WebHausdorff-Young theorem, and Young's inequality, where Fourier transforms and convolutions are used respectively. II* Diagram proof of the Hausdorίϊ-Young Theorem* As a corollary of the author's diagram proof [6] of Riesz's theorem, we 1 Terms used in the introduction will be defined in the paper. 97 WebHausdorff-Young theorem, and Young's inequality, where Fourier transforms and convolutions are used respectively. II* Diagram proof of the Hausdorίϊ-Young Theorem* …

MATH 247A : Fourier analysis - UCLA Mathematics

The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by William Henry Young (1913) and extended by Hausdorff (1923). It is now typically understood as a rather direct corollary of the Plancherel theorem, found in … See more Given a nonzero real number p, define the real number p' (the "conjugate exponent" of p) by the equation $${\displaystyle {\frac {1}{p}}+{\frac {1}{p'}}=1.}$$ If p is equal to one, … See more Equality is achieved in the Hausdorff-Young inequality for (multidimensional) Fourier series by taking See more Fourier series Given a function $${\displaystyle f:(0,1)\to \mathbb {C} ,}$$ one defines its "Fourier coefficients" as a … See more Here we use the language of normed vector spaces and bounded linear maps, as is convenient for application of the Riesz-Thorin … See more The condition p∈[1,2] is essential. If p>2, then the fact that a function belongs to $${\displaystyle L^{p}}$$, does not give any additional … See more Web2 Young’s Inequality 2 3 Minkowski’s Inequality 3 4 H older’s inequality 5 1 Introduction The Cauchy inequality is the familiar expression 2ab a2 + b2: (1) This can be proven very simply: noting that (a b)2 0, we have 0 (a b)2 = a2 2ab b2 (2) which, after rearranging terms, is precisely the Cauchy inequality. In this note, we prove low red eyes

On the vector valued Hausdorff-Young inequality SpringerLink

It has been shown in the first section that the Fourier transform maps L (R ) boundedly into L (R ) and L (R ) into itself. A similar argument shows that the Fourier series operator, which transforms periodic functions  f  : T → C into functions whose values are the Fourier coefficients The Hausdorff–Young inequality can also be established for the Fourier transform on locally compact Abelian groups. The norm estimate of 1 is not optimal. See the main article for references. WebMay 11, 2024 · In the proof of the proposition prior to this one (where B = R n ), we showed that the inequality ‖ f ^ ‖ L q ( R n) ≤ C ‖ f ‖ L p ( R n) gives us λ n λ − n / q ≤ C ~ λ n / p … Webrem (QSP), the quantum Hausdorff–Young inequality (QHY) with 1=p +1=q =1, the quantum Young inequality (QY) with 1=p +1=q =1+1=r, and the basic quantum … low red dunks