基本介紹
- 中文名:黎曼-勒貝格定理
- 外文名:Riemann-Lebesgue's Theory
- 所屬學科:數學
- 內容:函式的傅立葉展開均收斂於其自身
- 套用:信號處理、傅立葉分析
基本介紹,定理的證明,相關結論,
基本介紹
定理 設
,則當
時
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定理的證明
證明 這裡不妨假設f(x)具有周期b-a: f(x+b-a)=f(x)。對於正數ε,有全連續函式φ(x)適合
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顯然地,我們也不妨假設b-a=2π.假設
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相關結論
系1 設
是
的係數,則
。
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當f(x)∈L(a,b)時,稱
為f在區間[a,b]上的平均連續模數,假如f(x)∈C(a,b),那么稱
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系2 假如f(x+2π)≡f(x),
,那么
的傅立葉係數
都是O(n);當f∈Lip 1時nan和nbn都是o(1)。
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事實上,
,當f∈Lip 1時,
是一全連續函式,從而
也是傅立葉級數,
=o(1)。
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假如
是一個有界變差的函式,
,那么
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