An Entity of Type : yago:Quality104723816, within Data Space : dbpedia-live.openlinksw.com associated with source document(s)

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.

AttributesValues
rdf:type
sameAs
foaf:isPrimaryTopicOf
rdfs: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.
rdfs:label
• Hausdorff–Young inequality
has 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 f {\displaystyle f} on the unit circle and outputs the sequence of its Fourier coefficients f ^ ( n ) = 1 2 π ∫ 0 2 π e − i n x f ( x ) d x , n = 0 , ± 1 , ± 2 , … . {\displaystyle {\widehat {f}}(n)={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-inx}f(x)\,dx,\quad n=0,\pm 1,\pm 2,\dots .} Parseval's theorem shows that T is bounded from L 2 {\displaystyle L^{2}} to ℓ 2 {\displaystyle \ell ^{2}} with norm 1. On the other hand, clearly, | ( T f ) ( n ) | = | f ^ ( n ) | = | 1 2 π ∫ 0 2 π e − i n t f ( t ) d t | ≤ 1 2 π ∫ 0 2 π | f ( t ) | d t {\displaystyle |(Tf)(n)|=|{\widehat {f}}(n)|=\left|{\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-int}f(t)\,dt\right|\leq {\frac {1}{2\pi }}\int _{0}^{2\pi }|f(t)|\,dt} so T is bounded from L 1 {\displaystyle L^{1}} to ℓ ∞ {\displaystyle \ell ^{\infty }} with norm 1. Therefore we may invoke the Riesz–Thorin theorem to get, for any 1 < p < 2 that T, as an operator from L p {\displaystyle L^{p}} to ℓ q {\displaystyle \ell ^{q}} , is bounded with norm 1, where 1 p + 1 q = 1. {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1.} In a short formula, this says that ( ∑ n = − ∞ ∞ | f ^ ( n ) | q ) 1 / q ≤ ( 1 2 π ∫ 0 2 π | f ( t ) | p d t ) 1 / p . {\displaystyle \left(\sum _{n=-\infty }^{\infty }|{\widehat {f}}(n)|^{q}\right)^{1/q}\leq \left({\frac {1}{2\pi }}\int _{0}^{2\pi }|f(t)|^{p}\,dt\right)^{1/p}.} 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 L p {\displaystyle L^{p}} , does not give any additional information on the order of growth of its Fourier series beyond the fact that it is in ℓ 2 {\displaystyle \ell ^{2}} .
Link to the Wikipage edit URL
Link from a Wikipage to an external page
extraction datetime
Link to the Wikipage history URL
Wikipage page ID
page length (characters) of wiki page
Wikipage modification datetime
Wiki page out degree
Wikipage revision ID
Link to the Wikipage revision URL
• William Henry Young
dbp:first
• William Henry
dbp:last
• Young
dbp:wikiPageUsesTemplate
dbp:year
dct:subject
is foaf:primaryTopic of
is Wikipage redirect of
Faceted Search & Find service v1.17_git39 as of Aug 10 2019

Alternative Linked Data Documents: iSPARQL | ODE     Content Formats:       RDF       ODATA       Microdata      About

OpenLink Virtuoso version 08.03.3315 as of Sep 13 2019, on Linux (x86_64-generic-linux-glibc25), Single-Server Edition (61 GB total memory)