Let m be a positive integer and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> be a collection of closed subspaces in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^2(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Given the measurements <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {F}_Y=\left\{ \left\{ y_k^1 \right\} _{k\in \mathbb {Z}},\ldots , \left\{ y_k^m \right\} _{k\in \mathbb {Z}} \right\} \subset \ell ^2(\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">F</mi> <mi>Y</mi> </msub> <mo>=</mo> <mfenced close="}" open="{"> <msub> <mfenced close="}" open="{"> <msubsup> <mi>y</mi> <mi>k</mi> <mn>1</mn> </msubsup> </mfenced> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mfenced close="}" open="{"> <msubsup> <mi>y</mi> <mi>k</mi> <mi>m</mi> </msubsup> </mfenced> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </msub> </mfenced> <mo>⊂</mo> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of unknown functions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {F}=\left\{ f_1, \ldots ,f_m \right\} \subset L^2( \mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo>=</mo> <mfenced close="}" open="{"> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mi>m</mi> </msub> </mfenced> <mo>⊂</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, in this paper we study the problem of finding an optimal space S in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> that is “closest” to the measurements <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {F}_Y$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mi>Y</mi> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>. Since the class of finitely generated shift-invariant spaces (FSISs) is popularly used for modelling signals, we assume <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> consists of FSISs. We will be considering three cases. In the first case, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> consists of FSISs without any assumption on extra invariance. In the second case, we assume <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> consists of extra invariant FSISs, and in the third case, we assume <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> has translation-invariant FSISs. In all three cases, we prove the existence of an optimal space.

Optimal Shift-Invariant Spaces from Uniform Measurements

期刊论文
发表日期：

2026年1月31日

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Rohan Joy,
Radha Ramakrishnan

摘要

Let m be a positive integer and $\mathcal {C}$ $C$ be a collection of closed subspaces in $L^2(\mathbb {R})$ $L^{2} (R)$ . Given the measurements $\mathcal {F}_Y=\left\{ \left\{ y_k^1 \right\} _{k\in \mathbb {Z}},\ldots , \left\{ y_k^m \right\} _{k\in \mathbb {Z}} \right\} \subset \ell ^2(\mathbb {Z})$ $F_{Y} = \{{\{y_{k}^{1}\}}_{k \in Z}, \dots, {\{y_{k}^{m}\}}_{k \in Z}\} \subset ℓ^{2} (Z)$ of unknown functions $\mathcal {F}=\left\{ f_1, \ldots ,f_m \right\} \subset L^2( \mathbb {R})$ $F = \{f_{1}, \dots, f_{m}\} \subset L^{2} (R)$ , in this paper we study the problem of finding an optimal space S in $\mathcal {C}$ $C$ that is “closest” to the measurements $\mathcal {F}_Y$ $F_{Y}$ of $\mathcal {F}$ $F$ . Since the class of finitely generated shift-invariant spaces (FSISs) is popularly used for modelling signals, we assume $\mathcal {C}$ $C$ consists of FSISs. We will be considering three cases. In the first case, $\mathcal {C}$ $C$ consists of FSISs without any assumption on extra invariance. In the second case, we assume $\mathcal {C}$ $C$ consists of extra invariant FSISs, and in the third case, we assume $\mathcal {C}$ $C$ has translation-invariant FSISs. In all three cases, we prove the existence of an optimal space.