Approximation of rough functions

M. F. Barnsley; B. Harding; A. Vince; P. Viswanathan

doi:10.1016/j.jat.2016.04.003

Approximation of rough functions

M. F. Barnsley^*, B. Harding, A. Vince, P. Viswanathan

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

For given p∈[1,∞] and g∈L^p(R), we establish the existence and uniqueness of solutions f∈L^p(R), to the equation f(x)−af(bx)=g(x), where a∈R, b∈R∖{0}, and |a|≠|b|^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.

Original language	English
Pages (from-to)	23-43
Number of pages	21
Journal	Journal of Approximation Theory
Volume	209
DOIs	https://doi.org/10.1016/j.jat.2016.04.003
Publication status	Published - 1 Sept 2016

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title = "Approximation of rough functions",

abstract = "For given p∈[1,∞] and g∈Lp(R), we establish the existence and uniqueness of solutions f∈Lp(R), to the equation f(x)−af(bx)=g(x), where a∈R, b∈R∖{0}, and |a|≠|b|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.",

keywords = "Fourier series, Fractal geometry, Fractal interpolation, Functional equations, Iterated function system",

author = "Barnsley, {M. F.} and B. Harding and A. Vince and P. Viswanathan",

note = "Publisher Copyright: {\textcopyright} 2016 The Author(s)",

year = "2016",

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doi = "10.1016/j.jat.2016.04.003",

language = "English",

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TY - JOUR

T1 - Approximation of rough functions

AU - Barnsley, M. F.

AU - Harding, B.

AU - Vince, A.

AU - Viswanathan, P.

PY - 2016/9/1

Y1 - 2016/9/1

N2 - For given p∈[1,∞] and g∈Lp(R), we establish the existence and uniqueness of solutions f∈Lp(R), to the equation f(x)−af(bx)=g(x), where a∈R, b∈R∖{0}, and |a|≠|b|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.

AB - For given p∈[1,∞] and g∈Lp(R), we establish the existence and uniqueness of solutions f∈Lp(R), to the equation f(x)−af(bx)=g(x), where a∈R, b∈R∖{0}, and |a|≠|b|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.

KW - Fourier series

KW - Fractal geometry

KW - Fractal interpolation

KW - Functional equations

KW - Iterated function system

UR - http://www.scopus.com/inward/record.url?scp=84976593294&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2016.04.003

DO - 10.1016/j.jat.2016.04.003

M3 - Article

SN - 0021-9045

VL - 209

SP - 23

EP - 43

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

ER -

Approximation of rough functions

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