2016/06/17 by Xin-Rong Dai, Qiyu Sun · 3 citations
Mathematics · Engineering · Computer Science · #Mathematical Analysis and Transform Methods #Advanced Numerical Analysis Techniques #Image and Signal Denoising Methods
paper · doi:10.1090/memo/1152
A longstanding problem in Gabor theory is to identify time-frequency shifting lattices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a double-struck upper Z times b double-struck upper Z"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo> × </mml:mo> <mml:mi>b</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">a\mathbb Z× b\mathbb Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and ideal window functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi Subscript upper I"> <mml:semantics> <mml:msub> <mml:mi> χ </mml:mi> <mml:mi>I</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">χ I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on intervals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet e Superscript minus 2 pi i n b t Baseline chi Subscript upper I Baseline left-parenthesis t minus m a right-parenthesis colon left-parenthesis m comma n right-parenthesis element-of double-struck upper Z times double-struck upper Z EndSet"> <mml:semantics> <mml:mrow> <mml:mo></mml:mo> <mml:mspace width="thinmathspace"/> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mi>b</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:msub> <mml:mi> χ </mml:mi> <mml:mi>I</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo> − </mml:mo> <mml:mi>m</mml:mi> <mml:mi>a</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo> × </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mo></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex"> \ e-2π i n bt χ I(t- m a): (m, n)∈ \mathbb Z× \mathbb Z \</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are Gabor frames for the space of all square-integrable functions on the real line. In this paper, we create a time-domain approach for Gabor frames, introduce novel techniques involving invariant sets of non-contractive and non-measure-preserving transformations on the line, and provide a complete answer to the above <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a b c"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">abc</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -problem for Gabor systems.