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- Hausdorff–Young_inequality abstract "In mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. William Henry Young (1913) proved the inequality for some special values of q, and Hausdorff (1923) 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 Babenko (1961) and Beckner (1975) 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, whereIn a short formula, this says thatThis 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 .".
- Hausdorff–Young_inequality wikiPageID "22386332".
- Hausdorff–Young_inequality wikiPageRevisionID "593037879".
- Hausdorff–Young_inequality authorlink "William Henry Young".
- Hausdorff–Young_inequality first "William Henry".
- Hausdorff–Young_inequality last "Young".
- Hausdorff–Young_inequality year "1913".
- Hausdorff–Young_inequality subject Category:Fourier_analysis.
- Hausdorff–Young_inequality subject Category:Inequalities.
- Hausdorff–Young_inequality comment "In mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. William Henry Young (1913) proved the inequality for some special values of q, and Hausdorff (1923) proved it in general.".
- Hausdorff–Young_inequality label "Hausdorff–Young inequality".
- Hausdorff–Young_inequality sameAs Hausdorff%E2%80%93Young_inequality.
- Hausdorff–Young_inequality sameAs Q5682823.
- Hausdorff–Young_inequality sameAs Q5682823.
- Hausdorff–Young_inequality wasDerivedFrom Hausdorff–Young_inequality?oldid=593037879.