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<p><i>Recently, it has been discovered that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error being measured in the square norm. It was established that a simple greedy type algorithm – Weak Orthogonal Matching Pursuit – based on good points for universal discretization provides effective recovery in the square norm. In this paper, we extend these results by replacing the square norm with other integral norms. In this case, we need to conduct our analysis in a Banach space rather than in a Hilbert space, making the techniques more involved. In particular, we establish that a greedy type algorithm&#xa0;– the Weak Chebyshev Greedy Algorithm&#xa0;– based on good points for the</i> <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-<i>universal discretization provides good recovery in the</i> <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> <i>norm for</i> <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(2\le p&lt;\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>p</mi> <mo>&lt;</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. <i>Furthermore, we discuss the problem of stable recovery and demonstrate its close relationship with sampling discretization. Bibliography</i>: 15 <i>titles</i>.</p>

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LEBESGUE TYPE INEQUALITIES IN SPARSE SAMPLING RECOVERY

  • 期刊论文

  • 发表日期：

    2026年4月23日

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  • Feng Dai,
  • Vladimir Temlyakov

摘要

Recently, it has been discovered that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error being measured in the square norm. It was established that a simple greedy type algorithm – Weak Orthogonal Matching Pursuit – based on good points for universal discretization provides effective recovery in the square norm. In this paper, we extend these results by replacing the square norm with other integral norms. In this case, we need to conduct our analysis in a Banach space rather than in a Hilbert space, making the techniques more involved. In particular, we establish that a greedy type algorithm – the Weak Chebyshev Greedy Algorithm – based on good points for the \(L_p\) L p -universal discretization provides good recovery in the \(L_p\) L p norm for \(2\le p<\infty \) 2 ≤ p < ∞ . Furthermore, we discuss the problem of stable recovery and demonstrate its close relationship with sampling discretization. Bibliography: 15 titles.