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

By Michael F. Barnsley

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Our main concern is to prove that the extension y(t) is uniformly continuous on any finite interval whatever the base b, the even number 2N of nodes, and the initial values y( n ). For each base b and each number N, there is a fundamental function F(t), the iterative interpolation of the sequence F(n), which is equal to 1 at n = 0 and equal to 0 elsewhere. The main properties of the interpolation process come from its fundamental function. The basic functional equation for F is F(tl b)= I F(nl b)F(t- n).

M. F. A. J. H. A. P. A. Constr. Approx. (1989) 5: 33-48 CONSTRUCTIVE APPROXIMAnON © 1989 Springer-Verlag New York Inc. Holder Exponents and Box Dimension for Self-Affine Fractal Functions Tim Bedford Abstract. We consider some self-affine fractal functions previously studied by Barnsley et al. The graphs of these functions are invariant under certain affine scalings, and we extend their definition to allow the use of nonlinear scalings. The Holder exponent, h, for these fractal functions is calculated and we show that there is a larger Holder exponent, h,, defined at almost e~ery point (with respect to Lebesgue measure).

Lemma 1. hx =sup{ a: lf(x)- f(y )j ::5 jx- yja for ally in some neighbourhood of x}. Proof. Let y equal the right-hand side of the latter equation. We first show that hx ~ y. Choose 5 > 0 and let a= hx- 5. In some small e-neighbourhood of x we have y E B(x; e) => a< (logjf(x)- f(y )i)/logjx- yj and so if(x )- f(y )j ::5 jx- yja. Thus a = hx- 5 ::5 '}' for any 5 > 0, and so hx ::5 ')'. In order to prove that hx::::; y first choose some 5 > 0. s holds. Taking logarithms gives us that (logjf(x)- f(y )j)/logjx- yj ~ y- 5 in some neighbourhood of x and so • we must have hx ~ y- 5 for any 5 > 0 which implies that hx ~ y.

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