Bessel tengsizligi

Matematikada, ayniqsa funksional tahlil(analiz)da, Bessel tengsizligi ortonormal ketma-ketlikka nisbatan Gilbert fazosida $x$ elementning koeffitsientlari haqidagi jumladir. Tengsizlik 1828-yilda nemis olimi Friedrich Bessel (astronomiya, matematika, fizika va geodeziya fanlari olimi) tomonidan keltirib chiqarilgan.

$H$ Gilbert fazosi va $e_1,e_2,...$ $H$ dagi ortonormal ketma-ketlik bo'lsin. U holda, $H$ dagi har qanday vektor $x$ uchun

$\sum _{k=1}^{\mathrm{\infty }}{\left|⟨x,e_k⟩\right|}^2\le {\Vert x\Vert }^2,$

munosabat o'rinli bo'ladi, bu yerda ⟨·,·⟩ belgi $H$ Gilbert fazosidagi skalyar ko'paytmani ifodalaydi. Agar $e_k$ yo'nalishdagi $x$ vektor proyeksiyaning "cheksiz yig'indi" sidan tuzilgan

$x^{\prime }=\sum _{k=1}^{\mathrm{\infty }}⟨x,e_k⟩e_k,$

cheksiz summani aniqlasak, u holda Bessel tengsizligi bu qatorning yaqinlashuvchi ekanligini ta'kidlaydi. Bu haqda quyidagicha ham o'ylash mumkin: potensial bazis $e_1,e_2,\dots$ orqali ifodalanishi mumkin bo'lgan $x^{\prime }\in H$ mavjud bo'ladi.

To'liq ortonormal ketma-ketlik uchun (ya'ni, bazis bo'lgan ortonormal ketma-ketlik uchun) biz Parseval ayniyatiga egamiz. Bunda tengsizlikni tenglik bilan almashtiriladi (va natijada $x^{\prime }$ ham $x$ bilan almashtirilishi kerak bo'ladi).

Bessel tengsizligi quyidagi, har qanday natural son n uchun bajariladigan ayniyatdan kelib chiqadi

$\begin{array}{cc}0\le {\Vert x-\sum _{k=1}^n⟨x,e_k⟩e_k\Vert }^2& =\Vert x{\Vert }^2-2\sum _{k=1}^n\mathrm{Re}⟨x,⟨x,e_k⟩e_k⟩+\sum _{k=1}^n|⟨x,e_k⟩{|}^2\\ & =\Vert x{\Vert }^2-2\sum _{k=1}^n|⟨x,e_k⟩{|}^2+\sum _{k=1}^n|⟨x,e_k⟩{|}^2\\ & =\Vert x{\Vert }^2-\sum _{k=1}^n|⟨x,e_k⟩{|}^2.\end{array}$

Yana qarang

• Koshi-Shvarz tengsizligi
• Parseval teoremasi

Manbalar

1. "Bessel inequality - Encyclopedia of Mathematics".
2. Saxe, Karen (2001-12-07). Beginning Functional Analysis. Springer Science & Business Media. p. 82. ISBN 9780387952246.
3. Zorich, Vladimir A.; Cooke, R. (2004-01-22). Mathematical Analysis II. Springer Science & Business Media. pp. 508-509. ISBN 9783540406334.
4. Vetterli, Martin; Kovačević, Jelena; Goyal, Vivek K. (2014-09-04). Foundations of Signal Processing. Cambridge University Press. p. 83. ISBN 9781139916578.

Havolalar

 Bessel's Inequality the article on Bessel's Inequality on MathWorld.

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