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arXiv · 1412.3871

Approximation of Rough Functions

M. F. Barnsley·B. Harding·A. Vince·P. Viswanathan
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Abstract

For given $p\in\lbrack1,\infty]$ and $g\in L^{p}\mathbb{(R)}$, we establish the existence and uniqueness of solutions $f\in L^{p}(\mathbb{R)}$, to the equation \[ f(x)-af(bx)=g(x), \] where $a\in\mathbb{R}$, $b\in\mathbb{R} \setminus \{0\}$, and $\left\vert a\right\vert \neq\left\vert b\right\vert ^{1/p}$. Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established.

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BibTeXRIS

M. F. Barnsley, B. Harding, A. Vince, P. Viswanathan. 2014-12-12. Approximation of Rough Functions. https://arxiv.org/abs/1412.3871

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