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

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Fourier Bases on Fractals

Keri Kornelson

University of Oklahoma - Norman

February Fourier Talks February 21, 2013

K. Kornelson (U. Oklahoma)

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Coauthors

This is joint work with Palle Jorgensen (University of Iowa) Karen Shuman (Grinnell College).

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Outline

1 Bernoulli convolution measures

2 Fourier bases

3 Families of ONBs

4 Operator-fractal

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Bernoulli convolution measures

Convolution measure

Let λ ∈ (0, 1). A fractal Cantor subset of R is the unique set Xλ satisfying the invariance relation: Xλ = λ(Xλ + 1) ∪ λ(Xλ − 1). (1)

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Bernoulli convolution measures

Convolution measure

Let λ ∈ (0, 1). A fractal Cantor subset of R is the unique set Xλ satisfying the invariance relation: Xλ = λ(Xλ + 1) ∪ λ(Xλ − 1). (1) The Bernoulli convolution measure µλ is the unique probability measure satisfying:

f dµλ = 1

2

f(λx + λ) + f(λx − λ) dµλ(x).

(2)

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Bernoulli convolution measures

Convolution measure

Let λ ∈ (0, 1). A fractal Cantor subset of R is the unique set Xλ satisfying the invariance relation: Xλ = λ(Xλ + 1) ∪ λ(Xλ − 1). (1) The Bernoulli convolution measure µλ is the unique probability measure satisfying:

f dµλ = 1

2

f(λx + λ) + f(λx − λ) dµλ(x).

(2) Historical note: The Bernoulli measures date back to work of Erdös and others in the 1930s and 40s. µλ is the distribution of the random variable

k ±λk where + and −

have equal probability.

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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].

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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 µλ is supported on a fractal with Lebesgue measure zero.

Thus, the measures are singular when λ < 1

2.

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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 µλ is supported on a fractal with Lebesgue measure zero.

Thus, the measures are singular when λ < 1

2.

When λ ≥ 1

2, the set Xλ is an interval.

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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 µλ is supported on a fractal with Lebesgue measure zero.

Thus, the measures are singular when λ < 1

2.

When λ ≥ 1

2, the set Xλ is an interval.

[Erdös, 1939] If λ is the inverse of a Pisot number, e.g. the golden ratio φ, the measure µλ is singular with respect to Lebesgue measure even though the attractor is an interval.

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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 µλ is supported on a fractal with Lebesgue measure zero.

Thus, the measures are singular when λ < 1

2.

When λ ≥ 1

2, the set Xλ is an interval.

[Erdös, 1939] If λ is the inverse of a Pisot number, e.g. the golden ratio φ, the measure µλ is singular with respect to Lebesgue measure even though the attractor is an interval. [Hutchinson, 1981] Existence of (X, µ), from an IFS perspective.

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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 µλ is supported on a fractal with Lebesgue measure zero.

Thus, the measures are singular when λ < 1

2.

When λ ≥ 1

2, the set Xλ is an interval.

[Erdös, 1939] If λ is the inverse of a Pisot number, e.g. the golden ratio φ, the measure µλ 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), µλ is absolutely

continuous.

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Fourier bases

µλ as an infinite product

Consider the Hilbert space L2(µλ). Is it possible for L2(µλ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions?

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Fourier bases

µλ as an infinite product

Consider the Hilbert space L2(µλ). Is it possible for L2(µλ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? Recall that µλ satisfies the invariance

f dµλ = 1

2

f(λx + λ) + f(λx − λ) dµλ(x).
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Fourier bases

µλ as an infinite product

Consider the Hilbert space L2(µλ). Is it possible for L2(µλ) to have a Fourier basis, i.e. an orthonormal basis (ONB) of complex exponential functions? Recall that µλ satisfies the invariance

f dµλ = 1

2

f(λx + λ) + f(λx − λ) dµλ(x).

Then the Fourier transform of µλ is:

µλ(t)

=

e2πixt dµλ(x)

= 1 2

e2πi(λx+λ)t dµλ(x) + 1

2

e2πi(λx−λ)t dµλ(x)

= cos(2πλt) µλ(λt) = cos(2πλt) cos(2πλ2t) µλ(λ2t) =

∞

k=1

cos(2πλkt)

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Fourier bases

Orthogonality Condition

Denote by eγ the exponential function e2πiγ· in L2(µλ).

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Fourier bases

Orthogonality Condition

Denote by eγ the exponential function e2πiγ· in L2(µλ). eγ, e˜

γL2

=

eγ−˜

γ dµλ

=

µλ(γ − ˜

γ) =

∞

k=1

cos

2π
λ

k (γ − ˜ γ)

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Fourier bases

Orthogonality Condition

Denote by eγ the exponential function e2πiγ· in L2(µλ). eγ, e˜

γL2

=

eγ−˜

γ dµλ

=

µλ(γ − ˜

γ) =

∞

k=1

cos

2π
λ

k (γ − ˜ γ)

Lemma

The two exponentials eγ, e˜

γ are orthogonal if and only if one of the factors in

the infinite product above is zero. This is equivalent to γ − ˜ γ ∈ 1 4λ−k(2m + 1) : k ∈ N, m ∈ Z

=: Zλ.
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Fourier bases

Test for ONB

We use the zero set Zλ 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 {eγ : γ ∈ Γ}.

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Fourier bases

Test for ONB

We use the zero set Zλ 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 {eγ : γ ∈ Γ}. If E(Γ) is an ONB, then for every value of t ∈ R, we have 1 = et2

µλ

=

γ∈Γ

|et, eγ|2 =

γ∈Γ

| µλ(t − γ)|2

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Fourier bases

Test for ONB

We use the zero set Zλ 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 {eγ : γ ∈ Γ}. If E(Γ) is an ONB, then for every value of t ∈ R, we have 1 = et2

µλ

=

γ∈Γ

|et, eγ|2 =

γ∈Γ

| µλ(t − γ)|2 It other words, cΓ(t) =

γ∈Γ

[ µλ(t − γ)]2 =

γ∈Γ

∞

k=1

cos2

2π
λ

k (t − γ)

≡ 1.

The function cΓ is sometimes called a spectral function .

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Fourier bases

First results

Theorem (Jorgensen, Pedersen 1998) L2(µ 1

4 ) has an ONB of exponential functions.

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Fourier bases

First results

Theorem (Jorgensen, Pedersen 1998) L2(µ 1

4 ) has an ONB of exponential functions.

Example E(Γ 1

4 ) is an ONB for L2(µ 1 4 ), where

Γ 1

4 =

  

p

j=0

aj4j : aj ∈ {0, 1}, p finite    = {0, 1, 4, 5, 16, 17, 20, 21, 64, . . .}.

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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:

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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

2n ) but

when n is odd, there can be only finitely many elements in any orthogonal collection of exponentials.

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Fourier bases

More recent progress

Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008) For λ = a

b, if b is odd, then any orthonormal collection of exponentials in

L2(µλ) must be finite. If b is even, then there exists countable collections of

rthonormal exponentials in L2(µλ).
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Fourier bases

More recent progress

Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008) For λ = a

b, if b is odd, then any orthonormal collection of exponentials in

L2(µλ) must be finite. If b is even, then there exists countable collections of

rthonormal exponentials in L2(µλ).

Theorem (Dutkay, Han, Jorgensen 2009) If λ > 1

2, i.e. there is essential overlap, then L2(µλ) does not have an ONB (or

even a frame) of exponential functions.

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Fourier bases

More recent progress

Theorem (Jorgensen, K, Shuman 2008; Hu, Lau 2008) For λ = a

b, if b is odd, then any orthonormal collection of exponentials in

L2(µλ) must be finite. If b is even, then there exists countable collections of

rthonormal exponentials in L2(µλ).

Theorem (Dutkay, Han, Jorgensen 2009) If λ > 1

2, i.e. there is essential overlap, then L2(µλ) 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.

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Fourier bases

“Canonical” ONBs

[Jorgensen, Pedersen 1998]

Definition Let λ =

1 2n and consider the set from Jorgensen & Pedersen

Γ 1

2n =

  

p

j=0

aj(2n)j : aj ∈

0, n

2

, p finite

   . We call Γ 1

2n the canonical spectrum and E(Γ 1 2n ) the canonical ONB for L2(µ 1 2n ).

Note: We will justify the nomenclature by describing alternate bases for the same spaces.

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Families of ONBs

Let λ =

1

2n. We can construct alternate orthogonal families of exponentials

from the canonical ONBs E(Γ 1

2n ). We then determine whether these alternate

sets are ONBs.

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Families of ONBs

Let λ =

1

2n. We can construct alternate orthogonal families of exponentials

from the canonical ONBs E(Γ 1

2n ). We then determine whether these alternate

sets are ONBs. Recall the zero set, which gives the test for orthogonality of exponentials: Zλ = 1 4λ−k(2m + 1) : k ∈ N, m ∈ Z

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Families of ONBs

Let λ =

1

2n. We can construct alternate orthogonal families of exponentials

from the canonical ONBs E(Γ 1

2n ). We then determine whether these alternate

sets are ONBs. Recall the zero set, which gives the test for orthogonality of exponentials: Zλ = 1 4λ−k(2m + 1) : k ∈ N, m ∈ Z

Observe that if γ − ˜

γ ∈ Zλ then pγ − p˜ γ ∈ Zλ as well, for any odd integer p. So scaling a canonical spectrum by p yields at least an orthogonal set, and sometimes an ONB.

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Families of ONBs

Let λ =

1

2n. We can construct alternate orthogonal families of exponentials

from the canonical ONBs E(Γ 1

2n ). We then determine whether these alternate

sets are ONBs. Recall the zero set, which gives the test for orthogonality of exponentials: Zλ = 1 4λ−k(2m + 1) : k ∈ N, m ∈ Z

Observe that if γ − ˜

γ ∈ Zλ then pγ − p˜ γ ∈ Zλ 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

2n ) is not maximal, hence is not an ONB.

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Families of ONBs

Theorem (Jorgensen, K, Shuman 2010) The set E(3Γ 1

8 ) is an (alternate) orthonormal basis for L2(µ 1 8 ).

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Families of ONBs

Theorem (Jorgensen, K, Shuman 2010) The set E(3Γ 1

8 ) is an (alternate) orthonormal basis for L2(µ 1 8 ).

Theorem (Jorgensen, K, Shuman 2010) If p < 2(2n−1)

π

, then E(pΓ 1

2n ) is an ONB for L2(µ 1 2n ).

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Families of ONBs

Theorem (Jorgensen, K, Shuman 2010) The set E(3Γ 1

8 ) is an (alternate) orthonormal basis for L2(µ 1 8 ).

Theorem (Jorgensen, K, Shuman 2010) If p < 2(2n−1)

π

, then E(pΓ 1

2n ) is an ONB for L2(µ 1 2n ).

Laba/Wang and Dutkay/Jorgensen have described many other values of p for which pΓ 1

2n is a spectrum, particularly in the 1

4 case.

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Families of ONBs

Examples of pΓ 1

2n ONBs

n λ p Canonical Γλ 3

1 6

1, 3 {0, 3

2, 9, 21 2 , . . .}

4

1 8

1, 3 {0, 2, 16, 18, . . .} 5

1 10

1, 3, 5 {0, 5

2, 25, 55 2 , . . .}

6

1 12

1, 3, 5, 7 {0, 3, 36, 39, . . .}

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Operator-fractal

Operators on L2

Dutkay, Jorgensen, 2009: When λ = 1

4, both Γ 1

4 and 5Γ 1 4 are spectra for L2(µ 1 4 ).

Γ 1

4 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}

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Operator-fractal

Operators on L2

Dutkay, Jorgensen, 2009: When λ = 1

4, both Γ 1

4 and 5Γ 1 4 are spectra for L2(µ 1 4 ).

Γ 1

4 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}

For each γ ∈ Γ 1

4 , define

S0 : eγ → e4γ S1 : eγ → e4γ+1 U : eγ → e5γ

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Operator-fractal

Operators on L2

Dutkay, Jorgensen, 2009: When λ = 1

4, both Γ 1

4 and 5Γ 1 4 are spectra for L2(µ 1 4 ).

Γ 1

4 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}

For each γ ∈ Γ 1

4 , define

S0 : eγ → e4γ S1 : eγ → e4γ+1 U : eγ → e5γ S0 and S1 map between ONB elements, so are both isometries.

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Operator-fractal

Operators on L2

Dutkay, Jorgensen, 2009: When λ = 1

4, both Γ 1

4 and 5Γ 1 4 are spectra for L2(µ 1 4 ).

Γ 1

4 = {0, 1, 4, 5, 16, 17, 20, 21, 64, 65, . . .}

For each γ ∈ Γ 1

4 , define

S0 : eγ → e4γ S1 : eγ → e4γ+1 U : eγ → e5γ S0 and S1 map between ONB elements, so are both isometries. U maps one ONB to another, so U is a unitary operator.

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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) = 42Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ) . . . = 4kΓ ∪ 4k−1(1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ) S0(H) is the span of the exponentials E(4Γ)

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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) = 42Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ) . . . = 4kΓ ∪ 4k−1(1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ) S0(H) is the span of the exponentials E(4Γ) Sk

0(H) is the span of E(4kΓ)

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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) = 42Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ) . . . = 4kΓ ∪ 4k−1(1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ) S0(H) is the span of the exponentials E(4Γ) Sk

0(H) is the span of E(4kΓ)

H ⊃ S0(H) ⊃ S2

0(H) ⊃ · · ·

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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) = 42Γ ∪ 4(1 + 4Γ) ∪ (1 + 4Γ) . . . = 4kΓ ∪ 4k−1(1 + 4Γ) · · · 4(1 + 4Γ) ∪ (1 + 4Γ) S0(H) is the span of the exponentials E(4Γ) Sk

0(H) is the span of E(4kΓ)

H ⊃ S0(H) ⊃ S2

0(H) ⊃ · · ·

If we define Wk = Sk

0(H) ⊖ Sk+1

(H), then H = sp(e0) ⊕

∞

k=0

Wk

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Operator-fractal

The operator U

Recall U : eγ → e5γ. 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)

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Operator-fractal

The operator U

Recall U : eγ → e5γ. 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 Wk is invariant under U.

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Operator-fractal

The operator U

Recall U : eγ → e5γ. 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 Wk is invariant under U. With respect to the Wk ordering of Γ, the matrix of U has block diagonal form...and the infinite blocks are all the same!

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Operator-fractal

The operator U

Recall U : eγ → e5γ. 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 Wk is invariant under U. With respect to the Wk 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: U = (e0 ⊗ e0) ⊕

∞

k=1

Me1U. We call U an operator-fractal.

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Operator-fractal

Matrix of U

Γ0 Γ1 Γ2 Γ3 · · · 1 · · · Γ0 Me1U · · · Γ1 Me1U · · · Γ2 Me1U · · · Γ3 Me1U · · · . . . . . . . . . . . . . . . ...

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Operator-fractal

Spectral properties of U

Theorem (Jorgensen, K, Shuman 2012) U has the following properties.

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Operator-fractal

Spectral properties of U

Theorem (Jorgensen, K, Shuman 2012) U has the following properties. 1 is the only eigenvalue of U. (Ue0 = e0)

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SLIDE 53

Operator-fractal

Spectral properties of U

Theorem (Jorgensen, K, Shuman 2012) U has the following properties. 1 is the only eigenvalue of U. (Ue0 = e0) 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 02/21/2013 20 / 21

SLIDE 54

Operator-fractal

Spectral properties of U

Theorem (Jorgensen, K, Shuman 2012) U has the following properties. 1 is the only eigenvalue of U. (Ue0 = e0) 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 = ce0 for some c ∈ C. In other words, U is an ergodic operator.

K. Kornelson (U. Oklahoma)

Fractal Fourier Bases FFT 02/21/2013 20 / 21

SLIDE 55

Thank You

K. Kornelson (U. Oklahoma)

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