Constructive Approximation: Special Issue: Fractal by Michael F. Barnsley

By Michael F. Barnsley

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As before we estimate IEx(m)IH· Defining Sx(m) as we did in the proof of Theorem 3(ii) we obtain, by induction, m ll i=l m ax(i)(uJ:S:IEx(m)lw:::; ll i=l axu)(vj), m m j-1 m m m j-1 m /Sx(m)/H:S:/S/H n Cx(i)(zi)+/S/wb i=i I n Cx(k)(zd n ax(k)(vd, i=i k=i k=j+i /Sx(m)IH ~ /S/H n Cx(i)(zi) -/S/wb i=i I n Cx(k)(zk) n ax(k)(vk)· i=l k=i k=j+l and Following the argument of Theorem 3(ii) we find a d > 0 such that d · Cxolzt) · · · Cx(m)(Zm) :51Ex(m)IH :5 d · Cx(l)(zl) · · · Cx(m)(Zm)· We can now apply Lemmas 6 and 7 to show that, for A almost all x, (1/m)logiEx(m)lw~A(A) and (1/m)logiEx(m)IH~C(A) as m ~ oo.

If Lk F(k/ b)= b, then the sequence of distributions Tn converges weakly toward a distribution T. If G( ~) is the Fourier transform ofT, then G( ~) = Dk, 1 [P( -gj bk)/ b]. 2) G(bg) = G(g) ( ~ F(k/ b) e-ikg) I b. transform of Tn is in fact Gn(g) = Pn( -gj bn)/ bn = D1"'k"'n [P(-gjbk)/b]. The sequence Gn(g) converges pointwise to G(g). The Fourier Definition. process. The function G is called the Fourier transform of the interpolation 6. Continuity of the Fundamental Function We consider an interpolation process of type (b, N).

The condition 0 < a; < C; that we imposed is required to ensure that the values of b; do not asymptotically influence IEx(m)IH for any x, m. There is an interesting example suggested to me by Benoit Mandelbrot for which b; 0 so that the condition of a; < C; can be relaxed, namely when f is the cumulative distribution function of a Bernoulli measure. To be more precise, let JL be the measure of Corollary 5. For any Borel set B and 0:::::; i < n, = p;· JL(B)=JL(a· i+a· B) (where a = 1/ n ). j

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