Skip navigation
Skip navigation

Approximation of rough functions

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

Description

For given p ∈ [1,∞] and g ∈ Lᵖ(R), we establish the existence and uniqueness of solutions f ∈ Lᵖ(R), to the equation f (x) − a f (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.

dc.contributor.authorBarnsley, M. F.
dc.contributor.authorHarding, B.
dc.contributor.authorVince, A.
dc.contributor.authorViswanathan, P.
dc.date.accessioned2017-02-13T00:14:46Z
dc.date.available2017-02-13T00:14:46Z
dc.identifier.issn0021-9045
dc.identifier.urihttp://hdl.handle.net/1885/112240
dc.description.abstractFor given p ∈ [1,∞] and g ∈ Lᵖ(R), we establish the existence and uniqueness of solutions f ∈ Lᵖ(R), to the equation f (x) − a f (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.
dc.description.sponsorshipWe acknowledge support for this work by Australian Research Council grant DP13 0101738. This work was partially supported by a grant from the Simons Foundation (322515 to Andrew Vince).
dc.format21 pages
dc.format.mimetypeapplication/pdf
dc.publisherElsevier
dc.rights© 2016 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
dc.sourceJournal of Approximation Theory
dc.subjectFunctional equations
dc.subjectFractal interpolation
dc.subjectIterated function system
dc.subjectFractal geometry
dc.subjectFourier series
dc.titleApproximation of rough functions
dc.typeJournal article
local.identifier.citationvolume209
dcterms.dateAccepted2016-04-25
dc.date.issued2016-09
local.publisher.urlhttps://www.elsevier.com/
local.type.statusPublished Version
local.contributor.affiliationBarnsley, Michael F., Mathematical Sciences Institute, CPMS Mathematical Sciences Institute, The Australian National University
local.contributor.affiliationHarding, Brendan, Mathematical Sciences Institute, CPMS Mathematical Sciences Institute, The Australian National University
local.contributor.affiliationViswanathan, P., Mathematical Sciences Institute, CPMS Mathematical Sciences Institute, The Australian National University
dc.relationhttp://purl.org/au-research/grants/arc/DP13 0101738
local.bibliographicCitation.startpage23
local.bibliographicCitation.lastpage43
local.identifier.doi10.1016/j.jat.2016.04.003
dcterms.accessRightsOpen Access
CollectionsANU Research Publications

Download

File Description SizeFormat Image
Barnsley M F et al Approximation of rough 2016.pdf230.19 kBAdobe PDFThumbnail


Items in Open Research are protected by copyright, with all rights reserved, unless otherwise indicated.

Updated:  19 May 2020/ Responsible Officer:  University Librarian/ Page Contact:  Library Systems & Web Coordinator