Fourier Bases on Fractals Keri Kornelson University of Oklahoma - - PowerPoint PPT Presentation

Fourier Bases on Fractals Keri Kornelson University of Oklahoma - Norman February Fourier Talks February 21, 2013 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 1 / 21

Coauthors This is joint work with Palle Jorgensen (University of Iowa) Karen Shuman (Grinnell College). K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 2 / 21

Outline 1 Bernoulli convolution measures 2 Fourier bases Families of ONBs 3 4 Operator-fractal K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 3 / 21

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 02/21/2013 4 / 21

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 d µ λ = 1 f ( λ x + λ ) + f ( λ x − λ ) d µ λ ( x ) . (2) 2 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 4 / 21

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 d µ λ = 1 f ( λ x + λ ) + f ( λ x − λ ) d µ 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 ± λ k where + and − have equal probability. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 4 / 21

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 02/21/2013 5 / 21

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 02/21/2013 5 / 21

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 02/21/2013 5 / 21

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 02/21/2013 5 / 21

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 02/21/2013 5 / 21

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 02/21/2013 5 / 21

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 02/21/2013 6 / 21

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 d µ λ = 1 f ( λ x + λ ) + f ( λ x − λ ) d µ λ ( x ) . 2 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 6 / 21

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 d µ λ = 1 f ( λ x + λ ) + f ( λ x − λ ) d µ λ ( x ) . 2 Then the Fourier transform of µ λ is: � e 2 π ixt d µ λ ( x ) µ λ ( t ) = � � � e 2 π i ( λ x + λ ) t d µ λ ( x ) + 1 e 2 π i ( λ x − λ ) t d µ λ ( x ) 1 = 2 2 = cos ( 2 πλ t ) � µ λ ( λ t ) cos ( 2 πλ t ) cos ( 2 πλ 2 t ) � µ λ ( λ 2 t ) = ∞ � cos ( 2 πλ k t ) = k = 1 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 6 / 21

Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2 π i γ · in L 2 ( µ λ ) . K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 7 / 21

Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2 π i γ · in L 2 ( µ λ ) . � γ d µ λ � e γ , e ˜ γ � L 2 = e γ − ˜ = � µ λ ( γ − ˜ γ ) � � � � k � ∞ = cos 2 π λ ( γ − ˜ γ ) k = 1 K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 7 / 21

Fourier bases Orthogonality Condition Denote by e γ the exponential function e 2 π i γ · in L 2 ( µ λ ) . � γ d µ λ � e γ , e ˜ γ � L 2 = e γ − ˜ = � µ λ ( γ − ˜ γ ) � � � � k � ∞ = cos 2 π λ ( γ − ˜ 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 � 4 λ − k ( 2 m + 1 ) : k ∈ N , m ∈ Z γ − ˜ γ ∈ =: Z λ . K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 7 / 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 } . K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 8 / 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 |� e t , e γ �| 2 = µ λ γ ∈ Γ � µ λ ( t − γ ) | 2 = | � γ ∈ Γ K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 8 / 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 |� e t , e γ �| 2 = µ λ γ ∈ Γ � µ λ ( t − γ ) | 2 = | � γ ∈ Γ It other words, � � � � k � � � ∞ µ λ ( t − γ )] 2 = cos 2 c Γ ( t ) = [ � ( t − γ ) ≡ 1 . 2 π λ γ ∈ Γ γ ∈ Γ k = 1 The function c Γ is sometimes called a spectral function . K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 8 / 21

Fourier bases First results Theorem (Jorgensen, Pedersen 1998) L 2 ( µ 1 4 ) has an ONB of exponential functions. K. Kornelson (U. Oklahoma) Fractal Fourier Bases FFT 02/21/2013 9 / 21