“Hausdorff–Young inequality”的意思、由来-开放百科全书

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Hausdorff–Young inequality

释义

Optimal estimates
References

In mathematics, the Hausdorff−Young inequality bounds the L^q-norm of the Fourier coefficients of a periodic function for q ≥ 2. {{harvs|txt|authorlink=William Henry Young|first=William Henry|last=Young|year=1913}} proved the inequality for some special values of q, and {{harvs|txt|authorlink=Felix Hausdorff|last=Hausdorff|year=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 {{harvtxt|Babenko|1961}} and {{harvtxt|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, 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 .

Optimal estimates

The constant involved in the Hausdorff–Young inequality can be made optimal by using careful estimates from the theory of harmonic analysis. If for , the optimal bound is

where is the Hölder conjugate of {{harv|Cifuentes|2010}}

References

{{Citation | last1=Babenko | first1=K. Ivan | title=An inequality in the theory of Fourier integrals |mr=0138939 | year=1961 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=25 | pages=531–542}} English transl., Amer. Math. Soc. Transl. (2) 44, pp. 115–128
{{Citation | doi=10.2307/1970980 | last1=Beckner | first1=William | author1-link=William Beckner (mathematician) | title=Inequalities in Fourier analysis |mr=0385456 | year=1975 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=102 | issue=1 | pages=159–182 | jstor=1970980}}
{{citation|title=Harmonic Analysis and Partial Differential Equations|volume=505|series=Contemporary Mathematics|first=Patricio|last=Cifuentes|publisher=American Mathematical Society|year=2010|isbn=9780821858318|page=94|url=https://books.google.com/books?id=ern6j-9vjSgC&pg=PA94}}.
{{Citation | last1=Hausdorff | first1=Felix | author1-link=Felix Hausdorff | title=Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen | doi=10.1007/BF01175679 | year=1923 | volume=16 | pages=163–169 | journal=Mathematische Zeitschrift}}
{{Citation | last1=Young | first1=W. H. | author1-link=William Henry Young | title=On the Determination of the Summability of a Function by Means of its Fourier Constants | doi=10.1112/plms/s2-12.1.7 | year=1913 | journal=Proc. London Math. Soc. | volume=12 | pages=71–88}}

2 : Inequalities|Fourier analysis

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