Generalized eigenfunctions and a Borel Theorem on the Sierpinski Gasket.

Transcription

1 Generalized eigenfunctions and a Borel Theore on the Sierpinski Gasket. Kasso A. Okoudjou, Luke G. Rogers, and Robert S. Strichartz May 26, Introduction There is a well developed theory (see [5, 9]) of analysis on certain types of fractal sets, of which the Sierpinski Gasket (SG) is the siplest non-trivial exaple. In this theory the fractals are viewed as liits of graphs, and notions analogous to the Dirichlet energy and the Laplacian are constructed as renoralized liits of the corresponding objects on the approxiating graphs. The nature of this construction has naturally led to extensive study of the eigenfunctions of this Laplacian, and to functional-analytic notions based on the eigenfunctions. However, ore recent work [7, 2] has exained other eleentary functions on SG, including analogues of polynoials, analytic functions and certain exponentials. A forthcoing paper [8] will extend this investigation to study sooth bup functions and a ethod for partitioning sooth functions subordinate to an open cover. In the present work we prove there are exponentially decaying generalized eigenfunctions on a blow-up of S G with boundary (which we denote S G ), proving: Theore 1.1. For each < 0 and j N there is a sooth function E j on S G such that for each j we have ( + )E j = je. Moreover E j decays exponentially away fro the boundary point of S G and satisfies E j j! j. There are sufficiently any of these generalized eigenfunctions that they can be used to prove a Borel-type theore on S G, thereby answering a question asked in [7, 2]. Using the ter jet for the sequence of values of the natural derivatives at a junction point of S G our result ay be suarized as: Theore 1.2. Given an arbitrary jet there is a sooth function on S G with that jet at the boundary point. Our otivation for studying generalized eigenfunctions and for proving Theore 1.1 was to prove Theore 1.2. The structure of the paper reflects this otivation: apart MSC Priary 28A80; Secondary 31C45 Research supported in part by the National Science Foundation, Grant DMS

2 fro soe background in Section 2 our first results (in Section 3) are those showing that Theore 1.2 follows fro Theore 1.1 and soe known results about localized eigenfunctions on S G. Section 4 is then devoted to the construction of the generalized eigenfunctions and the proof of Theore 1.1. Acknowledgents The authors would like to thank A. Teplyaev, who provided source code for the figures. 2 Setting We give a brief description of soe parts of the theory of analysis on the Sierpinski Gasket, ore details of which are in [9]. For the general theory of analysis on fractals the standard reference is [5]. SG and SG The Sierpinski gasket S G is the unique non-epty copact set in R 2 that is invariant under the iterated function syste f i = 1 2 (x + q i), i = 0, 1, 2 in the sense that S G = 2i=0 f i (S G), where the points q i are the vertices of an equilateral triangle. For N and (i 1, i 2,..., i ) {0, 1, 2} we call f i1 f i2 f i (S G) a cell of level. The points V 0 = {q i : j = 0, 1, 2} are the boundary of S G and we view S G as the liit of graphs Γ with vertices defined inductively by V = 2 i=0 f i (V 1 ) and edge relation x y if x and y are in the sae -cell. The set of all vertices is V = V and the junction points are V \ V 0. We let µ be the usual self-siilar probability easure on S G with µ( f w (S G)) = 3 w, and also use µ to denote the obvious extension to S G. The infinite blow-up of S G with boundary point q 0 is denoted S G and defined by S G = =0 f ( n) 0 (S G) (2.1) This is the siplest blow-up of S G; we could also consider arbitrary sequences of blow-ups =1 fi 1 1 fi 1 2 fi 1 (S G), but all that have a boundary point necessarily have {i } to be constant after soe 0 and are isoetric, so aong those with boundary it suffices to consider S G (see Lea 2.3 in [11]). The work in this paper will crucially use that we are on a blow-up with boundary. We refer to [10, 11] for ore inforation about blow-ups and the Laplacian on S G. Laplacian, derivatives and jets Each graph approxiation Γ of S G supports a graph Laplacian defined at nonboundary points x V \ V 0, and we define a Laplacian at junction points of S G as 2

3 a renoralized liit of the graph Laplacians u(x) = (u(y) u(x)) (2.2) y x u(x) = 3 2 li 5 u(x). (2.3) A continuous function u is in the doain of the Laplacian, u do( ), if there is a continuous f such that the right side of (2.3) converges uniforly to f on V \ V 0. Then we write u = f, extending u to all points of S G by continuity. The factor 3/2 in (2.3) is for consistency with an alternative definition of the Laplacian using a renoralized Dirichlet energy (see [5, 9]). We ake the obvious definition of k u and do( k ), and call a function sooth if it is in do( ) = k do( k ). One additional property of the Laplacian that we will use extensively is its scaling; it is iediate fro (2.3) that for u do( ) (u f0 1 1 ) = 5( u) f0 on f 0 (S G) (2.4) In addition to the Laplacian there are two derivatives at boundary points, the noral derivative n and tangential derivative T, defined by ( 5 ) (2u(qi n u(q i ) = li ) u( f j (q i+1 )) u( f j (q i+2 )) ) (2.5) 3 T u(q i ) = li 5 ( u( f j (q j+1 )) u( f j (q j+2 )) ) (2.6) (with q i+3 = q i ). The forer exists for any u do( ) and the latter exists under the additional assuption that u is Hölder continuous. Both ay be localized to boundary points of cells. The noral derivatives are uch better understood than the tangential derivatives, and have considerable application; for this paper their iportant feature is the atching condition for the noral derivatives: if u do( ) then at any junction point of two cells the noral derivatives corresponding to these cells su to zero. Conversely, if u is continuous and u = f on each -cell then u = f on S G if and only if f is continuous and the atching condition holds at each point of V \ V 0. At a boundary point q we will call the values of k u(q), n k u(q) and T k u(q) the Laplacian powers, noral derivatives and tangential derivatives, respectively. The j-jet of u at q is ( u(q), n u(q), T u(q),..., j u(q), n j u(q), T j u(q) ) and the infinite jet is the corresponding sequence of Laplacian powers and derivatives. Spectral deciation A useful feature of the Laplacian on S G and S G is that restricting a Laplcaian eigenfunction E to the graph approxiation Γ produces an eigenfunction of the graph Laplacian, with a shift in the eigenvalue. This phenoenon is known as spectral deciation [4, 3, 11]. Specifically, if ( + )E = 0 on S G or S G then ( + )E = 0 3

4 on Γ, where 1 = (5 ) (2.7) = 3 2 li 5 (2.8) E (y i ) = 2E (x i ) + (4 )(E (x i+1 ) + E (x i+2 )) (2 )(5 ) (2.9) in which the points x i are the vertices of an ( 1)-cell, and each y i is the point fro V opposite to x i as shown in Figure 1. For proofs we refer to [3] or [9]; an explanation of spectral deciation in a ore general context is in [6]. y 1 y 0 y 2 x 1 x 0 x 2 Figure 1: Points x i in V 1 and y i in V. We will need one technical result about spectral deciation on S G that is wellknown but for which there does not appear to be a proof in the literature. Lea 2.1. For < 0 there is an entire function Ψ such that = Ψ(5 ). Proof. Considering (2.7) we define functions by 2φ ± (ζ) = 5 ± 25 4ζ, so that is one of φ ± ( 1 ). Observe that if 1 0 then 0, so fro (2.8) and < 0 we ust have < 0 and = φ ( 1 ) for all. The renoralized liit Φ(ζ) = 3 2 li 5 φ (ζ) is analytic in a neighborhood of the origin and has Φ (0) = 3/2 by virtue of the fact that φ (ζ) = ζ/5 + O(ζ 2 ) for sufficiently sall ζ. It follows that Φ has an analytic inverse Ψ(ζ) = Φ 1 (ζ) = k=0 α k ζ k in a neighborhood of 0. Using (2.7) and (2.8) we find Ψ(5 ) = for all sufficiently sall 5. This gives a recursion for the coefficients α k, beginning with α 0 = 0, α 1 = 2/3 and continuing according to k 1 (5 k 5)α k = α l α k l (2.10) for k 2. An alost identical recursion appears for a different purpose in [7] (as Equation 2.9 and in Theore 2.7) and their arguent shows that α k C(k!) log 5/ log 2. It follows iediately that Ψ is entire, while = Ψ(5 ) is true by construction. l=1 4

5 3 The Borel Theore In this section we prove Theore 1.2 under the assuption of Theore 1.1. First we construct sooth functions with finitely any prescribed values of the Laplacian powers and tangential derivatives at q 0 using known results about the existence of localized eigenfunctions. Then we use Theore 1.1 and linear algebra to prove that there are sooth functions with finitely any prescribed noral derivatives. Finally we state a precise version of Theore 1.2 and show that its validity for finite jets gives the full result by a scaling and convergence arguent. Localized eigenfunctions A curious feature of any highly syetric fractals is that their Laplacians have localized eigenfunctions. We will not need the details of the theory, for which we refer to [1, 5], but only the existence of two specific eigenfunctions u 1 and u 2 on S G. The values of u 1 on V 1 and u 2 on V 2 are shown in Figure 2. Fro the values shown we can Figure 2: The functions u 1 and u 2. copute u 1 and u 2 at any scale by the ethod of spectral deciation given in (2.7)- (2.9) (with the caveat that for u 1 the positive root ust be taken at the first step of the recursion (2.7)). What is iportant here is that the noral derivatives of both u i vanish at the points q 1 and q 2, which we see fro (2.5) and the antisyetry of the u i on the cells f 1 (S G) and f 2 (S G). Since the u i are eigenfunctions we then find that all of the values k u i and n k u i vanish at q 1 and q 2, and (2.4) shows the sae is true for u i f0 at f0 (q 1) and f0 (q 2) for any. It follows fro the atching condition that u i f0 on f0 (S G) u i, = 0 otherwise are sooth functions, and therefore are Laplacian eigenfunctions with eigenvalues 5 i. For obvious reasons they are called localized eigenfunctions. The jets of the u i, at q 0 are easily coputed. The eigenfunction equations u i = i u i give the higher order ters fro the initial ones, so by siple algebra fro (2.9) 5

6 and soe syetry arguents k 2( 1 ) k 5 k i = 1 u i, (q 0 ) = 0 i = 2 T k 0 i = 1 u i, (q 0 ) = 2( 2 ) k 5 k i = 2 n k u i, (q 0 ) = 0 i = 1, 2. With these functions as building blocks we show that there is a sooth function with finitely prescribed values of the Laplacian powers and tangential derivatives at q 0, and whose noral derivatives are all zero. Lea 3.1. For n N and values ζ 0,..., ζ n and θ 0,..., θ n there is u do( ) such that k u(q 0 ) = ζ k, n k u(q 0 ) = 0, and T k u(q 0 ) = θ k for all 0 k n. Proof. We observe that the vectors ( u1, (q 0 ), u 1, (q 0 ),..., n u 1, (q 0 ) ) = 2 ( 1, ( 1 )5,..., ( 1 ) n 5 n) are linearly independent with respect to. A siilar result is true for the vector of tangential derivatives of u 2,. We ay then obtain the desired u as a linear cobination of the functions u i, for 0 n by linear algebra. We reark that this ethod cannot be applied to prescribe values of the noral derivatives at q 0 using localized eigenfunctions. The structure of the localized eigenfunctions is well understood (see [9]) and non-zero noral derivatives can occur only in closed loops circling the holes in the gasket. As each junction point corresponds to a hole of precisely one size, our scaling arguents are not applicable. Siilar arguents are needed in Lea 3.4 below, so any proof of Theore 1.2 using only localized eigenfunctions would need to be quite different fro ours. Generalized eigenfunctions and noral derivatives The generalized eigenfunctions produced in Theore 1.1 are a sufficiently rich class that we can use a finite linear cobination of the to atch finitely any prescribed noral derivatives at q 0. Lea 3.2. For n N, < 0 and values η 0,..., η n there is u do( ) which is a finite linear cobination of the E j, 0 j n, and has n k u(q 0 ) = η k for all 0 k n. Proof. Let a j,k = n k E j (q 0). It clearly suffices to show that the atrix [ ] a n j,k j,k=0 is invertible, so we exaine its deterinant. Writing the generalized eigenfunction equation ( + )E j = je in ters of a j,k we have a j,k + a j,k 1 = ja,k 1, which suggests a colun operation on [a j,k ]. For all coluns k 1 we replace a j,k with ja,k 1, which akes the first row zero except in the first place siply because 6

7 j = 0 on this row. For concreteness the result of this coputation for the deterinant in the case n = 2 is given below. a 0,0 a 0,1 a 0,2 a a 1,0 a 1,1 a 1,2 a 2,0 a 2,1 a 2,2 = 0,0 0 0 a 1,0 a 0,0 a 0,1 a 2,0 2a 1,0 2a 1,1 = a a 0,0 a 0,1 0,0 2a 1,0 2a 1,1 This operation can be repeated inductively, because it shows det [ a j,k ] n 0 = a 0,0 det [ ja,k 1 ] n 1 = a 0,0 det [ ( j + 1)a j,k ] n 1 and the only change to the atrix at each stage is to ultiply each row by a constant and reduce the degree, so the sae colun operations apply each tie. We conclude that det [ ] a n j,k 0 = ( 1)n n!a n+1 0,0, and is non-zero by (4.2) below. Corollary 3.3. Given values (ζ k, η k, θ k ) for 0 k n there is a finite linear cobination u of localized eigenfunctions and generalized eigenfunctions such that k u(q 0 ) = ζ k, n k u(q 0 ) = η k and T k u(q 0 ) = θ k for all 0 k n. Proof. Apply Lea 3.2 to atch the noral derivatives and then Lea 3.1 to correct the Laplacian powers and tangential derivatives without affecting the values of the noral derivatives. Proof of the Borel theore Corollary 3.3 supplies the natural building blocks for obtaining a sooth function with any given jet at q 0. Define for each j functions F j fro which we will deterine the Laplacian powers, G j for the noral derivatives and H j for the tangential derivatives by requiring that for all 0 k j k F j (q 0 ) = δ j,k n k F j (q 0 ) = 0 T k F j (q 0 ) = 0 k G j (q 0 ) = 0 n k G j (q 0 ) = δ j,k T k G j (q 0 ) = 0 k H j (q 0 ) = 0 n k H j (q 0 ) = 0 T k H j (q 0 ) = δ j,k where δ j,k is the Kronecker delta. The natural goal is to construct a sooth function with prescribed (infinite) jet by using the ters of the jet as coefficients in a series with functions like the F j, G j and H j. To ake the series converge to a sooth function we will need soe estiates on these functions and their Laplacian powers. What we know so far is that they are finite linear cobinations of the localized eigenfunctions u i, with 0 j and i = 1, 2, and the generalized eigenfunctions E k for a fixed and 0 k j. All of these functions and their Laplacian powers of order at ost j are bounded: for the localized eigenfunctions this is obvious, while for the generalized eigenfunctions it follows fro the bound in Theore 1.1 and the recursion (4.3). We conclude that for each j there is a constant C( j) such that for all 0 k j k F j C( j) k G j C( j) k H j C( j) and turn now to a scaling arguent that allows us to ake these as sall as desired. 0 7

8 Lea 3.4. If N then the functions F j, = 5 j F j f0 G j, = 5 j G j f0 H j, = 5 j H j f0 have the sae j-jets at q 0 as F j, G j and H j respectively, but for 0 k j they satisfy the following estiates on S G k F j, C( j)5 (k j) k G j, C( j)5 (k j) k H j, C( j)5 (k j). Proof. The result is an eleentary consequence of the scaling property of the Laplacian. By induction fro (2.4) we see that k (u f0 ) = 5 k ( k u) f0. Both stateents of the lea are iediate consequences of this and the definitions (2.5) and (2.6) of the noral and tangential derivatives. Proof of Theore 1.2. We are supplied with values (ζ k, η k, θ k ) for k N and seek a sooth function u such that k u(q 0 ) = ζ k, n k u(q 0 ) = η k and T k u(q 0 ) = θ k for all k. This will be certainly be the case for the function u = ( ) ζ j F j, j + η j G j, j + θ j H j, j j=0 (3.1) provided only that applying any power of the Laplacian yields a uniforly convergent series. However by Lea 3.4 we ay choose the sequence j such that ters after the j-th have only a sall effect on the Laplacian powers of order at ost j. Specifically, given any ɛ > 0 we ay ake j so large that for 0 k j 1 k( ζ j 5 j j F j, j + η j 5 j j G j, j + θ j 5 j j ) H j, j C( j)5 (k j) j ax { ζ k, η k, θ k : 0 k j 1 } ɛ2 k j providing a bound on the tail of the series obtained by applying k to (3.1). We conclude that u is sooth, that it has the desired jet at q 0, and oreover that ost of the contribution to the k-th jet is fro the first k ters in the su: k u k ( ɛ + C(k) ζ j + η j + θ j ). j=0 4 Generalized eigenfunctions with decay In this section we prove Theore 1.1, showing that there are exponentially decaying generalized eigenfunctions E j of the Laplacian on S G. Our results depend on work in [7], where the negative-eigenvalue eigenfunctions of were studied using spectral deciation. Using notation fro (2.7), the results we need ay be suarized as: 8

9 Proposition 4.1 ([7], Section 6). For each < 0 there is an eigenfunction E on S G which is syetrical under the reflection that fixes q 0 and exchanges q 1 with q 2, and which satisfies ( + )E = 0. There is an explicit forula for E at the points z = f 0 (q 1 ) E (z ) = ( 4 ) j j=0 (4.1) which is uniforly continuous on copacta in V, and E is the liit of this on S G. These functions have exponential decay E (z ) = O( 1 ) = O ( 2 2 ) as and the noral derivative at q 0 is given by n E (q 0 ) = ( 4 ) ( 6 n ) n =0 n=1 > 0. (4.2) Our construction is otivated as follows. Forally setting E j = ( d j d) E we find that the E j satisfy the generalized eigenfunction equation ( + )E j = je (4.3) and we hope that the decay of E will ensure exponential decay for E j. This arguent is ade rigorous by Lea 4.2, but it will initially be sipler to construct the E j fro (4.3) than by proving E can be differentiated with respect to. Observe that on S G we can inductively obtain solutions E j of (4.3) for j N starting with E 0 = E, erely because < 0 and the spectru of is positive. The resulting functions are clearly in do( ), and depend on the boundary data we assign. Guided by the foral idea that E j should be ( d j d) E we set E j (z) = ( d ) je (z) (4.4) d at each of the three boundary points z = q 0, q 1, q 2. The definition is legitiate because = Ψ(5 ) and Ψ is entire (Lea 2.1), so the rapid growth of ensures the expression (4.1) is analytic with respect to. Lea 4.2. Using the supreu nor, the functions E j are differentiable with respect to and d d E = E j. Proof. We require a standard estiate (like Lea of [5]). Let u be in do( ) and subtract the haronic function u 0 with the sae boundary values. It is well known that then u u 0 c u 2, and by the axiu principle we conclude u c u 2 + ax V0 u. Let < 0, and let κ 1 > 0 be the first Dirichlet eigenvalue of. The spectral representation of iediately shows u c ( 1 + κ 1 ) ( + )u 2 + ax V 0 u. (4.5) 9

10 Now suppose inductively that the lea is true up to j 1. For the difference between E j and the Newton quotient for the derivative of E we have ( + ) ( E j 1 ( E t +t E )) ( ( j 1)E j 2 +t te+t = je 1 t = ( E +t E ) 1 + ( j 1) t 0 in L 2 (S G) ( E j 2 +t E j 2 ) j 2 + ( j 1)E ) ( j 1)E by induction. Fro (4.5) and the fact that the boundary data varies analytically with we conclude E j 1 ( E +t t E ) 0. The sae reasoning reduces the base case of the induction to showing E +t E 2 0, which is a consequence of the fact (fro Proposition 4.1) that E is uniforly approxiated by the analytic function of in (4.1). With this in hand we can describe the natural scaling behavior of the E j. Fro (2.4) we know that E 5 = E f 1 0, whence on f 0(S G) E j f 0 1 = ( d ) je f0 1 = ( d je5 = 5 d d) j E j 5 on f 1 0 (S G) is to set E j = 5 j E j 5 f 0. Induc- and therefore the natural definition of E j tively we let E j = 5 jn E 5 n f0 n on f0 n (S G) (4.6) for each n N to extend E j to all of S G. We reark that this gives the sae result as solving (4.3) on f0 n(s G) with boundary data (4.4) at the points z = q 0, f0 n(q 1), f0 n(z 2). Cobining the above results we have proven the existence stateents of Theore 1.1. What reains to be proven is the content of the following lea, which is regrettably but perhaps unavoidably technical. Lea 4.3. The generalized eigenfunctions satisfy E j j! j and have decay E j (z ) = O ( 2 2 ) as. Proof. We first prove the decay. Recall that the recursion (2.7) guarantees C2 2 as. Set β j = (d/d) j. It is eleentary to verify that β j = O( ) as fro the definition of Ψ and the recursion (2.10). Using the explicit forula (4.1) we write E (z ) = 1 + (P )/4, where P = ( 4 ) n n=0 10

11 and define S j by (d/d) j P = P S j. Exaining S 1 we have S 1 = β1 + n=0 4β 1 n (2 n )(6 n ) = β1 + 4β1 + O( 2 2 ) + = β1 + 4β1 + O( 2 2 ) n=1 4β 1 n (2 n )(6 n ) where the penultiate estiate uses the series expansion for 1/(2 )(6 ), and the final one uses the structure of the series. This series consists of ters which are rational functions of n and β 1 n, and in which the degree of the denoinator strictly exceeds that of the nuerator. The rapid growth of ensures this is bounded by a ultiple of the first ter, which is O( 1 1 ) = O( 2 ). We will call a series in which the ters are rational functions that depend on n, β 1 n,..., β j for n 1, but ust always have the degree of the denoinator to be strictly greater than the nuerator, a good rational series or GRS. Notice that the derivative of a GRS is a GRS, and that the product of a GRS with a GRS or a rational function in which the nuerator has the sae or lesser degree than the denoinator is also a GRS. Our estiates on n and β 1 n guarantee that any GRS sus to a value which is O( 2 ). Using induction over j we see that the function S j β/ j 4β/ j 2 is always a GRS, so is O( 2 ). Indeed, this is true for j = 1, and if we assue it to be true for j 1 and apply the recursion S j = S 1 S and S 1 S, = ( β 1 + 4β1 + GRS 2 = β1 β 2 + 8β1 β 3 + (ds ) β + + GRS /d), then both 4β 2 + GRS ds, d = β j β 2 β 1 + 4β j 2 8β β 1 + GRS 3 so we ay su the to coplete the induction. In particular we conclude that 4E j (z ) = d j P d j β j = P S j β j = P S j β j 4β j + β j (P + 4) + 4β j (P ) 2 = O( 1 ) = O(2 2 ) 2 11

12 where in the last step we used that the first bracketed ter is O( 2 ), and that P = O( ), (P + 4) = O( 1 ) and (P ) = O(1), all of which are fro the fact that E (z ) = O( 1 ) (see Proposition 4.1) and the definition of P. Now that we know E j has exponential decay it ust be the case that its axiu occurs at an interior point of soe f0 (S G). It is well known (see [7] Proposition 2.11) that E j and E j ust have opposite signs at any local extree point of E j. Since E j = E j je and < 0 we find that sgn( E j ) = sgn(e j ) iplies E j je. The bound on E j follows by induction and the fact that E E (q 0 ) = 1. References [1] Martin T. Barlow and Jun Kigai, Localized eigenfunctions of the Laplacian on p.c.f. self-siilar sets, J. London Math. Soc. (2) 56 (1997), no. 2, MR MR (99b:35162) [2] Nitsan Ben-Gal, Abby Shaw-Krauss, Robert S. Strichartz, and Clint Young, Calculus on the sierpinski gasket II: Point singularities, eigenfunctions, and noral derivatives of the heat kernel, Trans. Aer. Math. Soc., posted on April 17, 2006, PII: S (06) (to appear in print). [3] Kyallee Dalryple, Robert S. Strichartz, and Jade P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl. 5 (1999), no. 2-3, MR MR (2000k:31016) [4] M. Fukushia and T. Shia, On a spectral analysis for the Sierpiński gasket, Potential Anal. 1 (1992), no. 1, MR MR (95b:31009) [5] Jun Kigai, Analysis on fractals, Cabridge Tracts in Matheatics, vol. 143, Cabridge University Press, Cabridge, MR MR (2002c:28015) [6] Leonid Malozeov and Alexander Teplyaev, Self-siilarity, operators and dynaics, Math. Phys. Anal. Geo. 6 (2003), no. 3, MR MR (2004d:47012) [7] Jonathan Needlean, Robert S. Strichartz, Alexander Teplyaev, and Po-La Yung, Calculus on the Sierpinski gasket. I. Polynoials, exponentials and power series, J. Funct. Anal. 215 (2004), no. 2, MR MR [8] Luke Rogers, Robert S. Strichartz, and Alexander Teplyaev, Sooth bups, a Borel theore and partitions of unity on p.c.f. fractals, In preparation. [9] Robert S. Strichartz, Differential equations on fractals: A tutorial, Princeton University Press, to appear [10], Fractals in the large, Canad. J. Math. 50 (1998), no. 3, MR MR (99f:28015) [11] Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal. 159 (1998), no. 2, MR MR (99j:35153) 12