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<p>In this article we prove the existence of complex local fractal functions of the Hardy-Orlicz class. A local fractal function is the fixed point of the Read-Bajraktarević functional. The graph of the fixed point is the attractor of an appropriate iterated function system (IFS), whose construction is fairly standard. However, local fractal functions of the Hardy-Orlicz class enjoy of a few particular properties stemming from the complex conjugation. For example, both the fixed point and its graph are intrinsically real. The latter reflects as an embedding of the <i>realified</i> attractor of the induced complex iterated function system (CIFS) into the product <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13663_2025_802_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </math></EquationSource> <EquationSource Format="TEX">$[-1,1]\times [-1,1]$</EquationSource> </InlineEquation>. We provide a characterization of this type of IFSs via a quasi-integral representation of the Read-Bajraktarević functional. The construction of the associated CIFS in this case requires of a local mean value theorem for holomorphic functions dating back to 1965.</p>

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The Read-Bajraktarević functional on Hardy-Orlicz spaces

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  • 发表日期：

    2025年9月1日

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  • Waldo Arriagada

摘要

In this article we prove the existence of complex local fractal functions of the Hardy-Orlicz class. A local fractal function is the fixed point of the Read-Bajraktarević functional. The graph of the fixed point is the attractor of an appropriate iterated function system (IFS), whose construction is fairly standard. However, local fractal functions of the Hardy-Orlicz class enjoy of a few particular properties stemming from the complex conjugation. For example, both the fixed point and its graph are intrinsically real. The latter reflects as an embedding of the realified attractor of the induced complex iterated function system (CIFS) into the product [ − 1 , 1 ] × [ − 1 , 1 ] $[-1,1]\times [-1,1]$ . We provide a characterization of this type of IFSs via a quasi-integral representation of the Read-Bajraktarević functional. The construction of the associated CIFS in this case requires of a local mean value theorem for holomorphic functions dating back to 1965.