[數學分析] 三角多項式 與 三角級數 (2)- Generalized Fourier Series

現在我們定義廣義 Fourier Series ：

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Definition: (Orthogonal System of Functions)
令 $\{\phi_n\}$, $n \in \mathbb{N}$ 為在 $[a,b]$ 上 的 Complex-valued 函數 sequence 且滿足下列積分
\[
\int_a^b \phi_n(x) \phi_m^*(x) dx =0, \;\; \text{ if $n \neq m$}
\]那麼我們稱 $\{\phi_n\}$ 為在 $[a,b]$ 上 orthogonal 或稱 (orthogonal system of functions on $[a,b]$) 。除此之外，若積分
\[
\int_a^b \phi_n(x) \phi_n^*(x) dx =1
\]我們稱 $\{\phi_n\}$ 為在 $[a,b]$ 上 orthonormal 或稱 (orthonormal system of functions on $[a,b]$) 。
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Comments: 
一般而言，若我們取 $\{\phi_n\}$  $n \in \mathbb{N}$ 為在 $[0, 2\pi]$ 上 的 Complex-valued 函數 sequence 且滿足 $\phi_n(x):= exp(inx)$  則讀者可自行驗證此 函數 sequence 為 orthogonal


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Definition: (n-th Fourier Coefficient of $f$)
若 $\{\phi_n\}$ 為 orthonormal on $[a,b]$ 且 對任意 $n \in \mathbb{N}$，
\[
c_n:=\int_a^b f(x) \phi_n^*(x) dx
\]我們稱 $c_n$ 為 $n$-th Fourier coefficient of $f$ (relative to $\{\phi_n\}$)

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上述 $^*$ 為 complex conjugate。

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Definition: Generalized Fourier Series
Generalized Fourier Series of $f$ (relative to $\{\phi_n\}$) 定義為
\[
f(x) \sim \sum_{n=1}^\infty c_n \phi_n(x)
\]其中 $c_n=\int_a^b f(x) \phi_n^*(x) dx $。
====================

Theorem: Bessel's inequality
若 $\{\phi_n\}$ 為 orthnormal on $[a,b]$ 且若 $f(x) \sim \sum_{n=1}^\infty c_n \phi_n(x)$ 則 \[
\sum_{n=1}^\infty |c_n|^2 \le \int_a^b |f(x)|^2 dx
\]


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Theorem: Best Approximation of Fourier Series 
令 $\{\phi_n\}$ 為在 $[a,b]$ 上 的 Complex-valued 函數 sequence ，且 $\phi_n$ 為 orthogonal。現在定義 $n$-th partial sum of Fourier Series of $f$ 如下
\[
s_n (x):= \sum_{m=1}^n c_m \phi_m(x)
\]且令 $t_n$ 為任意 series 如下
\[
t_n(x) := \sum_{m=1}^n \gamma_m \phi_m(x)
\]其中 $\gamma_n \in \mathbb{C}$ 則
\[
\int_a^b |f (x)- s_n(x)|^2 dx \le \int_a^b |f(x) - t_n(x)|^2 dx
\] 且 上式等式成立 若且為若 $\gamma_n = c_n$ 對任意 $n$。
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Proof:
令
\[
s_n (x):= \sum_{m=1}^n c_m \phi_m(x);\;\;\; t_n(x) := \sum_{m=1}^n \gamma_m \phi_m(x)
\]
我們首先證明下列不等式成立
\[
\int_a^b |f (x)- s_n(x)|^2 dx \le \int_a^b |f(x) - t_n(x)|^2 dx
\] 首先觀察
\[\begin{array}{l}
\int_a^b | f(x) - {t_n}(x){|^2}dx = \int_a^b {\left( {f(x) - {t_n}(x)} \right){{\left( {f(x) - {t_n}(x)} \right)}^*}} dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {\left( {f(x){f^*}(x) - f(x){t_n}^*(x) - {t_n}(x){f^*}(x) + {t_n}(x){t_n}^*(x)} \right)} dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \underbrace {\int_a^b {{{\left| {f(x)} \right|}^2}} dx}_{term1} - \underbrace {\int_a^b {f(x){t_n}^*(x)} dx}_{term2} - \underbrace {\int_a^b {{t_n}(x){f^*}(x)} dx}_{term3} + \underbrace {\int_a^b {{{\left| {{t_n}(x)} \right|}^2}} dx}_{term4} \ \ \ \ \ \ (*)
\end{array}\]接著對上式逐項觀察，首先檢驗 term 2：
\[\begin{array}{l}
\int_a^b {f(x){t_n}^*(x)} dx = {\int_a^b {f(x)\left[ {\sum\limits_{m = 1}^n {{\gamma _m}} {\phi _m}(x)} \right]} ^*}dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {f(x)\sum\limits_{m = 1}^n {{\gamma _m}^*} {\phi _m}^*(x)} dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \sum\limits_{m = 1}^n {{\gamma _m}^*} \underbrace {\int_a^b {f(x){\phi _m}^*(x)} dx}_{ = {c_m}}\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \sum\limits_{m = 1}^n {{\gamma _m}^*} {c_m}
\end{array}\]
接著我們檢驗 term 3:
\[\begin{array}{l}
\int_a^b {{t_n}(x){f^*}(x)} dx = \int_a^b {\sum\limits_{m = 1}^n {{\gamma _m}} {\phi _m}(x){f^*}(x)} dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \sum\limits_{m = 1}^n {{\gamma _m}} \underbrace {\int_a^b {{f^*}(x){\phi _m}(x)} dx}_{ = c_m^*}\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \sum\limits_{m = 1}^n {{\gamma _m}} c_m^*
\end{array}\]
最後檢驗 term 4：
\[\begin{array}{l}
\int_a^b {{{\left| {{t_n}(x)} \right|}^2}} dx = \int_a^b {{t_n}(x)t_n^*(x)} dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = {\int_a^b {\sum\limits_{m = 1}^n {{\gamma _m}} {\phi _m}(x)\left[ {\sum\limits_{k = 1}^n {{\gamma _k}} {\phi _k}(x)} \right]} ^*}dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {\sum\limits_{m = 1}^n {{\gamma _m}} {\phi _m}(x)\sum\limits_{k = 1}^n {{\gamma _k}^*} {\phi _k}^*(x)} dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \sum\limits_{m = 1}^n {{\gamma _m}} \sum\limits_{k = 1}^n {{\gamma _k}^*} \underbrace {\int_a^b {{\phi _m}(x){\phi _k}^*(x)} dx}_{ = 1\begin{array}{*{20}{c}}
{}
\end{array}if\begin{array}{*{20}{c}}
{}
\end{array}m = k}\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \sum\limits_{m= 1}^n {{\gamma _m}{\gamma _m}^*}  = \sum\limits_{m = 1}^n {{{\left| {{\gamma _m}} \right|}^2}}
\end{array}\]
故現在將上述結果 帶回 $(*)$ 可得
\[\begin{array}{l}
\int_a^b | f(x) - {t_n}(x){|^2}dx = \underbrace {\int_a^b {{{\left| {f(x)} \right|}^2}} dx}_{term1} - \underbrace {\int_a^b {f(x){t_n}^*(x)} dx}_{term2} - \underbrace {\int_a^b {{t_n}(x){f^*}(x)} dx}_{term3} + \underbrace {\int_a^b {{{\left| {{t_n}(x)} \right|}^2}} dx}_{term4}\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {{{\left| {f(x)} \right|}^2}} dx - \sum\limits_{m = 1}^n {\gamma _m^*c_m^{}}  - \sum\limits_{m = 1}^n {{\gamma _m}c_m^*}  + \sum\limits_{m = 1}^n {{{\left| {{\gamma _m}} \right|}^2}}  \ \ \ \ (\star)
\end{array}\]
注意到下列 FACT：
\[\begin{array}{l}
\sum\limits_{m = 1}^n {{{\left| {{c_n} - {\gamma _m}} \right|}^2}}  = \sum\limits_{m = 1}^n {\left( {{c_n} - {\gamma _m}} \right){{\left( {{c_n} - {\gamma _m}} \right)}^*}}  = \sum\limits_{m = 1}^n {\left( {{c_n}{c_n}^* - {c_n}{\gamma _m}^* - {\gamma _m}{c_n}^* + {\gamma _m}{\gamma _m}^*} \right)} \\
 \Rightarrow \sum\limits_{m = 1}^n {{{\left| {{c_n} - {\gamma _m}} \right|}^2}}  = \sum\limits_{m = 1}^n {{c_n}{c_n}^*}  - \sum\limits_{m = 1}^n {{c_n}{\gamma _m}^*}  - \sum\limits_{m = 1}^n {{\gamma _m}{c_n}^*}  + \sum\limits_{m = 1}^n {{\gamma _m}{\gamma _m}^*} \\
 \Rightarrow \sum\limits_{m = 1}^n {{{\left| {{c_n} - {\gamma _m}} \right|}^2}}  - \sum\limits_{m = 1}^n {{{\left| {{c_n}} \right|}^2}}  =  - \sum\limits_{m = 1}^n {{c_n}{\gamma _m}^*}  - \sum\limits_{m = 1}^n {{\gamma _m}{c_n}^*}  + \sum\limits_{m = 1}^n {{{\left| {{\gamma _m}} \right|}^2}}
\end{array}\]與
\[\begin{array}{l}
\int_a^b | f(x) - {s_n}(x){|^2}dx = \int_a^b {\left( {f(x) - {s_n}(x)} \right){{\left( {f(x) - {s_n}(x)} \right)}^*}} dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {\left( {f(x){f^*}(x) - f(x){s_n}^*(x) - {s_n}(x){f^*}(x) + {s_n}(x){s_n}^*(x)} \right)} dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {f(x){f^*}(x)} dx - \int_a^b {f(x){s_n}^*(x)} dx - \int_a^b {{s_n}(x){f^*}(x)} dx + \int_a^b {{s_n}(x){s_n}^*(x)} dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {{{\left| {f(x)} \right|}^2}} dx - \int_a^b {\left( {f(x)\sum\limits_{m = 1}^n {{c_m}^*{\phi _m}^*\left( x \right)} } \right)} dx - \int_a^b {\left( {\sum\limits_{m = 1}^n {{c_m}{\phi _m}\left( x \right)} {f^*}(x)} \right)} dx + \int_a^b {\left( {\sum\limits_{m = 1}^n {{c_m}{\phi _m}\left( x \right)} \sum\limits_{k = 1}^n {{c_k}^*{\phi _k}^*\left( x \right)} } \right)} dx\\
{\rm{since}}\begin{array}{*{20}{c}}
{}
\end{array}{s_n}(x): = \sum\limits_{m = 1}^n {{c_m}{\phi _m}\left( x \right)}  \Rightarrow s_n^*(x): = {\left[ {\sum\limits_{m = 1}^n {{c_m}{\phi _m}\left( x \right)} } \right]^*} = \sum\limits_{m = 1}^n {{c_m}^*{\phi _m}^*\left( x \right)} \\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {{{\left| {f(x)} \right|}^2}} dx - \sum\limits_{m = 1}^n {{c_m}^*} \int_a^b {\left( {f(x){\phi _m}^*\left( x \right)} \right)} dx - \sum\limits_{m = 1}^n {{c_m}} \int_a^b {{f^*}(x){\phi _m}\left( x \right)} dx + \sum\limits_{m = 1}^n {{c_m}} \sum\limits_{k = 1}^n {{c_k}^*} \int_a^b {{\phi _m}\left( x \right){\phi _k}^*\left( x \right)} dx\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {{{\left| {f(x)} \right|}^2}} dx - \sum\limits_{m = 1}^n {{c_m}^*} {c_m} - \sum\limits_{m = 1}^n {{c_m}} c_m^* + \sum\limits_{m = 1}^n {{c_m}{c_m}^*} \\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {{{\left| {f(x)} \right|}^2}} dx - \sum\limits_{m = 1}^n {{c_m}^*} {c_m}\\
 \Rightarrow \int_a^b | f(x) - {s_n}(x){|^2}dx = \int_a^b {{{\left| {f(x)} \right|}^2}} dx - \sum\limits_{m = 1}^n {{{\left| {{c_m}} \right|}^2}}
\end{array}\]

故 $\star$ 可進一步改寫
\[\begin{array}{l}
\int_a^b | f(x) - {t_n}(x){|^2}dx = \int_a^b {{{\left| {f(x)} \right|}^2}} dx - \sum\limits_{m = 1}^n {\gamma _m^*c_m^{}}  - \sum\limits_{m = 1}^n {{\gamma _m}c_m^*}  + \sum\limits_{m = 1}^n {{{\left| {{\gamma _m}} \right|}^2}} \\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b {{{\left| {f(x)} \right|}^2}} dx + \sum\limits_{m = 1}^n {{{\left| {{c_n} - {\gamma _m}} \right|}^2}}  - \sum\limits_{m = 1}^n {{{\left| {{c_n}} \right|}^2}} \\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \underbrace {\int_a^b {{{\left| {f(x)} \right|}^2}} dx - \sum\limits_{m = 1}^n {{{\left| {{c_n}} \right|}^2}} }_{ = \int_a^b | f(x) - {s_n}(x){|^2}dx} + \sum\limits_{m = 1}^n {{{\left| {{c_n} - {\gamma _m}} \right|}^2}} \\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_a^b | f(x) - {s_n}(x){|^2}dx + \underbrace {\sum\limits_{m = 1}^n {{{\left| {{c_n} - {\gamma _m}} \right|}^2}} }_{ \ge 0}\\
 \Rightarrow \int_a^b | f(x) - {t_n}(x){|^2}dx \ge \int_a^b | f(x) - {s_n}(x){|^2}dx\\

\end{array}\]
且若 $c_m = \gamma_m$ 則等式成立。 $\square$