Multiwavelet Analysis on Fractals
av Andreas Brodin
Akademisk avhandling
som med vederb¨orligt tillst˚and av rektors¨ambetet vid Ume˚a universitet f¨or avl¨aggande av filosofie doktorsexamen framl¨agges till offentligt f¨orsvar i MA 121, MIT-huset,
torsdagen den 7 juni 2007, kl. 10.15.
Fakultetsopponent: Docent Per-Anders Ivert, Matematiska institutio-nen, Lunds Universitet.
Department of Mathematics and Mathematical Statistics, Ume˚a University
Organization Document type
UME˚A UNIVERSITY DOCTORAL DISSERTATION
Department of Mathematics and Mathematical Statistics
SE-901 87 Ume˚a, Sweden Date of publication
May 2007 Author
Andreas Brodin
Title
Wavelet Analysis on Fractals
Abstract
This thesis consists of an introduction and a summary, followed by two papers, both of them on the topic of function spaces on fractals.
Paper I, Andreas Brodin, Pointwise Convergence of Haar type Wavelets on Self-Similar Sets, Manuscript.
Paper II, Andreas Brodin, Regularization of Wavelet Expansion characterizes Besov Spaces on Fractals, Manuscript.
Properties of wavelets, originally constructed by A. Jonsson, are studied in both papers. The wavelets are piecewise polynomial functions on self-similar fractal sets.
In Paper I, pointwise convergence of partial sums of the wavelet expansion is investigated. On a specific fractal set, the Sierpinski gasket, pointwise convergence of the partial sums is shown by calculating the kernel explicitly, when the wavelets are piecewise constant functions. For more general self-similar fractals, pointwise convergence of the partial sums and their derivatives, in case the expanded function has higher regularity, is shown using a different technique based on Markov’s inequality.
A. Jonsson has shown that on a class of totally disconnected self-similar sets it is possible to characterize Besov spaces by means of the magnitude of the coefficients in the wavelet expansion of a function. M. Bodin has extended his results to a class of graph directed self-similar sets introduced by Mauldin and Williams. Unfortunately, these results only holds for fractals such that the sets in the first generation of the fractal are disjoint. In Paper II we are able to characterize Besov spaces on a class of fractals not necessarily sharing this condition by making the wavelet expansion smooth. We create continuous regularizations of the partial sums of the wavelet expansion and show that properties of these regularizations can be used to characterize Besov spaces.
Key words: Wavelets, fractals, pointwise convergence, Besov spaces, regularization.
Language: English ISBN 91-7264-334-5 Number of pages: 25+2 papers
ISSN1102-8300 (doktorsavhandlingar vid Institutionen f¨or matematik och matematisk statistik, UMU), nr. 38 (2007).
Multiwavelet Analysis on Fractals
Andreas Brodin
Department of Mathematics and Mathematical Statistics
Ume˚
a University
Department of Mathematics and Mathematical Statistics Ume˚a University
SE-901 87 Ume˚a, Sweden
Andreas Brodin: Multiwavelet Analysis on Fractals c
2007 Andreas Brodin
Printed by Print & Media, Ume˚a University, Ume˚a, 2007 ISBN 91-7264-334-5
ISSN 1102-8300 (doktorsavhandlingar vid Institutionen f¨or matematik och matematisk statistik, UMU), nr. 38 (2007).
Contents
1 Introduction 1
1.1 Standard Notation . . . 1
1.2 Trigonometric Series . . . 1
1.3 Haar’s Orthonormal System . . . 4
1.4 Besov Spaces in Euclidean Space . . . 12
2 Summary of Papers 14 2.1 Paper I: Pointwise Convergence of Haar type Wavelets on Self-Similar Sets . . . 15
2.2 Paper II: Regularization of Wavelet Expansion characterizes Besov Spaces on Fractals . . . 17 Papers I and II
Acknowledgements
I would like to extend many thanks to Alf Jonsson, my supervisor, for in-troducing me to an interesting field of mathematics and for always being supportive. Many thanks also to my wife, Kristen, and my daughters, Heidi and Ella, who let me spend so much time doing math, and at other times, give me other things to think about. My parents, Kjell och Inger, should also each have a medallion for their helpfulness and for never ever having stopped any of my ideas or projects.
1
Introduction
1.1
Standard Notation
Throughout this thesis R will denote the real numbers, C the complex num-bers and Z the integers. For points x in R or C, |x| is the ordinary absolute value of x. We shall be working on Rn, and n will always denote the dimen-sion. For points x = (x1, . . . , xn) ∈ Rn, |x| denotes the Euclidean norm:
|x| = q
x2
1+ . . . + x2n. (1) With the metric induced by this norm, the distance between two points x and y in Rn is |x − y|. If X and Y are subsets of Rn, the distance between a point x and X is dist(x, X) = inf{|x − y| : y ∈ X}, the distance between X and Y is dist(X, Y ) = inf{|x − y| : x ∈ X, y ∈ Y } and the diameter of X is diam(X) = sup{|x − y| : x, y ∈ X}.
If U ⊂ Rn is open and k ∈ {0, 1, 2, . . .}, we denote by Ck(U ) the class of all functions on U possessing continuous partial derivatives of order ≤ k, and we put C∞(U ) = ∩∞
k=1Ck(U ). It will be convenient to have a compact language for partial derivatives and we will use the multi-index notation. A multi-index is an ordered n-tuple of non-negative integers. If j = (j1, . . . , jn) is a multi-index, we define |j| = j1+ . . . + jn, j! = j1! · · · jn!, ∂j = ∂ ∂x1 j1 · · · ∂ ∂xn jn , (2) and for x = (x1, . . . , xn) ∈ Rn, xj= xj11· · · xjnn. For example, in this notation, Leibniz’ product rule for derivatives becomes
∂j(f g) = (f g)(j) =X k≤j j k f(k)g(j−k), (3) where j k
= k!(j−k)!j! and k ≤ j means ki≤ ji for i = 1, . . . , n.
1.2
Trigonometric Series
A trigonometric polynomial is a finite sum of the form p(t) = a0+
τ X k=1
where t is a real number, and a0, a1, . . . , aτ and b1, b2, . . . , bτ are complex numbers. On account of Euler identities, (4) can also be written in the form
p(t) = τ X k=−τ
ckeikt. (5)
It is clear that every trigonometric polynomial has period 2π, that is, satisfies p(t) = p(t + 2π) for each t ∈ R, and also, every 2π periodic function on R is completely determined by its values on the interval [−π, π).
For 1 ≤ p ≤ ∞, let Lp(Π) be the space of 2π periodic complex valued Lebesgue measurable functions on R for which the norm
kfkp,Π= 1 2π Z π −π|f(t)| pdt 1/p <∞. (6) Since the Lebesgue measure of [−π, π) is finite it is easy to see that Lp2(Π) ⊂
Lp1(Π) if p
1≤ p2. Put
uk(t) = eikt (7) for each k ∈ Z. If we define the inner product for f and g in L2(Π) by
< f, g >= 1 2π
Z π −π
f(t)g(t)dt, (8) then an easy computation shows that
< uk, ul >= 0 if k 6= l,1 if k = l (9) which means that the family {uk}k∈Z is an orthonormal system of functions in L2(Π), usually called the trigonometric system.
For any f ∈ L1(Π) we define the Fourier coefficients by < f, u
k > for k∈ Z and f’s Fourier series is
∞ X k=−∞
< f, uk> uk. (10)
If τ ∈ {0, 1, 2 . . .}, Sτ will denote the τ :th (symmetric) partial sum of the Fourier series of f ∈ L1(Π): Sτ(f, t) = τ X k=−τ < f, uk > uk(t). (11)
One celebrated result in the theory of Fourier series says that if f ∈ L2(Π), then its Fourier series converges to f , in the L2 norm.
The remainder of this section is devoted to a definitely trickier business, to a discussion of pointwise convergence of the Fourier series, that is, the convergence of Sτ(f, t) for any fixed t in [−π, π).
A few mathematicians of the eighteenth century such as Bernoulli, Euler and Lagrange knew experimentally that for some simple functions f ,
f(t) = ∞ X k=−∞
< f, uk > uk(t), (12)
and for a long time it was believed that this was the case for all continuous functions.
The Frenchman, Jean Baptiste Joseph Fourier (1768-1830), claimed that (12) always was true and showed, in a book of outstanding importance in the history of physics (Analytic Theory of Heat 1822), how expressions of the formP∞
−∞< f, uk > uk could be used to solve linear partial differential equations of the kind which dominated 19th century physics.
Fourier’s work stimulated Peter-Lejeune Dirichlet to study in more detail the notion of a function and Bernhard Riemann to develop the concept known today as the Riemann integral. In a paper (1829) which set up new and previously undreamed of standards in rigour and clarity in analysis, Dirichlet was able to prove convergence under quite general conditions. For example, one consequence of his results is that if f is continuous and has a bounded continuous derivative except, possibly, at a finite number of points, then Sτ(f, t) → f(t) at all points t where f is continuous.
However, the conditions on f , except being continuous, could not be relaxed completely for in 1876 Du Bois-Reymond constructed a counterex-ample, a continuous function f ∈ L1(Π) such that the sequence (S
τ(f, 0)) diverges.
The end of the story was told by Lennart Carleson in 1966 when he showed that if f ∈ L2(Π), then S
τ(f, t) → f(t) for almost every t. This result was extended to hold for any f ∈ Lp(Π) where p > 1 by Richard Hunt in 1968. At around the same time, Jean-Pierre Kahane and Yitzhak Katznelson proved a theorem which complements Carleson’s and Hunt’s result. If E is a set of measure zero, then there exists a continuous function f ∈ L1(Π) such that lim supτ →∞|Sτ(f, t)| = ∞ for all t ∈ E.
In 1909 Alfred Haar asked in an article, [Haa10], whether some other orthonormal system than the trigonometric system would actually guarantee pointwise convergence of the Fourier series of a continuous function. He also
gave an example of a system of piecewise constant functions on [0, 1] which indeed does the job.
1.3
Haar’s Orthonormal System
In this section we will show how Haar in [Haa10] proved convergence by calculating the so called kernel. The intention with this is twofold. One reason is beauty, as Haar’s work on [0, 1] is quite elegant. In Paper I, the same technique is used to show convergence on the Sierpinski gasket. The work on the Sierpinski gasket is much more technical, so the second goal for this section is to present the idea of what is going on in the proof for the convergence on the Sierpinski gasket.
For p ≥ 1, let Lp([0, 1]) be the space of real valued Lebesgue measurable functions f on [0, 1] such that for p < ∞, the norm
kfkp,[0,1] = Z 1 0 |f(t)| pdt 1/p (13) is finite, and for p = ∞, the norm
kfk∞,[0,1]= ess sup x∈[0,1]|f(x)|
(14) is finite. We also equip L2([0, 1]) with the inner product
< f, g >= Z 1
0
f(t)g(t)dt. (15) Haar’s orthonormal system in L2([0, 1]) consists of the constant function φ(x) = 1 and functions ψk,l for integers k and l with k ≥ 0 and 0 ≤ l < 2k. The support of ψk,l is [2lk, l+1 2k ]. For x ∈ ( 2l 2k+1,22l+1k+1), ψk,l(x) = 2k/2, and for x∈ (2l+12k+1,2l+22k+1), ψk,l(x) = −2k/2. If x ∈ ∪2 k−1 l=0 {2k+12l ,22l+1k+1,22l+2k+1} \ {0, 1}, then
ψk,l(x) = 12(ψkl(x−) + ψkl(x+)), that is, the arithmetic mean of the function on the nearby intervals of length 1
2k+1. At x = 0 and x = 1, ψk,0(0) = 2k/2 and ψk,2k−1(1) = −2k/2, that is, the ψk,l become constant on [0, 1
2k+1) and
(1 − 1
2k+1,1]. To more easily get the idea of what they look like, graphs of
@ @ @@ 1 1 −1 x y s s y= φ(x) @ @ @@ 1 1 −1 x y s c s c s y= ψ0,0(x) @ @ @@ 1 1 −1 x y s c s c c s c s y= ψ1,0(x) @ @ @@ 1 1 −1 x y c c s c s s c s y= ψ1,1(x)
@ @ @@ 1 1 −1 x y s c s c c s c s y= ψ2,0(x) @ @ @@ 1 1 −1 x y c c s c c s c s s c s y= ψ2,1(x) @ @ @@ 1 1 −1 x y c c s c c s c s s c s y= ψ2,2(x) @ @ @@ 1 1 −1 x y c c s c s s c s y= ψ2,3(x)
To see that these functions are orthonormal is not difficult. We are going to show that partial sums of a function’s expansion in this family of functions converges in some sense. For this we need a couple of definitions. For δ > 0 and f ∈ L1([0, 1]) we define
w[δ, f ] = sup t1,t2∈[0,1]
0<|t1−t2|≤δ
|f(t1) − f(t2)|. (16) The function w is called the modulus of continuity of f . It is easy to check that f is continuous on [0, 1] if and only if w[δ, f ] → 0 as δ → 0. For f ∈ L1([0, 1]) we define for τ ≥ 0 and −1 ≤ ρ < 2τ − 1 the (τ, ρ):th partial
sum of f ’s Fourier series with respect to the Haar system at x ∈ [0, 1] by Sτρ(f, x) = < f, φ > φ(x) + X 0≤k<τ X 0≤l<2k < f, ψk,l> ψk,l(x) + X 0≤l≤ρ < f, ψτ,l> ψτ,l(x). (17)
PropositionLet τ ≥ 0 and −1 ≤ ρ < 2τ − 1. If f ∈ L1([0, 1]), then
kSτρ(f, ·) − fk∞,[0,1]≤ w[21−τ, f], (18) and, if in addition f is continuous, then
sup x∈[0,1]|S
ρ
τ(f, x) − f(x)| ≤ w[21−τ, f]. (19) Corollary If f is continuous, then Sρ
τ(f, x) converges uniformly to f (x) on [0, 1].
If f is continuous, then w[21−τ, f] → 0 as τ → ∞, and we see that the corollary follows from statement (19) in the proposition.
Remark. Haar did not use the modulus of continuity nor the norm notation in his article, and his claim looked more like the corollary.
A closer look at the Fourier coefficients < f, φ > and < f, ψk,l >gives us the so called kernel function of this system.
Sτρ(f, x) = < f, φ > φ(x) + X 0≤k<τ X 0≤l<2k < f, ψk,l > ψk,l(x) + X 0≤l≤ρ < f, ψτ,l > ψτ,l(x) (20) = Z 1 0 f(t)φ(t)dt φ(x) + X 0≤k<τ X 0≤l<2k Z 1 0 f(t)ψk,l(t)dt ψk,l(x) + X 0≤l≤ρ Z 1 0 f(t)ψτ,l(t)dt ψτ,l(x) (21) = Z 1 0 Kτρ(x, t)f (t)dt (22)
where the kernel is
Kτρ(s, t) = φ(s)φ(t) + X 0≤k<τ X 0≤l<2k ψk,l(s)ψk,l(t) + X 0≤l≤ρ ψτ,l(s)ψτ,l(t). (23)
It is an easy exercise to calculate the kernel explicitly. One may do this directly for small values of τ and ρ and then inductively for any τ and ρ.
1 1 1 s t K0−1(s, t) 1 1 2 2 0 0 s t 1 r r r r K1−1(s, t)
The kernel has the value within each region according to the figures. On the boundary between a region and the outside of the square [0, 1]×[0, 1], the value is the same as within the region. At a boundary point of two regions, the kernel value will be the arithmetic mean of the values of the neighbouring regions. At a boundary point of four regions, for example (14,14) and (12,12) for K0
1(s, t), the kernel value is the arithmetic mean of the values on the neighbouring boundaries. 1 1 4 4 2 0 0 s t 2 1 1 2 r r r r r r h Q QQ 3 2 K0 1(s, t) 1 1 4 4 4 4 0 0 s t 2 r r r r r r r r K2−1(s, t)
1 1 2τ 2τ 0 0 2τ −1 s t 2τ r rr 2−τ 2−τ r r r r K−1 τ (s, t)
1 1 2τ +1 2τ +1 0 0 2τ s t h Q QQ 2τ −1+ 2τ −2 2τ +1 2τ 2τ +1 2τ 2τ −1 r rr r rr (ρ + 1)2−τ (ρ + 1)2−τ r r r r Kρ τ(s.t)
For any x ∈ [0, 1] one may use the plot of Kρ
τ(s.t) and consider cases of the location of x to show that Sρ
τ(f, x) is equal to 1 l(Ix) Z Ix f(t)dt (24)
or equal to an average of two integrals of this kind. Here Ix is a certain interval containing x and l(Ix) = length(Ix) which is equal to one of the values 21−τ, 2−τ or 2−1−τ.
If x < ρ+12τ and x 6= 2τ+1l for 1 ≤ l ≤ 2ρ + 1, then Sρτ(f, x) = Z Ix 2τ +1f(t) dt = 1 l(Ix) Z Ix f(t) dt. (25)
If x > ρ+12τ and x 6= 2lτ for ρ + 1 < l < 2τ, then
Sτρ(f, x) = Z Ix 2τf(t) dt = 1 l(Ix) Z Ix f(t) dt. (26) If x = 2τ+1l with 1 ≤ l ≤ 2ρ + 1, then Sτρ(f, x) = Z l 2τ +1 l−1 2τ +1 2τf(t) dt + Z l+1 2τ +1 l 2τ +1 2τf(t) dt = 1 l(Ix) Z Ix f(t) dt. (27) Sτρ(f,ρ+ 1 2τ ) = Z ρ+12τ 2ρ+1 2τ +1 2τf(t) dt + Z ρ+22τ ρ+1 2τ 2τ −1f(t) dt = 1 2l(Ii x) Z Ii x f(t) dt + 1 2l(Iii x) Z Iii x f(t) dt. (28) If x = 2lτ with ρ + 1 < l < 2τ, then Sτρ(f, x) = Z 2τl l−1 2τ 2τ −1f(t) dt + Z l+12τ l 2τ 2τ −1f(t) dt = 1 l(Ix) Z Ix f(t) dt. (29)
If x is a point where f (x) is defined and x 6= ρ+12τ , then we have
|Sτρ(f, x) − f(x)| = 1 l(Ix) Z Ix f(t)dt − 1 l(Ix) Z Ix f(x)dt (30) ≤ l(I1 x) Z Ix |f(t) − f(x)| dt (31) ≤ w[21−τ, f]. (32) It is easy to check that (32) is also valid for x = ρ+12τ . Now, if f ∈ L1([0, 1]),
then (32) holds for almost all x ∈ [0, 1], and if f is continuous, then (32) holds for each x ∈ [0, 1], and the statements in the proposition follow.
Remark. Haar also used the fact that the size of the interval Ix in (24) is proportional to 2−τ. More precisely, he noticed that
lim τ →∞S p τ(f, x) = τ →∞lim 1 l(Ix) Z Ix f(t)dt (33) = d ds Z s 0 f(t)dt s=x (34) if the derivative exists. By a theorem of Lebesgue, for f ∈ L1([0, 1]), the derivative exists and equals f (x) except on a set of measure zero. If in addition f is continuous, then the derivative exists for all x ∈ [0, 1] and equals f (x).
Suggested reading: Katz book [Kat93] is a well-written text on the history of mathematics. For the theory of Fourier analysis, we mention K¨orner’s, [K¨or88], and Dym and McKean’s, [DM72] books as good starters and Zyg-mund’s Trigonometric Series, [Zyg77], and Katznelson’s An introduction to Harmonic Analysis, [Kat04], for the more advanced reader. For a deeper investigation of and more results on the Haar system, we refer to Novikov and Semenov’s text [NS97].
1.4
Besov Spaces in Euclidean Space
A natural way to define smoothness of order α, for 0 < α ≤ 1, of a function f defined on Rn is to require that f satisfies a Lipschitz condition of order α. That is, there is a constant M such that for x and h in Rn
|∆hf(x)| ≤ M|h|α, (35) where ∆hdenotes the first difference with step h: ∆hf(x) = f (x + h) − f(x). Rudolf Lipschitz (1832-1903) is remembered chiefly for simplifying and clarifying the existing theory of the existence and uniqueness of solutions of differential equations. One of his1 results is: if f (x, y) is continuous and satisfies a Lipschitz condition of order α = 1 in the variable y on a strip [a, b] × (−∞, ∞), and (x0, y0) is any point on the strip, then the initial value problem y′= f (x, y), y(x
0) = y0 has one and only one solution y = y(x) on [a, b].
As was pointed out in Section 1.2, the Fourier series of a continuous 2π periodic function need not converge pointwise. A simple criterion for
1
Charles Picard, Ernst Lindel¨of, and Augustin Cauchy should be mentioned in this context too.
the Fourier series to converge absolutely and hence pointwise uniformly was given by Sergei Bernstein in 1914: If f ∈ L1(Π) is continuous and satisfies a Lipschitz condition of order α > 12, then Sτ(f, ·) converges absolutely. Also, for α = 1
2 this is not necessarily true.
Results of this kind make it natural to classify functions according to this smoothness condition, and for this, Lipschitz spaces of order α > 0 were introduced. Let α > 0 and k be the non-negative integer satisfying k < α ≤ k + 1. A function f belongs to the Lipschitz space Lip(α, Rn) if f ∈ Ck(Rn) and there is a constant M such that |f(j)| ≤ M for all j with |j| ≤ k, and such that f(j) with |j| = k, satisfies a Lipschitz condition of order α − k with the constant M. The Lipschitz norm of f is defined as the infimum of the possible constants M .
More general Lipschitz spaces were defined by Antoni Zygmund in the 1940’s by replacing, in the Lipschitz condition (35), the first difference ∆h by the second difference ∆2
h= ∆h∆h. When α is an integer, this gives larger function spaces, and when α is not an integer we get the same function spaces with first and second order differences. It has also been shown that higher order differences than second order do not give new function spaces.
About 1900 Henri L´eon Lebesgue introduced the Lebesgue integral and Lebesgue spaces, Lp(Rn) for 1 ≤ p ≤ ∞, consisting of Lebesgue measurable functions such that for p < ∞ the norm
kfkp= Z Rn|f(t)| pdt 1/p (36) is finite, and for p = ∞ the norm kfk∞= ess supx∈Rn|f(x)| is finite.
When studying partial differential equations, Lipschitz spaces are not often suitable settings, as it is usually hard to show that solutions actually belong to such spaces. In the 1930’s Sergei Lvovich Sobolev found the right balance in the level of smoothness in the function spaces we now call the Sobolev spaces. For 1 ≤ p ≤ ∞ and k a non-negative integer, the Sobolev space Wkp(Rn) consists of all functions f on Rnsuch that for each multi-index j with |j| ≤ k, the derivative f(j) exists in the weak sense and is in Lp(Rn). The function f(j) is the j:th derivative of f in the weak or distributional sense if Z Rn f(j)(t)φ(t)dt = (−1)|j| Z Rn f(t)φ(j)(t)dt (37) holds for all φ ∈ C∞(Rn) with compact support.
A generalization of Sobolev spaces are Bessel2 potential spaces, Lp α(Rn). For 1 ≤ p ≤ ∞ and α > 0, f ∈ Lp
α(Rn) if there is a function g ∈ Lp(Rn)
2
Friedrich Wilhelm Bessel (1784 - 1846), who used what is now called Bessel functions to treat problems of perturbation in the planetary system.
such that f is equal to the convolution of the so called Bessel kernel on Rn, Gα with g. The norm of f ∈ Lpα(Rn) is the Lp(Rn) norm of the function g. These spaces were introduced by N. Aronzajn, K.T. Smith and A.P. Calderon in the early 1960’s. In 1961 Calderon proved that for positive integers k and 1 < p < ∞, Lpk(Rn) = W
p k(Rn).
In order to get a nice theory for certain boundary value problems, the need for for a new function space arose. The new function spaces are today known as Besov spaces named after Oleg Vladimirovich Besov who made a systematic study of these spaces in the beginning of the 1960’s. For 1 ≤ p, q ≤ ∞, α > 0 and k as the integer with k < α ≤ k + 1, the Besov space Λp,q
α (Rn) consists of those functions on Rn such that the norm
kfkΛp,qα (R n) = X |j|≤k kf(j)k p+ X |j|=k Z Rn k∆2 hf(j)kqp |h|n+(α−k)q dh !1/q (38)
is finite. It should be mentioned that many mathematicians have contributed to the definition and early theory of Besov spaces, such as G.H. Hardy, J.E. Littlewood, S. M. Nikolskii, E.M. Stein and M.H. Taibleson.
Suggested reading: The textbook Differential Equations with Applications and Historical Notes by G.F. Simmons, [Sim72], is a really good mixture of the three ingredients. For the elementary theory of partial differential equations and Sobolev spaces Lawrence C. Evans’ book, [Eva98], is recom-mended. E.M. Stein’s Singular Integrals and Differentiability Properties of Functions, [Ste70], is a standard text on the functions spaces mentioned in this section. Also, in this monograph by Stein, many references to original papers may be found. Hans Wallin’s article, Function spaces on fractals, [Wal95], is a survey of results on function spaces. The Dirichlet problem on a domain with fractal boundary is discussed, and the article also has plenty of references.
2
Summary of Papers
In both papers, we use multiwavelets of Haar type of higher order m on self-similar fractals, consisting of the families {ψσ
i}i∈Ik,k≥0,1≤σ≤M and {φl}1≤l≤M0,
first introduced by Alf Jonsson in [Jon98]. These wavelets are piecewise polynomials of degree m instead of piecewise constants. We assume that K is the invariant set (the fractal) generated by similitudes T1, . . . , TN on Rn satisfying the open set condition, and K is not a subset of an n − 1 dimensional subspace of Rn. Also, µ is the invariant Borel measure with
support K and µ(K) = 1. In [Jon98] it was pointed out that any f ∈ Lp(µ) may be represented by its wavelet expansion,
f = M0 X l=1 < f, φl > φl+ ∞ X k=0 M X σ=1 X i∈Ik < f, ψiσ> ψiσ, (39)
where the coefficients are defined by < f, g >=R
Kf g dµ.
2.1
Paper I: Pointwise Convergence of Haar type Wavelets
on Self-Similar Sets
In Paper I we study convergence properties of the partial sums of the wavelet expansion (39), Sλ= M0 X l=1 < f, φl > φl+ λ−1 X k=0 M X σ=1 X i∈Ik < f, ψiσ> ψiσ. (40)
A systematic definition of the wavelets at all points x ∈ K where they are not already defined is outlined so that pointwise convergence of the wavelet expansion may be discussed.
On a specific fractal, the Sierpinski gasket S, the following result is at-tained by calculating the kernel function explicitly. The wavelet basis consists of piecewise constant functions in this case, (m = 0).
Proposition 3.3. If f ∈ L1(S, µ), then
kSλΛ− fk∞,S ≤ 3w[2−λ−
1
2, f], (41)
and if in addition f is continuous, then sup x∈S|S Λ λ(x) − f(x)| ≤ 3w[2−λ− 1 2, f]. (42)
The function w is the modulus of continuity of f on S; replace [0, 1] by S in (16). The superscript Λ in SΛ
λ indicates the possibility of having additional terms in the partial sum from functions of the next layer (Def. 3.2., Paper I).
One consequence of Proposition 3.3. is Corollary 3.4. which says that if f is continuous, then SΛ
λ converges uniformly to f .
When the wavelets are piecewise constant functions, it seems likely that there would be a lower bound for how small the pointwise distance between
a continuous function and its wavelet expansion can actually be. One result with the Sierpinski gasket is presented in Proposition 4.9.
Proposition 4.9. If f is a continuous function on S, then sup x∈S|Sλ(x) − f(x)| ≥ 1 4w[2 −λ+1 2, f]. (43)
Functions with higher regularity on a closed set F ⊂ Rn, Cl(F ) functions, are defined in Definition 4.5. The Whitney classes Cl(F ) were first defined in [Whi34].
Definition 4.2. For a closed F ⊂ Rn, a function f belongs to Cl(F ) for a non-negative integer l if there are functions f(j), 0 ≤ |j| ≤ l defined on F , with f(0)= f , which satisfy the following property. Let ¯x ∈ F and ǫ > 0 be given. If f(j)(x) = X |j+k|≤l f(j+k)(y) k! (x − y) k+ R j(x, y), (44)
then there is a δ > 0 such that if |¯x − x| < δ ,|¯x − y| < δ, with x, y ∈ F , then |Rj(x, y)| ≤ ǫ|x − y|l−|j|. (45) Remark. It is to be noticed that for a general closed set F , the function f does not necessarily determine the f(j) functions for |j| ≥ 1 uniquely. Consider, for example, the case of an f defined on a finite set F . So when speaking of an element in Cl(F ) we shall mean in fact a family {f(j)}
|j|≤l. However, the invariant set K under discussion in Papers I and II satisfies the so called Markov’s inequality3, and as shown in [JW84], on a closed set K ⊂ Rn with that property, the functions f(j) for |j| > 0 are uniquely determined by f = f(0). We may therefore, in this thesis, speak of functions f ∈ Cl(K).
For functions in Cl(K) a generalized module of continuity is defined in Definition 4.6.
3
Definition 4.6 If f ∈ Cl(K), then for any multi-index j with |j| ≤ l, the j:th order modulus of continuity is
wjl[δ, f ] = sup x,y∈K, 0<|x−y|≤δ f(j)(x) −P |j+k|≤l f(j+k)(y) k! (x − y) k (x − y)l−|j| . (46)
An expansion of f ∈ Cl(K) in a wavelet series should, to be most bene-ficial, have the wavelets of higher order, m, instead of just being piecewise constants. In this case, Haar’s method of calculating the kernel is hard to apply, and a different technique based on Markov’s inequality is used. This technique also makes it easier to reach results for general self-similar sets. The main result is Theorem 4.7. Let Dλ= max{rλidiamK} where the maxi-mum is taken over i ∈ {1, . . . , N} and ri is the contraction factor similitude Ti has.
Theorem 4.7. Let the wavelets be of order m ≥ 0. There is a constant c such that if f ∈ L1(µ), then
kSΛ
λ − fk∞,K ≤ cw[Dλ, f]. (47) If in addition f ∈ Cl(K) for some integer l with 0 ≤ l ≤ m, then for |ν| ≤ l,
sup x∈K|∂ νSΛ λ(x) − ∂νf(x)| ≤ cD l−|ν| λ (wl0[Dλ, f] + wνl[Dλ, f]). (48) The superscript Λ in SΛ
λ indicates the possibility of having additional terms in the partial sum from functions of the next layer (See Def. 4.3. and the remark following Theorem 4.7., Paper I). As a consequence of Theorem 4.7. we have that ∂νSΛ
λ converges pointwise uniformly to ∂νf for each |ν| ≤ l≤ m if f ∈ Cl(K). This is the content of Corollary 4.8.
2.2
Paper II: Regularization of Wavelet Expansion
char-acterizes Besov Spaces on Fractals
A. Jonsson and H. Wallin defined Besov spaces, Bp,q
α (F ), for d-sets F in the early 1980’s, see [JW84]. One example of a d-set is the invariant set K generated by a family of similarity contractions on Rn satisfying the open set condition. In [Jon98], Jonsson uses multiwavelets consisting of piecewise polynomial functions on self-similar fractals and in the same article he shows that on a class of totally disconnected self-similar sets it is possible to char-acterize Besov spaces Bp,q
α (K) by means of the magnitude of the coefficients in the wavelet expansion of a function.
In Paper II we also use the multiwavelet expansion as a tool to charac-terize Bp,q
α (K). To get results for more general self-similar sets than the ones that are totally disconnected we apply a different method than to look at the coefficients in the expansion. For each positive integer λ we construct a C∞(Rn) regularization E
λ out of the partial sum, Sλ given in (40), of the wavelet expansion of a function f .
Propositions 4.2, 4.3 and 4.4 are partial results leading toward the main Theorem 4.5:
Theorem 4.5. Let K be the invariant set with respect to similitudes T1, . . . , TN with the same contraction factor r. We assume that the similitudes satisfy the open set condition and that K is not a subset of an (n − 1)-dimensional subspace of Rn. Let κ be the integer with 2κ−1 <24(1 +√n) ≤ 2κ and define the sequence of integers (νλ)∞λ=1 by 2−νλ ≤ 2κrλdiamK < 21−νλ.
If α > 0 and 1 ≤ p, q ≤ ∞, then f ∈ Bp,q
α (K) if and only if there is a sequence (Aν)∞ν=ν1 with non-negative terms such that
P∞ ν=ν1A
q
ν <∞, and kEλ− fkp,K ≤ 2−νλαAνλ, (49)
and the norms kEλkBp,q
α (K) are uniformly bounded. Moreover, the norm of
f ∈ Bαp,q(K) is equivalent to the sum inf P∞ ν=ν1A q ν 1/q + supλkEλkBp,q α (K)
where the infimum is taken over all possible sequences (Aν)∞ν=ν1.
The author believes that the result in Theorem 4.5. may be extended to hold for self-similar fractals generated by any family of similitudes satisfying the open set condition, not just the ones with similitudes having the same contraction factor.
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