From 9ec4eea83923b8933e5acbd290ee13a1322b4a32 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 10 Jun 2026 23:39:46 -0400 Subject: [PATCH] Added the method of complex interpolation. --- refs.bib | 13 +++++ src/fa/interpolation/complex.tex | 91 +++++++++++++++++++++++++++++++ src/fa/interpolation/functors.tex | 2 +- src/fa/interpolation/index.tex | 1 + src/fa/notation.tex | 4 ++ 5 files changed, 110 insertions(+), 1 deletion(-) diff --git a/refs.bib b/refs.bib index 7a5e46a..595aed8 100644 --- a/refs.bib +++ b/refs.bib @@ -183,4 +183,17 @@ author={Pietsch, Albrecht}, year={2007}, publisher={Springer} +} + +@book{BerghInterpolation, + author = {Bergh, Jöran and Löfström, Jörgen}, + title = {Interpolation Spaces: An Introduction}, + series = {Grundlehren der Mathematischen Wissenschaften}, + volume = {223}, + publisher = {Springer-Verlag}, + address = {Berlin--New York}, + year = {1976}, + pages = {x+207}, + isbn = {3-540-07875-4}, + mrnumber = {0482275} } \ No newline at end of file diff --git a/src/fa/interpolation/complex.tex b/src/fa/interpolation/complex.tex index c7d9d5b..c52acdf 100644 --- a/src/fa/interpolation/complex.tex +++ b/src/fa/interpolation/complex.tex @@ -1,3 +1,94 @@ \section{The Complex Interpolation Method} \label{section:complex-interpolation} +\begin{definition}[Calderón Space] +\label{definition:calderon-space} + Let $S = \bracs{z \in \complex| \text{Re}(z) \in (0, 1)}$ and $(E_0, E_1)$ be a compatible couple of Banach spaces over $\complex$, then the \textbf{Calderón space} $\cf(E_0, E_1)$ is the Banach space of functions $f: \ol S \to E_0 + E_1$ such that: + \begin{enumerate} + \item $f$ is holomorphic on $S$. + \item $f$ is continuous on $\ol S$. + \item For each $t \in \real$, $f(it) \in E_0$, and $\lim_{|t| \to \infty}\norm{f(it)}_{E_0} = 0$. + \item For each $t \in \real$, $f(1 + it) \in E_1$, and $\lim_{|t| \to \infty}\norm{f(1 + it)}_{E_1} = 0$. + \end{enumerate} + + equipped with the norm + \[ + \norm{f}_{\cf(E_0, E_1)} = \max\braks{\sup_{t \in \real}\norm{f(it)}_{E_0}, \sup_{t \in \real}\norm{f(1 + it)}_{E_1}} + \] +\end{definition} +\begin{proof} + By the \hyperref[Maximum Modulus Theorem]{theorem:maximum-modulus-theorem} applied to $f$ as a function in $H(S; E_0 + E_1)$, $\norm{\cdot}_{\cf(E_0, E_1)}$ is a norm. + + By the \hyperref[Maximum Modulus Theorem]{theorem:maximum-modulus-theorem}, \autoref{proposition:holomorphic-complete}, and \autoref{proposition:uniform-limit-continuous}, $\cf(E_0, E_1)$ is complete. +\end{proof} + +\begin{definition}[The Complex Interpolation Method] +\label{definition:complex-interpolation-method} + Let $(E_0, E_1)$ be a compatible couple of Banach spaces over $\complex$, $\cf(E_0, E_1)$ be their \hyperref[Calderón space]{definition:calderon-space}, $\theta \in [0, 1]$, and + \[ + [E_0, E_1]_\theta = \bracsn{f(\theta)| f \in \cf(E_0, E_1)} + \] + + with the norm + \[ + \norm{x}_{[E_0, E_1]_\theta} = \inf\bracsn{\norm{f}_{\cf(E_0, E_1)}| f \in \cf(E_0, E_1), x = f(\theta)} + \] + + then: + \begin{enumerate} + \item $[E_0, E_1]_\theta$ is an intermediate Banach space between $E_0$ and $E_1$. + \item The mapping $C_\theta$ defined by $(E_0, E_1) \mapsto [E_0, E_1]_\theta$ is an interpolation functor of exact exponent $\theta$. + \end{enumerate} + + and functor $C_\theta$ is the \textbf{method of complex interpolation}. +\end{definition} +\begin{proof}[Proof, {{\cite[Theorem 4.1.2]{BerghInterpolation}}}. ] + (1): Let $\seq{x_n} \subset [E_0, E_1]_\theta$ with $\sum_{n \in \natp}\norm{x_n}_{[E_0, E_1]_\theta} < \infty$, then there exists $\seq{f_n} \subset \cf(E_0, E_1)$ such that for each $n \in \natp$, $f_n(\theta) = x_n$ and $\norm{f_n}_{\cf(E_0, E_1)} \le 2\norm{x_n}_{[E_0, E_1]_\theta}$. Since $\cf(E_0, E_1)$ is complete, there exists $f \in \cf(E_0, E_1)$ such that $f = \sum_{n = 1}^\infty f_n$. Let $x = f(\theta)$, then since $\sum_{n = 1}^N f_n \to f$ in $\cf(E_0, E_1)$ as $N \to \infty$, $\sum_{n = N}^\infty x_n \to x$ in $[E_0, E_1]_\theta$ as $N \to \infty$. Therefore $[E_0, E_1]_\theta$ is a Banach space by \autoref{lemma:banach-criterion}. + + For any $x \in E_0 \cap E_1$ and $\delta > 0$, let $f_\delta(z) = x_0 e^{(z - \theta)^2}$, then $f_\delta \in \cf(E_0, E_1)$ with $\norm{f_\delta}_{\cf(E_0, E_1)} \le e^\delta\norm{x}_{E_0 \cap E_1}$. Thus $x \in [E_0, E_1]_\theta$ with + \[ + \norm{x}_{[E_0, E_1]_\theta} \le \norm{f}_{\cf(E_0, E_1)} \le e^\delta\norm{x}_{E_0 \cap E_1} + \] + + As the above holds for all $\delta > 0$, $E_0 \cap E_1$ is continuously embedded in $[E_0, E_1]_\theta$. + + Let $x \in [E_0, E_1]_\theta$ and $f \in \cf(E_0, E_1)$ with $f(\theta) = x$, then by the \hyperref[Maximum Modulus Theorem]{theorem:maximum-modulus-theorem}, + \begin{align*} + \norm{x}_{E_0 + E_1} &= \norm{f(\theta)}_{E_0 + E_1} \\ + &\le \max\braks{\sup_{t \in \real}\norm{f(it)}_{E_0 + E_1}, \sup_{t \in \real}\norm{f(1 + it)}_{E_0 + E_1}} \\ + &\le \max\braks{\sup_{t \in \real}\norm{f(it)}_{E_1}, \sup_{t \in \real}\norm{f(1 + it)}_{E_1}} = \norm{f}_{\cf(E_0, E_1)} + \end{align*} + + so $[E_0, E_1]_\theta$ is continuously embedded in $E_0 + E_1$. + + + (2): Let $(F_0, F_1)$ be a compatible couple of Banach spaces over $\complex$, and $T \in L(E_0 + E_1; F_0 + F_1)$ such that $T|_{E_0} \in L(E_0; F_0)$ and $T|_{E_1} \in L(E_1; F_1)$. Let $x \in [E_0, E_1]_\theta$ and $f \in \cf(E_0, E_1)$ such that $f(\theta) = x$. For each $z \in \bracs{y \in \complex| \text{Re}(y) \in [0, 1]}$, let + \[ + g(z) = \norm{T}_{L(E_0; F_0)}^{z - 1} \norm{T}_{L(E_1; F_1)}^{-z} \cdot T \circ f(z) + \] + + then for each $t \in \real$, + \begin{align*} + \norm{g(it)}_{F_0} &= \norm{T}_{L(E_0; F_0)}^{-1} \cdot \norm{T \circ f(it)}_{F_0} \le \norm{f(it)}_{E_0} \\ + \norm{g(1 + it)}_{F_0} &= \norm{T}_{L(E_1; F_1)}^{-1} \cdot \norm{T \circ f(1 + it)}_{F_1} \le \norm{f(1 + it)}_{E_1} + \end{align*} + + so $g \in \cf(F_0, F_1)$ with $\norm{g}_{\cf(F_0, F_1)} \le \norm{f}_{\cf(E_0, E_1)}$. Thus + \begin{align*} + g(\theta) &= \norm{T}_{L(E_0; F_0)}^{\theta - 1} \norm{T}_{L(E_1; F_1)}^{-\theta} \cdot T \circ f(\theta) \\ + &= \norm{T}_{L(E_0; F_0)}^{\theta - 1} \norm{T}_{L(E_1; F_1)}^{-\theta} \cdot Tx + \end{align*} + + and $Tx \in [F_0, F_1]_\theta$ with + \begin{align*} + \norm{Tx}_{[F_0, F_1]_\theta} &\le \norm{T}_{L(E_0; F_0)}^{1 - \theta} \norm{T}_{L(E_1; F_1)}^{\theta}\norm{g}_{\cf(F_0, F_1)} \\ + &\le \norm{T}_{L(E_0; F_0)}^{1 - \theta} \norm{T}_{L(E_1; F_1)}^{\theta} \norm{f}_{\cf(E_0, E_1)} + \end{align*} + + Since the above holds for all $f \in \cf(E_0, E_1)$ with $f(\theta) = x$, + \[ + \norm{Tx}_{[F_0, F_1]_\theta} \le \norm{T}_{L(E_0; F_0)}^{1 - \theta} \norm{T}_{L(E_1; F_1)}^{\theta} \norm{x}_{[E_0, E_1]_{\theta}} + \] +\end{proof} + + diff --git a/src/fa/interpolation/functors.tex b/src/fa/interpolation/functors.tex index ad1937c..c32a7b2 100644 --- a/src/fa/interpolation/functors.tex +++ b/src/fa/interpolation/functors.tex @@ -104,7 +104,7 @@ \label{definition:interpolation-functor-exponent} Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $\catc_1$ be its categories of compatible couples, $F: \catc_1 \to \catc$ be an interpolation functor, and $\theta \in [0, 1]$, then $F$ is \textbf{of exponent $\theta$} if there exists $C \ge 0$ such that for every $(E_0, E_1), (F_0, F_1) \in \catc_1$ and $T \in \text{Mor}_{\catc_1}((E_0, E_1); (F_0, F_1))$ \[ - \norm{F(T)}_{L(F((E_0, E_1)); F((F_0, F_1)))} \le C\norm{T}_{L(E_0; E_1)}^\theta\norm{T}_{L(F_0; F_1)}^{1 - \theta} + \norm{F(T)}_{L(F((E_0, E_1)); F((F_0, F_1)))} \le C\norm{T}_{L(E_0; F_0)}^{1 - \theta}\norm{T}_{L(E_1; F_1)}^{\theta} \] If $C = 1$, then $F$ is \textbf{of exact exponent $\theta$}. diff --git a/src/fa/interpolation/index.tex b/src/fa/interpolation/index.tex index 11fc25d..0dfbeb2 100644 --- a/src/fa/interpolation/index.tex +++ b/src/fa/interpolation/index.tex @@ -2,5 +2,6 @@ \label{chap:interpolation} \input{./functors.tex} +\input{./complex.tex} diff --git a/src/fa/notation.tex b/src/fa/notation.tex index 0756355..aee88af 100644 --- a/src/fa/notation.tex +++ b/src/fa/notation.tex @@ -49,5 +49,9 @@ $\mu_G$ & Lebesgue-Stieltjes measure associated with $G$. & \autoref{definition:riemann-lebesgue-stieltjes} \\ $\int_\gamma f$, $\int_\gamma f(z)dz$ & Path integral of $f$ with respect to $\gamma$. & \autoref{definition:path-integral} \\ $PI([a, b], \gamma; E)$ & Space of path integrable functions with respect to $\gamma$. & \autoref{definition:path-integral} + % ---- Interpolation Spaces ---- \\ + $\catc_1$ & Category of compatible couples in $\catc$. & \autoref{definition:compatible-category} \\ + $\cf(E_0, E_1)$ & Calderón space of $(E_0, E_1)$ & \autoref{definition:calderon-space} \\ + $[E_0, E_1]_\theta$ & Complex interpolation space of exponent $\theta$ for the couple $(E_0, E_1)$. & \autoref{definition:complex-interpolation-method} \end{tabular}