In mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. William Henry Young () proved the inequality for some special values of q, and Hausdorff () proved it in general. More generally the inequality also applies to the Fourier transform of a function on a locally compact group, such as Rn, and in this case and gave a sharper form of it called the Babenko–Beckner inequality. We consider the Fourier operator, namely let T be the operator that takes a function Parseval's theorem shows that T is bounded from to to to .

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• In mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. William Henry Young () proved the inequality for some special values of q, and Hausdorff () proved it in general. More generally the inequality also applies to the Fourier transform of a function on a locally compact group, such as Rn, and in this case and gave a sharper form of it called the Babenko–Beckner inequality. We consider the Fourier operator, namely let T be the operator that takes a function on the unit circle and outputs the sequence of its Fourier coefficients Parseval's theorem shows that T is bounded from to with norm 1. On the other hand, clearly, so T is bounded from to with norm 1. Therefore we may invoke the Riesz–Thorin theorem to get, for any 1 < p < 2 that T, as an operator from to , is bounded with norm 1, where In a short formula, this says that This is the well known Hausdorff–Young inequality. For p > 2 the natural extrapolation of this inequality fails, and the fact that a function belongs to , does not give any additional information on the order of growth of its Fourier series beyond the fact that it is in . (en)
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• 22386332 (xsd:integer)
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• 626892195 (xsd:integer)
• William Henry Young
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• William Henry
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• Young
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• 1913 (xsd:integer)
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• In mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. William Henry Young () proved the inequality for some special values of q, and Hausdorff () proved it in general. More generally the inequality also applies to the Fourier transform of a function on a locally compact group, such as Rn, and in this case and gave a sharper form of it called the Babenko–Beckner inequality. We consider the Fourier operator, namely let T be the operator that takes a function Parseval's theorem shows that T is bounded from to to to . (en)
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• Hausdorff–Young inequality (en)
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