<p>In this paper, we investigate the scalability of a given frame in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_809_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> by using graphs. For each frame <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_809_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_809_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, we associate a simple undirected graph <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_809_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$G(\phi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and use it to verify the scalability of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_809_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation>. We provide some necessary conditions to test the scalability of a given frame. Finally, we study the scalability of some special classes of frames by using graphs.</p>

Frame Scaling by Graphs

期刊论文
发表日期：

2025年7月7日

查看全文

K. Ayyanar,
P. Sam Johnson,
A. Senthil Thilak

摘要

In this paper, we investigate the scalability of a given frame in $\mathbb {R}^n$ $R^{n}$ by using graphs. For each frame $\phi $ $ϕ$ in $\mathbb {R}^n$ $R^{n}$ , we associate a simple undirected graph $G(\phi )$ $G (ϕ)$ and use it to verify the scalability of $\phi $ $ϕ$ . We provide some necessary conditions to test the scalability of a given frame. Finally, we study the scalability of some special classes of frames by using graphs.