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<p>In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemarié wavelet system of nonnegative integer order, <i>n</i>. We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(s &lt; n+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>&lt;</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, which includes values of <i>s</i> where the Battle-Lemarié system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(s=n+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. This builds off of earlier work by G. Garrigós, A. Seeger, and T. Ullrich, where they proved the case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemarié system to give an unconditional basis of the Triebel-Lizorkin spaces.</p>

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A Battle-Lemarié Frame Characterization of Besov and Triebel-Lizorkin Spaces

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  • 发表日期：

    2026年4月20日

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  • Andrew Haar

摘要

In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemarié wavelet system of nonnegative integer order, n. We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to \(s < n+1\) s < n + 1 , which includes values of s where the Battle-Lemarié system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case \(s=n+1\) s = n + 1 . This builds off of earlier work by G. Garrigós, A. Seeger, and T. Ullrich, where they proved the case \(n=0\) n = 0 , i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemarié system to give an unconditional basis of the Triebel-Lizorkin spaces.