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

{\displaystyle H} Gilbert fazosi va {\displaystyle e\_{1},e\_{2},...} {\displaystyle H} dagi ortonormal ketma-ketlik bo'lsin. U holda, {\displaystyle H} dagi har qanday vektor {\displaystyle x} uchun 

{\displaystyle \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 {\displaystyle H} Gilbert fazosidagi skalyar ko'paytmani ifodalaydi. Agar {\displaystyle e\_{k}} yo'nalishdagi {\displaystyle x} vektor proyeksiyaning "cheksiz yig'indi" sidan tuzilgan

{\displaystyle 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 {\displaystyle e\_{1},e\_{2},\dots } orqali ifodalanishi mumkin bo'lgan {\displaystyle 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 {\displaystyle x'} ham {\displaystyle x} bilan almashtirilishi kerak bo'ladi).

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

{\displaystyle {\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
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* Koshi-Shvarz tengsizligi
* Parseval teoremasi

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

Ushbu maqola Creative Commons Attribution/Share-Alike litsenziyasi ostida litsenziyalangan PlanetMath-dagi Bessel tengsizligidan olingan materiallarni o'z ichiga oladi.

1. ↑  „Bessel inequality - Encyclopedia of Mathematics".
2. ↑  Saxe, Karen*. Beginning Functional Analysis* (en). Springer Science & Business Media, 2001-12-07 — 82 bet. ISBN 9780387952246.
3. ↑  Zorich, Vladimir A.*. Mathematical Analysis II* (en). Springer Science & Business Media, 2004-01-22 — 508–509 bet. ISBN 9783540406334.
4. ↑  Vetterli, Martin*. Foundations of Signal Processing* (en). Cambridge University Press, 2014-09-04 — 83 bet. ISBN 9781139916578.