Fourier Bases on Fractals
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1 Fourier Bases on Fractals Keri Kornelson University of Oklahoma - Norman February Fourier Talks February 21, 2013 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
2 Coauthors This is joint work with Palle Jorgensen (University of Iowa) Karen Shuman (Grinnell College). K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
3 Outline 1 Bernoulli convolution measures 2 Fourier bases 3 Families of ONBs 4 Operator-fractal K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
4 Bernoulli convolution measures Convolution measure Let λ (0, 1). A fractal Cantor subset of R is the unique set X λ satisfying the invariance relation: X λ = λ(x λ + 1) λ(x λ 1). (1) K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
5 Bernoulli convolution measures Convolution measure Let λ (0, 1). A fractal Cantor subset of R is the unique set X λ satisfying the invariance relation: X λ = λ(x λ + 1) λ(x λ 1). (1) The Bernoulli convolution measure µ λ is the unique probability measure satisfying: f dµ λ = 1 f(λx + λ) + f(λx λ) dµ λ (x). (2) 2 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
6 Bernoulli convolution measures Convolution measure Let λ (0, 1). A fractal Cantor subset of R is the unique set X λ satisfying the invariance relation: X λ = λ(x λ + 1) λ(x λ 1). (1) The Bernoulli convolution measure µ λ is the unique probability measure satisfying: f dµ λ = 1 f(λx + λ) + f(λx λ) dµ λ (x). (2) 2 Historical note: The Bernoulli measures date back to work of Erdös and others in the 1930s and 40s. µ λ is the distribution of the random variable k ±λk where + and have equal probability. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
7 Bernoulli convolution measures Some properties of Bernoulli measures Given Bernoulli measure with scale factor λ: When λ = 1 2, the measure µ 1 2 is scaled Lebesgue measure on [ 1, 1]. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
8 Bernoulli convolution measures Some properties of Bernoulli measures Given Bernoulli measure with scale factor λ: When λ = 1 2, the measure µ 1 2 is scaled Lebesgue measure on [ 1, 1]. If λ < 1 2, then µ λ is supported on a fractal with Lebesgue measure zero. Thus, the measures are singular when λ < 1 2. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
9 Bernoulli convolution measures Some properties of Bernoulli measures Given Bernoulli measure with scale factor λ: When λ = 1 2, the measure µ 1 2 is scaled Lebesgue measure on [ 1, 1]. If λ < 1 2, then µ λ is supported on a fractal with Lebesgue measure zero. Thus, the measures are singular when λ < 1 2. When λ 1 2, the set X λ is an interval. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
10 Bernoulli convolution measures Some properties of Bernoulli measures Given Bernoulli measure with scale factor λ: When λ = 1 2, the measure µ 1 2 is scaled Lebesgue measure on [ 1, 1]. If λ < 1 2, then µ λ is supported on a fractal with Lebesgue measure zero. Thus, the measures are singular when λ < 1 2. When λ 1 2, the set X λ is an interval. [Erdös, 1939] If λ is the inverse of a Pisot number, e.g. the golden ratio φ, the measure µ λ is singular with respect to Lebesgue measure even though the attractor is an interval. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
11 Bernoulli convolution measures Some properties of Bernoulli measures Given Bernoulli measure with scale factor λ: When λ = 1 2, the measure µ 1 2 is scaled Lebesgue measure on [ 1, 1]. If λ < 1 2, then µ λ is supported on a fractal with Lebesgue measure zero. Thus, the measures are singular when λ < 1 2. When λ 1 2, the set X λ is an interval. [Erdös, 1939] If λ is the inverse of a Pisot number, e.g. the golden ratio φ, the measure µ λ is singular with respect to Lebesgue measure even though the attractor is an interval. [Hutchinson, 1981] Existence of (X, µ), from an IFS perspective. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
12 Bernoulli convolution measures Some properties of Bernoulli measures Given Bernoulli measure with scale factor λ: When λ = 1 2, the measure µ 1 2 is scaled Lebesgue measure on [ 1, 1]. If λ < 1 2, then µ λ is supported on a fractal with Lebesgue measure zero. Thus, the measures are singular when λ < 1 2. When λ 1 2, the set X λ is an interval. [Erdös, 1939] If λ is the inverse of a Pisot number, e.g. the golden ratio φ, the measure µ λ is singular with respect to Lebesgue measure even though the attractor is an interval. [Hutchinson, 1981] Existence of (X, µ), from an IFS perspective. [Solomyak, 1995] For almost every λ ( 1 2, 1), µ λ is absolutely continuous. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
13 Fourier bases µ λ as an infinite product Consider the Hilbert space L 2 (µ λ ). Is it possible for L 2 (µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
14 Fourier bases µ λ as an infinite product Consider the Hilbert space L 2 (µ λ ). Is it possible for L 2 (µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? Recall that µ λ satisfies the invariance f dµ λ = 1 f(λx + λ) + f(λx λ) dµ λ (x). 2 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
15 Fourier bases µ λ as an infinite product Consider the Hilbert space L 2 (µ λ ). Is it possible for L 2 (µ λ ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? Recall that µ λ satisfies the invariance f dµ λ = 1 f(λx + λ) + f(λx λ) dµ λ (x). 2 Then the Fourier transform of µ λ is: µ λ (t) = e 2πixt dµ λ (x) = 1 e 2πi(λx+λ)t dµ λ (x) e 2πi(λx λ)t dµ λ (x) = cos(2πλt) µ λ (λt) = cos(2πλt) cos(2πλ 2 t) µ λ (λ 2 t) = cos(2πλ k t) k=1 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
16 Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2πiγ in L 2 (µ λ ). K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
17 Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2πiγ in L 2 (µ λ ). e γ, e γ L 2 = e γ γ dµ λ = µ λ (γ γ) ( ( ) ) k(γ = cos 2π λ γ) k=1 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
18 Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2πiγ in L 2 (µ λ ). e γ, e γ L 2 = e γ γ dµ λ = µ λ (γ γ) ( ( ) ) k(γ = cos 2π λ γ) k=1 Lemma The two exponentials e γ, e γ are orthogonal if and only if one of the factors in the infinite product above is zero. This is equivalent to { } 1 γ γ 4 λ k (2m + 1) : k N, m Z =: Z λ. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
19 Test for ONB Fourier bases We use the zero set Z λ to check orthogonality. Parseval s identity provides a test for an ONB. Let Γ R be a set and let E(Γ) be the set {e γ : γ Γ}. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
20 Fourier bases Test for ONB We use the zero set Z λ to check orthogonality. Parseval s identity provides a test for an ONB. Let Γ R be a set and let E(Γ) be the set {e γ : γ Γ}. If E(Γ) is an ONB, then for every value of t R, we have 1 = e t 2 µ λ = γ Γ e t, e γ 2 = γ Γ µ λ (t γ) 2 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
21 Fourier bases Test for ONB We use the zero set Z λ to check orthogonality. Parseval s identity provides a test for an ONB. Let Γ R be a set and let E(Γ) be the set {e γ : γ Γ}. If E(Γ) is an ONB, then for every value of t R, we have 1 = e t 2 µ λ = γ Γ e t, e γ 2 = γ Γ µ λ (t γ) 2 It other words, c Γ (t) = [ µ λ (t γ)] 2 = γ Γ γ Γ k=1 ( ) ) k(t cos (2π 2 λ γ) 1. The function c Γ is sometimes called a spectral function. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
22 First results Fourier bases Theorem (Jorgensen, Pedersen 1998) L 2 (µ 1 ) has an ONB of exponential functions. 4 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
23 First results Fourier bases Theorem (Jorgensen, Pedersen 1998) L 2 (µ 1 ) has an ONB of exponential functions. 4 Example E(Γ 1 4 Γ 1 = 4 ) is an ONB for L 2 (µ 1 ), where 4 p a j 4 j j=0 : a j {0, 1}, p finite = {0, 1, 4, 5, 16, 17, 20, 21, 64,...}. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
24 Fourier bases Another surprise: λ = 1 3 Theorem (Jorgensen, Pedersen 1998) Not only is there no Fourier basis when λ = 1 3, but orthogonal collections of exponential functions can have at most 2 elements. There is also a more general version in the [JP1998] paper: K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
25 Fourier bases Another surprise: λ = 1 3 Theorem (Jorgensen, Pedersen 1998) Not only is there no Fourier basis when λ = 1 3, but orthogonal collections of exponential functions can have at most 2 elements. There is also a more general version in the [JP1998] paper: Theorem (Jorgensen, Pedersen 1998) Given λ = 1 n, if n is even, there is an ONB of exponentials for L2 (µ 1 ) but 2n when n is odd, there can be only finitely many elements in any orthogonal collection of exponentials. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
26 More recent progress Fourier bases Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008) For λ = a b, if b is odd, then any orthonormal collection of exponentials in L 2 (µ λ ) must be finite. If b is even, then there exists countable collections of orthonormal exponentials in L 2 (µ λ ). K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
27 More recent progress Fourier bases Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008) For λ = a b, if b is odd, then any orthonormal collection of exponentials in L 2 (µ λ ) must be finite. If b is even, then there exists countable collections of orthonormal exponentials in L 2 (µ λ ). Theorem (Dutkay, Han, Jorgensen 2009) If λ > 1 2, i.e. there is essential overlap, then L2 (µ λ ) does not have an ONB (or even a frame) of exponential functions. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
28 More recent progress Fourier bases Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008) For λ = a b, if b is odd, then any orthonormal collection of exponentials in L 2 (µ λ ) must be finite. If b is even, then there exists countable collections of orthonormal exponentials in L 2 (µ λ ). Theorem (Dutkay, Han, Jorgensen 2009) If λ > 1 2, i.e. there is essential overlap, then L2 (µ λ ) does not have an ONB (or even a frame) of exponential functions. Theorem (Xinrong Dai 2012) The only spectral Bernoulli measures are for λ = 1 2n. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
29 Canonical ONBs [Jorgensen, Pedersen 1998] Fourier bases Definition Let λ = 1 2n and consider the set from Jorgensen & Pedersen Γ 1 2n = p a j (2n) j j=0 { : a j 0, n } 2, p finite. We call Γ 1 the canonical spectrum and E(Γ 1 ) the canonical ONB for L 2 (µ 1 ). 2n 2n 2n Note: We will justify the nomenclature by describing alternate bases for the same spaces. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
30 Families of ONBs Families of ONBs Let λ = 1 2n. We can construct alternate orthogonal families of exponentials from the canonical ONBs E(Γ 1 ). We then determine whether these alternate 2n sets are ONBs. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
31 Families of ONBs Families of ONBs Let λ = 1 2n. We can construct alternate orthogonal families of exponentials from the canonical ONBs E(Γ 1 ). We then determine whether these alternate 2n sets are ONBs. Recall the zero set, which gives the test for orthogonality of exponentials: Z λ = { } 1 4 λ k (2m + 1) : k N, m Z K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
32 Families of ONBs Families of ONBs Let λ = 1 2n. We can construct alternate orthogonal families of exponentials from the canonical ONBs E(Γ 1 ). We then determine whether these alternate 2n sets are ONBs. Recall the zero set, which gives the test for orthogonality of exponentials: Z λ = { } 1 4 λ k (2m + 1) : k N, m Z Observe that if γ γ Z λ then pγ p γ Z λ as well, for any odd integer p. So scaling a canonical spectrum by p yields at least an orthogonal set, and sometimes an ONB. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
33 Families of ONBs Families of ONBs Let λ = 1 2n. We can construct alternate orthogonal families of exponentials from the canonical ONBs E(Γ 1 ). We then determine whether these alternate 2n sets are ONBs. Recall the zero set, which gives the test for orthogonality of exponentials: Z λ = { } 1 4 λ k (2m + 1) : k N, m Z Observe that if γ γ Z λ then pγ p γ Z λ as well, for any odd integer p. So scaling a canonical spectrum by p yields at least an orthogonal set, and sometimes an ONB. Note: Not every p yields an ONB, e.g. p = 2n 1 for λ = 1 2n. The set E((2n 1)Γ 1 ) is not maximal, hence is not an ONB. 2n K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
34 Families of ONBs Families of ONBs Theorem (Jorgensen, K, Shuman 2010) The set E(3Γ 1 ) is an (alternate) orthonormal basis for L 2 (µ 1 ). 8 8 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
35 Families of ONBs Families of ONBs Theorem (Jorgensen, K, Shuman 2010) The set E(3Γ 1 ) is an (alternate) orthonormal basis for L 2 (µ 1 ). 8 8 Theorem (Jorgensen, K, Shuman 2010) If p < 2(2n 1) π, then E(pΓ 1 ) is an ONB for L 2 (µ 1 ). 2n 2n K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
36 Families of ONBs Families of ONBs Theorem (Jorgensen, K, Shuman 2010) The set E(3Γ 1 ) is an (alternate) orthonormal basis for L 2 (µ 1 ). 8 8 Theorem (Jorgensen, K, Shuman 2010) If p < 2(2n 1) π, then E(pΓ 1 ) is an ONB for L 2 (µ 1 ). 2n 2n Laba/Wang and Dutkay/Jorgensen have described many other values of p for which pγ 1 is a spectrum, particularly in the 1 2n 4 case. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
37 Families of ONBs Examples of pγ 1 2n ONBs n λ p Canonical Γ λ , 3 {0, 3 2, 9, 21 2,...} , 3 {0, 2, 16, 18,...} , 3, 5 {0, 5 2, 25, 55 2,...} , 3, 5, 7 {0, 3, 36, 39,...} K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
38 Operators on L 2 Operator-fractal Dutkay, Jorgensen, 2009: When λ = 1 4, both Γ 1 4 and 5Γ 1 4 are spectra for L 2 (µ 1 ). 4 Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} 4 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
39 Operators on L 2 Operator-fractal Dutkay, Jorgensen, 2009: When λ = 1 4, both Γ 1 4 and 5Γ 1 4 are spectra for L 2 (µ 1 ). 4 Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} 4 For each γ Γ 1, define 4 S 0 : e γ e 4γ S 1 : e γ e 4γ+1 U : e γ e 5γ K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
40 Operators on L 2 Operator-fractal Dutkay, Jorgensen, 2009: When λ = 1 4, both Γ 1 4 and 5Γ 1 4 are spectra for L 2 (µ 1 ). 4 Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} 4 For each γ Γ 1, define 4 S 0 : e γ e 4γ S 1 : e γ e 4γ+1 U : e γ e 5γ S 0 and S 1 map between ONB elements, so are both isometries. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
41 Operators on L 2 Operator-fractal Dutkay, Jorgensen, 2009: When λ = 1 4, both Γ 1 4 and 5Γ 1 4 are spectra for L 2 (µ 1 ). 4 Γ 1 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} 4 For each γ Γ 1, define 4 S 0 : e γ e 4γ S 1 : e γ e 4γ+1 U : e γ e 5γ S 0 and S 1 map between ONB elements, so are both isometries. U maps one ONB to another, so U is a unitary operator. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
42 Operator-fractal Structure of Γ Example: Let λ = 1 4 and denote H = L2 (µ 1 ). 4 Γ = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} Γ = 4Γ (1 + 4Γ) = 4 2 Γ 4(1 + 4Γ) (1 + 4Γ).. = 4 k Γ 4 k 1 (1 + 4Γ) 4(1 + 4Γ) (1 + 4Γ) S 0 (H) is the span of the exponentials E(4Γ) K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
43 Operator-fractal Structure of Γ Example: Let λ = 1 4 and denote H = L2 (µ 1 ). 4 Γ = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} Γ = 4Γ (1 + 4Γ) = 4 2 Γ 4(1 + 4Γ) (1 + 4Γ).. = 4 k Γ 4 k 1 (1 + 4Γ) 4(1 + 4Γ) (1 + 4Γ) S 0 (H) is the span of the exponentials E(4Γ) S k 0 (H) is the span of E(4k Γ) K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
44 Operator-fractal Structure of Γ Example: Let λ = 1 4 and denote H = L2 (µ 1 ). 4 Γ = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} Γ = 4Γ (1 + 4Γ) = 4 2 Γ 4(1 + 4Γ) (1 + 4Γ).. = 4 k Γ 4 k 1 (1 + 4Γ) 4(1 + 4Γ) (1 + 4Γ) S 0 (H) is the span of the exponentials E(4Γ) S k 0 (H) is the span of E(4k Γ) H S 0 (H) S 2 0 (H) K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
45 Operator-fractal Structure of Γ Example: Let λ = 1 4 and denote H = L2 (µ 1 ). 4 Γ = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65,...} Γ = 4Γ (1 + 4Γ) = 4 2 Γ 4(1 + 4Γ) (1 + 4Γ).. = 4 k Γ 4 k 1 (1 + 4Γ) 4(1 + 4Γ) (1 + 4Γ) S 0 (H) is the span of the exponentials E(4Γ) S k 0 (H) is the span of E(4k Γ) H S 0 (H) S 2 0 (H) If we define W k = S0 k (H) Sk+1 0 (H), then H = sp(e 0 ) k=0 W k K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
46 The operator U Operator-fractal Recall U : e γ e 5γ. How does that scaling by ( 5) in the ONB frequencies interact with the inherent scaling ( 4) of the fractal measure space? Theorem (Jorgensen,K,Shuman 2012) K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
47 The operator U Operator-fractal Recall U : e γ e 5γ. How does that scaling by ( 5) in the ONB frequencies interact with the inherent scaling ( 4) of the fractal measure space? Theorem (Jorgensen,K,Shuman 2012) Each subspace W k is invariant under U. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
48 The operator U Operator-fractal Recall U : e γ e 5γ. How does that scaling by ( 5) in the ONB frequencies interact with the inherent scaling ( 4) of the fractal measure space? Theorem (Jorgensen,K,Shuman 2012) Each subspace W k is invariant under U. With respect to the W k ordering of Γ, the matrix of U has block diagonal form...and the infinite blocks are all the same! K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
49 The operator U Operator-fractal Recall U : e γ e 5γ. How does that scaling by ( 5) in the ONB frequencies interact with the inherent scaling ( 4) of the fractal measure space? Theorem (Jorgensen,K,Shuman 2012) Each subspace W k is invariant under U. With respect to the W k ordering of Γ, the matrix of U has block diagonal form...and the infinite blocks are all the same! Even more, in the ( 4, 5) case U actually has a self-similar structure: We call U an operator-fractal. U = (e 0 e 0 ) M e1 U. k=1 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
50 Matrix of U Operator-fractal 0 Γ 0 Γ 1 Γ 2 Γ Γ 0 0 M e1 U Γ M e1 U 0 0 Γ M e1 U 0 Γ M e1 U K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
51 Operator-fractal Spectral properties of U Theorem (Jorgensen, K, Shuman 2012) U has the following properties. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
52 Operator-fractal Spectral properties of U Theorem (Jorgensen, K, Shuman 2012) U has the following properties. 1 is the only eigenvalue of U. (Ue 0 = e 0 ) K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
53 Operator-fractal Spectral properties of U Theorem (Jorgensen, K, Shuman 2012) U has the following properties. 1 is the only eigenvalue of U. (Ue 0 = e 0 ) U is not spatially implemented; i.e. is not of the form Uf = f τ for τ a point transformation on [0, 1]. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
54 Operator-fractal Spectral properties of U Theorem (Jorgensen, K, Shuman 2012) U has the following properties. 1 is the only eigenvalue of U. (Ue 0 = e 0 ) U is not spatially implemented; i.e. is not of the form Uf = f τ for τ a point transformation on [0, 1]. U only fixes the constant functions; if Uv = v then v = ce 0 for some c C. In other words, U is an ergodic operator. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
55 Thank You K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/ / 21
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