Fourier Bases on the Skewed Sierpinski Gasket Calvin Hotchkiss University Nebraska-Iowa Functional Analysis Seminar November 5, 2016
Part 1: A Fast Fourier Transform for Fractal Approximations
The Fractal Approximation S N Begin with an iterated function system generated by contractions {ψ 0, ψ 1,..., ψ K 1 } on R d of the following form: ψ j (x) = A(x + b j ) where A is a d d invertible matrix with A < 1. We require A 1 to have integer entries, the vectors b j Z d, and b 0 = 0. Define the finite orbit: S N ( 0) := {ψ jn 1 ψ jn 2 ψ j1 ψ j0 ( 0) : j k {0, 1,..., K 1}}. This orbit is our Nth fractal approximation.
A Fourier Basis on S N We then choose a second iterated function system generated by {ρ 0, ρ 1,..., ρ K 1 } of the form ρ j (x) = Bx + c j where B = (A T ) 1, with c j Z d, and c 0 = 0. Define the finite orbit: T N ( 0) := {ρ jn 1 ρ jn 2 ρ j1 ρ j0 ( 0) : j k {0, 1,..., K 1}}. These are the frequencies for an exponential basis on L 2 (µ n ), where µ n = 1 K N s S n δ s.
Example: The Skewed Sierpinski Gasket Let: ( 1 ) / A = 3 0 0 1 / 3 c 1 = So that ( ) x ψ 0 = 1 ( ) x ( y ) 3 y x ψ 1 = 1 ( ) x + 2 ( y ) 3 y x ψ 2 = 1 ( ) x y 3 y + 2 ( ) 1 2 b 1 = ( ) 2 0 c 2 = ( ) 2 1 b 2 = ( ) 0 2 ( ) ( ) x x ρ 0 = 3 ( y ) ( y ) ( ) x x 1 ρ 1 = 3 + ( y ) ( y ) ( 2 ) x x 2 ρ 2 = 3 + y y 1
Approximations and Frequencies for Small N 8 0.8 6 0.6 0.4 4 0.2 2 0.2 0.4 0.6 0.8 2 4 6 8 1.0 S 2 (0, 0) 25 T 2 (0, 0) 0.8 20 0.6 15 0.4 10 0.2 5 0.2 0.4 0.6 0.8 5 10 15 20 25 S 3 (0, 0) T 3 (0, 0)
1.0 80 0.8 60 0.6 0.4 40 0.2 20 0.2 0.4 0.6 0.8 1.0 S 4 (0, 0) 20 40 60 80 T 4 (0, 0)
Matrices The { c j } must be chosen so that the matrix is invertible (or Hadamard). M 1 = (e 2πi c j A b k ) j,k We then define ) M N = (e 2πi s j t k t k T N, s j S N and show that M N is a discrete Fourier Transform matrix for S N.
Diţǎ s Construction If A is a K K Hadamard matrix, B is an M M Hadamard matrix, and E 1,..., E K 1 are M M unitary diagonal matrices, then the KM KM block matrix H defined by: a 00 B a 01 E 1 B... a 0(K 1) E K 1 B a 10 B a 11 E 1 B... a 1(K 1) E K 1 B...... a (K 1)0 B a (K 1)1 E 1 B... a (K 1)(K 1) E K 1 B is a Hadamard matrix.
Similarly, if A, B, E 1,..., E K 1 invertible, H will also be invertible. For C = A 1, H 1 is: c 00 B 1 c 01 B 1... c 0(K 1) B 1 c 10 B 1 E1 1 c 11 B 1 E1 1... c 1(K 1) B 1 E1 1........ c (K 1)0 B 1 E 1 K 1 c 1(K 1) B 1 E 1 K 1... c (K 1)(K 1) B 1 E 1 K 1.
Complexity of Matrix Multiplication Let v be a vector of length KM. Consider H v where H is the block matrix as in Equation (9). Utilizing the block form of the matrix H, we obtain that the computational complexity of H v is O(M 2 K + MK 2 ), whereas for a generic KM KM matrix, the computational complexity is O(K 2 M 2 ). Thus, the block form of H reduces the computational complexity of the matrix multiplication.
A Fast Fourier Transform on S N The matrix M N representing the exponentials with frequencies given by T N ( 0) on the fractal approximation S N ( 0), both ordered in a particular manner, has the form: m 00 M N 1 m 01 D N,1 M N 1... m 0(K 1) D N,K 1 M N 1 m 10 M N 1 m 11 D N,1 M N 1... m 1(K 1) D N,K 1 M N 1..... m (K 1)0 M N 1 m (K 1)1 D N,1 M N 1... m (K 1)(K 1) D N,K 1 M N 1 Where m jk = [M 1 ] jk and the D N,m are unitary diagonal matrices. Therefore, by Diţǎ s construction, M N is invertible, and if M 1 is Hadamard, then M N is also Hadamard..
Block form for Inverse The matrix M N representing the exponentials with frequencies given by T N on the fractal approximation S N, both ordered in a different manner, has the form: m 00 MN 1 m 01 MN 1... m 0(K 1) MN 1 m 10 MN 1 DN,1 m 11 MN 1 DN,1... m 1(K 1) MN 1 DN,1......... m (K 1)0 MN 1 DN,K 1 m (K 1)1 MN 1 DN,K 1... m (K 1)(K 1) MN 1 DN,K 1 Where D N,l are also unitary diagonal matrices. Therefore, M 1 N has the form of Diţǎ s construction.
Complexity is O(n log n) By utilizing the block forms above, the computational complexity of multiplying a vector v by either M N or MN 1 is reduced to O(N K N ), or, in terms of the matrix size n = K N, O(n log n). This is the same computation time as the standard Fast Fourier Transform.
Part 2: Fourier Bases on the Skewed Sierpinski Gasket
The full Skewed Sierpinski Gasket! For any iterated function system on R d generated by contractions {ψ 0, ψ 1,..., ψ K 1 } of the form: ψ j (x) = A(x + b j ) where A is a d d invertible matrix with A < 1, J.E. Hutchinson (1981, [5]) proved there is a unique closed bounded set S with S = K 1 j=0 ψ j (S) That is, the compact attractor set S is invariant under {ψ 0, ψ 1,..., ψ K 1 } and their compositions.
In the same paper, Hutchinson also showed that there is a Borel probability measure ν on S with the property that, for all continuous f : f ( x)dν(x) = 1 K 1 f (ψ j ( x))dν(x). K S j=0 For our example of {ψ 0, ψ 1, ψ 2 } above, the invariant set S = {(x, y) x C 3, y C 3, x + y C 3 } where C 3 is the standard middle-third Cantor set. We call the corresponding probability measure ν 3.
Is T = N T N(0, 0) a complete set of frequencies for S? To (try to) show that E = {e 2πi(t x) t N T N (0, 0)} is a Fourier basis for L 2 (ν 3 ), we reconstruct it using a representation of the Cuntz algebra O 3 on L 2 (ν 3 ). First, choose filters for L 2 (ν 3 ): m 0 (x, y) = 1 3 e 2πi(x,y) (0,0) = 1 3 m 1 (x, y) = 1 3 e 2πi(x,y) (2,1) m 2 (x, y) = 1 3 e 2πi(x,y) (1,2)
An Orthonormal Set of Exponentials on S Then for j = 0, 1, 2, R(x, y) = 3(x, y) mod 1: S j f (x, y) = M = m j (x, y)f (R(x, y)) satisfy the Cuntz relations: S i S j = δ i,j I and 2 j=0 S js j = I. E is the orbit of the constant function 1 under specific powers of these {S j }; thus E is an orthonormal set for L 2 (ν 3 ). We tried to show that E was complete; however...
Answer: No! Lemma (Jorgensen and Pedersen, 1998) Let Q 1 (t) := λ P ˆµ(t λ) 2, for t R d and ˆµ the inverse Fourier transform ˆµ(t) = e 2πi(t x) dµ(x). Then {e 2πi(λ x) : λ P} is an orthonormal basis for L 2 (µ) if and only if Q 1 1 on R d. In our case, for P = T = N T N(0, 0), ν 3 (( 1/2, 1) (a, b)) = 0 for all (a, b) T. Therefore, Q 1 ( 1/2, 1) = 0 and E is NOT a Fourier basis for L 2 (ν 3 )!
Completing our spectrum Theorem (Dutkay and Jorgensen, 2006) Let Λ R d be the smallest set that contains C for every W B -cycle C, and such that SΛ + L Λ. Then {e 2πiλ x λ Λ} is an orthonormal basis for L 2 (µ B ). For the case of µ B = ν 3, S = 3I 2 and L = {(0, 0), (1, 2), (2, 1)}; SΛ + L = {ρ j (Λ) j (0, 1, 2)}. The W B -cycles are (0, 0), (1, 1/2), and (1/2, 1).
A Complete Spectrum Therefore, for then T N (x, y) := {ρ jn 1 ρ jn 2 ρ j1 ρ j0 (x, y) : j k {0, 1, 2}}, T = N (T N (0, 0) T N ( 1/2, 1) T N ( 1, 1/2)) forms a complete set of frequencies for S.
T 5 (0, 0) T 5 ( 1/2, 1) T 5 ( 1, 1/2)
Another spectrum Theorem (H. and Weber, 2016) {e 2πi(u,u/2) (x,y) u Z} is an orthonormal basis for L 2 (ν 3 ). This spectrum comes from the dual iterated function system: ( ) ( ) ( ) ( ) ( ) ( ) x x x x 2 x η 0 = 3 η y y 1 = 3 + η y y 1 2 = 3 y which has W B -cycles (0, 0), (2, 1), and (1, 1/2). ( x y ) + ( ) 4 2
Third iteration of the dual function system {η j } applied to (0, 0), ( 2, 1), and ( 1, 1/2).
Thank you.
References P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A 37 (2004), no. 20, 5355 5374. Dorin Ervin Dutkay and Palle E. T. Jorgensen, Iterated Function Systems, Ruelle Operators, and Invariant Projective Measures, Mathematics of Computation, 75 (2006), no. 256, 1931-1970., Analysis of orthogonality and of orbits in affine iterated function systems, Math. Z. (2007) no. 256, 801-823. Calvin Hotchkiss and Eric Weber, A Fast Fourier Transform for Fractal Approximations. Excursions in Harmonic Analysis: the February Fourier Talks at the Norbert Weiner Center, forthcoming. John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713 747. Palle E. T. Jorgensen and Steen Pedersen, Dense analytic subspaces in fractal L 2 -spaces, J. Anal. Math. 75 (1998), 185 228.