Riesz-Thorinの内挿定理を用いてHausdorff-Young の不等式を証明する
Riesz-Thorinの内挿定理についてchatGPTに英語で質問
Why they cite it interpolation theory
答えは
chatGPT:
The Riesz-Thorin theorem is commonly cited in the context of interpolation theory because it provides a useful tool for obtaining estimates for the boundedness of linear operators in the process of constructing interpolating functions.
In interpolation theory, one is often concerned with finding a function that interpolates a given set of data points and has desirable properties. The Riesz-Thorin theorem can be used to estimate the behavior of the interpolating function on different parts of the domain, based on its behavior on other parts of the domain. This allows one to obtain bounds on the error in the interpolation and to control the rate of convergence of the interpolating function.
Thus, the Riesz-Thorin theorem plays an important role in the study of interpolation and in many related areas, such as numerical analysis and partial differential equations.
つぎは文献から
Riesz-Thorinの内挿定理
$${{1\le {{p}_{0}},{{p}_{1}},{{q}_{0}},{{q}_{1}}\le \infty }}$$ とする。
$${{0<\theta <1}}$$ にたいして、
$${{\frac{1}{p}=\frac{1-\theta }{{{p}_{0}}}+\frac{\theta }{{{p}_{1}}}}}$$ , $${{\frac{1}{q}=\frac{1-\theta }{{{q}_{0}}}+\frac{\theta }{{{q}_{1}}}}}$$
により、$${{p,q}}$$ を定義する。
線形作用素$${{T}}$$ を$${{T:{{L}^{{{p}_{0}}}}+{{L}^{{{p}_{1}}}}\to {{L}^{{{q}_{0}}}}+{{L}^{{{q}_{1}}}}}}$$
であり
$${{f\in {{L}^{{{p}_{0}}}}}}$$に対して $${{{{\left\| Tf \right\|}_{{{q}_{0}}}}\le {{M}_{0}}{{\left\| f \right\|}_{{{p}_{0}}}}}}$$
かつ
$${{f\in {{L}^{{{p}_{1}}}}}}$$に対して $${{{{\left\| Tf \right\|}_{{{q}_{1}}}}\le {{M}_{1}}{{\left\| f \right\|}_{{{p}_{1}}}}}}$$
を満たすと仮定する。このとき、
$${{f\in {{L}^{p}}}}$$に対して $${{{{\left\| Tf \right\|}_{q}}\le M_{0}^{1-\theta }M_{1}^{\theta }{{\left\| f \right\|}_{p}}}}$$
が成立する。
また今度は日本語で質問
Housdorff-Young について説明して
chatGPT:
The Hausdorff-Young theorem, also known as the Hausdorff-Young inequality, is a result in harmonic analysis that provides conditions for the boundedness of certain Fourier transforms. It relates the norm of a function in the Lp space to the norm of its Fourier transform in the Lq space, where p and q are conjugate exponents.
The theorem states that if f is a function in the Lp space (1 ≤ p ≤ 2), then its Fourier transform F(ω) belongs to the Lq space (2 ≤ q ≤ ∞), and there exists a constant C such that:
||F||q <= C ||f||p
where ||.||p and ||.||q denote the Lp and Lq norms, respectively. The Hausdorff-Young theorem is a cornerstone of modern harmonic analysis, and it plays a fundamental role in many areas of mathematics, including signal processing, image processing, and wavelet theory.
次は文献から
Hausdorff-Young の不等式
Hausdorff-Youngの不等式 もし、$${{f\in {{L}^{p}}}}$$, $${{1\le p\le 2}}$$のとき、
$${{\hat{f}\in {{L}^{q}}}}$$であり、
$${{{{\left\| {\hat{f}} \right\|}_{q}}\le {{\left\| f \right\|}_{p}}}}$$
がなりたつ。ただし、$${{\frac{1}{p}+\frac{1}{q}=1}}$$ 。
Riesz-Thorinの定理の線形作用素$${{T}}$$ をフーリエ変換と置いて適用すれば、
Hausdorff-Youngの不等式は容易に証明できる。
$${{{{\left\| {\hat{f}} \right\|}_{\infty }}\le {{\left\| f \right\|}_{1}}}}$$
$${{{{\left\| {\hat{f}} \right\|}_{2}}={{\left\| f \right\|}_{2}}}}$$ （プランシュレル）
$${{\frac{1}{p}=\frac{1-\theta }{{{p}_{0}}}+\frac{\theta }{{{p}_{1}}}}}$$ , $${{\frac{1}{q}=\frac{1-\theta }{{{q}_{0}}}+\frac{\theta }{{{q}_{1}}}}}$$
において、
$${{{{q}_{0}}={{p}_{0}}=2}}$$ ,$${{{{p}_{1}}=1}}$$ ,$${{{{q}_{1}}=\infty }}$$とおくと$${{{{M}_{0}}={{M}_{1}}}}$$として仮定をみたすので、$${{\frac{1}{p}=\frac{1-\theta }{2}+\frac{\theta }{1}}}$$、$${{\frac{1}{q}=\frac{1-\theta }{2}+\frac{\theta }{\infty }}}$$
から、$${{\theta }}$$を$${{0<\theta <1}}$$の範囲でどのように選ぼうと
$${{{{\left\| {\hat{f}} \right\|}_{q}}\le {{\left\| f \right\|}_{p}}}}$$
が$${{\frac{1}{p}+\frac{1}{q}=1}}$$として成り立つことが証明される。
このように、Riesz-Thorin の定理はきわめて強力なものである。
そこで、もうひとつchatGPTに次の質問をしてみた。
Riesz-Thorin の定理の証明はどうするの
Riesz-Thorin theorem is a fundamental result in interpolation theory that provides a way to interpolate between two different norms on a Banach space. The proof of the theorem involves using Hölder's inequality and some properties of complex numbers.
A detailed proof of the Riesz-Thorin theorem can be found in many books and articles on functional analysis, for example, in "Interpolation of Linear Operators" by G. G. Lorentz, S. Marcinkiewicz, and J. Orlicz. It is a rather technical proof and requires a solid understanding of functional analysis and complex analysis.
たしかにcomplex analysis を用いるらしい。次の定理ではないだろうか
Doetschの3線定理
$${{f\left( z \right)}}$$ , $${{z=x+iy}}$$ は帯状領域$${{\alpha < x< \beta}}$$ ,$${{-\infty < y< \infty }}$$で正則、かつ有界とし、
$${{\underset{-\infty < y< \infty }{\mathop{\sup }}\,\left| f\left( x+iy \right) \right|=L\left( x \right)}}$$ とおくと$${{\log L\left( x \right)}}$$は$${{x}}$$ の凸関数である。すなわち、
$${{L\left( {{\sigma }_{2}} \right)\le L{{\left( {{\sigma }_{1}} \right)}^{\frac{{{\sigma }_{3}}-{{\sigma }_{2}}}{{{\sigma }_{3}}-{{\sigma }_{1}}}}}L{{\left( {{\sigma }_{2}} \right)}^{\frac{{{\sigma }_{2}}-{{\sigma }_{1}}}{{{\sigma }_{3}}-{{\sigma }_{1}}}}}}}$$,
$${{\alpha <{{\sigma }_{1}}<{{\sigma }_{2}}<{{\sigma }_{3}}<\beta }}$$