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

$\sum _{k=1}^{\infty }\left\vert \left\langle x,e_{k}\right\rangle \right\vert ^{2}\leq \left\Vert x\right\Vert ^{2},$

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

$x'=\sum _{k=1}^{\infty }\left\langle x,e_{k}\right\rangle e_{k},$

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 $x'\in H$ 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 $x'$ ham $x$ bilan almashtirilishi kerak bo'ladi).

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

${\begin{aligned}0\leq \left\|x-\sum _{k=1}^{n}\langle x,e_{k}\rangle e_{k}\right\|^{2}&=\|x\|^{2}-2\sum _{k=1}^{n}\operatorname {Re} \langle x,\langle x,e_{k}\rangle e_{k}\rangle +\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}\\&=\|x\|^{2}-2\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}+\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}\\&=\|x\|^{2}-\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}.\end{aligned}}$

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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