic Analysis and Fractal Geometry p. 2/24

3 Fractal Interpolation Functions Fractal Interpolation Functions f introduced by Michael Barnsley in 986 Construction based on Affine Iterated Function Systems graph f is the limit (in the Hausdorff metric) of sequence of compact sets More general construction based on function spaces via Read-Bajraktarević operators (Dubuc, Bedford, M.) Fractal function is limit (in the metric of the underlying function space) of a sequence of functions (M. 995) Regularity properties of fractal functions studied by M. in the setting of Besov and Triebel-Lizorkin spaces (997, 25) Harmonic Analysis and Fractal Geometry p. 3/24

4 Geometric Construction Harmonic Analysis and Fractal Geometry p. 4/24

5 Geometric Construction Harmonic Analysis and Fractal Geometry p. 4/24

6 Geometric Construction Harmonic Analysis and Fractal Geometry p. 4/24

7 Geometric Construction Harmonic Analysis and Fractal Geometry p. 4/24

8 Geometric Construction Harmonic Analysis and Fractal Geometry p. 4/24

9 Geometric Construction Harmonic Analysis and Fractal Geometry p. 4/24

10 Geometric Construction Harmonic Analysis and Fractal Geometry p. 4/24

11 Fractal Functions Theorem. Let Ω R be compact and < N N. Assume u i : Ω Ω are contractive homeomorphisms, λ i : R R bounded functions and s i real numbers, i =,...,N. Let T (f) := N i= [ λi u i + s i f u ] i χui (Ω) If max{ s i } <, then the operator T is contractive on L (Ω) and its unique fixed point F : Ω R satisfies F = N i= [ λi u i + s i F u ] i χui (Ω) F is called an (R-valued) fractal function. Harmonic Analysis and Fractal Geometry p. 5/24

12 (Affine) Fractal Surfaces Systematically defined first by M. (99); Geronimo & Hardin 993; Hardin & M. (993) Defined on (triangular) regions of E n Mappings u i :, x A i x + b i, i =,...,N = i u i and u i u j =, i j λ i : R n R, continuous (affine) functions, i =,...,N < s i < real numbers, i =,...,N linear isomorphism Λ := (λ,...,λ N ) F Λ Harmonic Analysis and Fractal Geometry p. 6/24

13 Example I Harmonic Analysis and Fractal Geometry p. 7/24

14 w w 2 w 3 w 4 x y = z x y = z x y = z x y = z Example II /2 x /2 /2 y + z z 2 z s z z /2 x /2 /2 y + z z 2 z s z z /2 x /2 y + /2 z 2 z 3 z 3 s z z 3 /2 x /2 y + /2 z 2 z 3 z 3 s z z 3 Harmonic Analysis and Fractal Geometry p. 8/24

15 Example III Harmonic Analysis and Fractal Geometry p. 9/24

16 Example III z.5 y x.75 Harmonic Analysis and Fractal Geometry p. 9/24

17 Example III z.5 y x.75 Harmonic Analysis and Fractal Geometry p. 9/24

18 Example III.75 y z x.75 Harmonic Analysis and Fractal Geometry p. 9/24

19 Example III.75 y z.25.5 x.75 Harmonic Analysis and Fractal Geometry p. 9/24

20 Example III x z.75.5y.25 Harmonic Analysis and Fractal Geometry p. 9/24

21 Fractal Surface Basis I A fractal surface basis is obtained by using the isomorphism Λ F Λ : If F interpolates the set {(x i,y j,z ij )} on, then F = i,j z ij ϕ ij, where ϕ ij (x,y) = {, (x,y) = (x i,y j ), otherwise Harmonic Analysis and Fractal Geometry p. /24

22 Fractal Surface Basis II.5 y x y x z.5 z.5.5 z.5.25 x y Harmonic Analysis and Fractal Geometry p. /24

23 Foldable Figures Definition. [Hoffman & Withers 988] A compact connected subset F of R n is called a foldable figure iff finite set S of hyperplanes that cuts F into finitely many congruent subfigures F,...,F m, each similar to F, so that reflection in any of the hyperplanes in S bounding F k takes it into some F l. Theorem. [Hoffman & Withers 988] A foldable figure F E n is a convex polytope that tessellates E n by reflections in hyperplanes. Harmonic Analysis and Fractal Geometry p. 2/24

24 Coxeter Groups Let H E n be a (linear) hyperplane. A linear transformation ρ is called a reflection about H if ρ(h) = H and ρ(x) = x, if x H. ρ r (x) = x 2 x,r r,r r, for fixed r H. Coxeter Group: A discrete group with a finite set of generators {r i : i =,...,k} satisfying C := r,...,r k (r i r j ) m ij =, i,j k where m ii =, for all i, and m ij 2, for all i j. (m ij = is used to indicate that no relation exists.) Finite Coxeter groups = finite Euclidean reflection groups Harmonic Analysis and Fractal Geometry p. 3/24

25 Root Systems and Weyl Groups I Root System R: finite set of nonzero vectors r,...,r k E n satisfying E n = span {r,...,r k } r,αr R iff α = ± r,s R: s 2 s,r r,r r R r R, the root system R is closed with respect to the reflection through the hyperplane orthogonal to r. r,s R: 2 s,r r,r Z ρ r(s) s Z Weyl Group W of R: group generated by the set of reflections {ρ r : r R}. W <. Harmonic Analysis and Fractal Geometry p. 4/24

26 Root Systems and Weyl Groups II r R is positive (negative) r,x > ( r,x < ) for some x E n. Every R has a basis B = {b i } consisting of positive (negative) roots. Weyl Chamber: C i := {x E n : x,b i > } W acts simply transitive on the Weyl chambers. C := i C i is a noncompact fundamental domain for the Weyl group W. C is convex and connected. Every Weyl group = finite Coxeter group has a fundamental domain that is a simplicial cone. Harmonic Analysis and Fractal Geometry p. 5/24

27 Affine Reflection Groups Reflection about affine hyperplanes: r R,k Z, H r,k := {x E n : x,r = k} ρ r,k (x) = x 2 x,r k r,r r = ρ r (x) + k r Affine Weyl Group: W := ρ r,k r R,k Z W = Theorem. The affine Weyl group W of a root system R is the semi-direct product W Γ, where Γ is the abelian group generated by the coroots r. Moreover, Γ is the subgroup of translations of W and W the isotropy group (stabilizer) of the origin. Harmonic Analysis and Fractal Geometry p. 6/24

28 Essential Reflection Groups Let G be a reflection group and O n the group of linear isometries of E n. There exists a homomorphism φ : G O n φ(g)(x) = g(x) g(), g G, x E n. G is called essential if φ(g) only fixes E n. The elements of ker φ are called translations. Harmonic Analysis and Fractal Geometry p. 7/24

29 Connection to Foldable Figures Theorem. [Bourbaki,968] The reflection group corresponding to a foldable figure F is the affine Weyl group of some root system. Theorem. [Bourbaki,968] If F is a foldable figure then F is the fundamental domain for the group generated by reflections through its bounding hyperplanes. Theorem. [Bourbaki,968] Let G be a reflection group with fundamental domain C. Then C is compact if G is essential and without fixed points. Theorem. There exists a bijection between foldable figures and reflection groups that are essential and without fixed points. Harmonic Analysis and Fractal Geometry p. 8/24

30 Fractal Surfaces on Foldable Figures I Let F E n be a foldable figure with as one of its vertices. Let H be the set of hyperplanes associated with F. Let Σ be the tessellation of F induced by H. Let W be the affine Weyl group generated by H. Then the following properties hold. H consists of the translates of a finite set of linear hyperplanes. W is simply-transitive on Σ, i.e., (σ,τ Σ)!r W : τ = rσ. κ N : κh H. Harmonic Analysis and Fractal Geometry p. 9/24

31 Fractal Surfaces on Foldable Figures II Fix < κ N and let := κf. is also a foldable figures, whose N subfigures i Σ. Let := F. Tessellation and set of hyperplanes for are κσ and κh, resp. By simple transitivity of W, define similitudes u i : i by: u := (/κ)( ) and j = 2,...,N : u j := r j, u. Choose functions λ,...,λ N C(R n, R) satisfying λ i (x) = λ j (x), whenever x = u i (e ij ) = u j (e ij ), where e ij = u i ( ) u j ( ). Harmonic Analysis and Fractal Geometry p. 2/24

32 Fractal Surfaces on Foldable Figures III Construct a fractal function F Λ on. Let C W := { C(R n, R)... C(R n, R) }{{} N factors For Λ C W, define F Λ by } : r W F Λ r := F Λ(r) r, r W, where Λ(r) = (Λ(r),...,Λ(r) N ) is the r-th coordinate of Λ. Harmonic Analysis and Fractal Geometry p. 2/24

33 MRA on Foldable Figures I Let V be a linear space of R-valued functions on R n, < κ N, and W an affine Weyl group. V is dilation - invariant with scale κ (f V = D κ f := f( /κ) V ) V is W - invariant (f V = f r V, r W) D κ - invariance of a global fractal function F Λ can be expressed in terms of an associated δ κ - invariance of Λ C W. Harmonic Analysis and Fractal Geometry p. 22/24

34 MRA on Foldable Figures II Choose a finite-dimensional subspace U of C(R n, R) such that U W is δ κ - invariant. { } Define V := F Λ : Λ U W and V m := Dκ m V If dim U = d, then dimv = Nd. Gram-Schmidt Orthogonalization = orthogonal basis {ϕ j : j =,...,Nd} of V. Let Φ := (ϕ,...,ϕ Nd ) T. Then V V = sequence of (Nd Nd)-matrices {P(r) : r W}, only a finite number of which are nonzero, such that Φ(x/κ) = r W P(r) (Φ r)(x) Harmonic Analysis and Fractal Geometry p. 23/24

35 MRA on Foldable Figures III For m N, define the wavelet spaces W m := V m V m. dim W = dimv dimv = (κ n )(Nd) =: K Gram-Schmidt Orthogonalization = orthogonal basis {ψ l : l =,...,K} of W. Let Ψ := (ψ,...,ϕ K ) T. W V = sequence of K Nd-matrices {Q(r) : r W}, only a finite number of which are nonzero, such that Ψ(x/κ) = r W Q(r) (Φ r)(x) Harmonic Analysis and Fractal Geometry p. 24/24