Abstract
While the Bedrosian identity for the Hilbert transform of a product does not hold for general Sobolev class functions, we show that the defect of this identity is more regular than would be naively expected. We use this result to give a stronger-than-expected estimate on the chain rule defect of the square root of the Laplacian.
Highlights
1 Introduction In 1963, Bedrosian [2] proved the following result on the Hilbert transform of the product of two functions: Let u, v ∈ L2(R) satisfy either supp u ⊂ [0, ∞) and supp v ⊂ [0, ∞) or supp u ⊂ [–a, a] and supp v ∩ [–a, a] = ∅ for some a > 0
This result is known as the Bedrosian theorem and (1) as the Bedrosian identity
The Bedrosian identity usually arises in the context of time-frequency analysis
Summary
Introduction In 1963, Bedrosian [2] proved the following result on the Hilbert transform of the product of two functions: Let u, v ∈ L2(R) satisfy either supp u ⊂ [0, ∞) and supp v ⊂ [0, ∞) or supp u ⊂ [–a, a] and supp v ∩ [–a, a] = ∅ for some a > 0, H(uv) = uHv. This result is known as the Bedrosian theorem and (1) as the Bedrosian identity.
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