AUTHOR=Aceska Roza , Kim Yeon H. 

TITLE=Scalability of Frames Generated by Dynamical Operators

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=3

YEAR=2017

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2017.00022

DOI=10.3389/fams.2017.00022

ISSN=2297-4687

ABSTRACT=<p>Let ℍ be a separable Hilbert space, let <italic>G</italic> ⊂ ℍ, and let <italic>A</italic> be an operator on ℍ. Under appropriate conditions on <italic>A</italic> and <italic>G</italic>, it is known that the set of iterations <inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:mstyle mathvariant="bold"><mml:mtext>g</mml:mtext></mml:mstyle><mml:mo>|</mml:mo><mml:mstyle mathvariant="bold"><mml:mtext>g</mml:mtext></mml:mstyle><mml:mo>∈</mml:mo><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mi>L</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>g</mml:mtext></mml:mstyle></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math></inline-formula> is a frame for ℍ. We call <italic>F</italic><sub><italic>G</italic></sub>(<italic>A</italic>) a dynamical frame for ℍ, and explore further its properties; in particular, we show that the canonical dual frame of <italic>F</italic><sub><italic>G</italic></sub>(<italic>A</italic>) also has an iterative set structure. We explore the relations between the operator <italic>A</italic>, the set <italic>G</italic> and the number of iterations <italic>L</italic> which ensure that the system <italic>F</italic><sub><italic>G</italic></sub>(<italic>A</italic>) is a scalable frame. We give a general statement on frame scalability, and study in detail the case when <italic>A</italic> is a normal operator, utilizing the unitary diagonalization. In addition, we answer the question of when <italic>F</italic><sub><italic>G</italic></sub>(<italic>A</italic>) is a scalable frame in several special cases involving block-diagonal and companion operators.</p>