Ziyosfera

Bessel tengsizligi

Matematikada, ayniqsa funksional tahlil(analiz)da, Bessel tengsizligi ortonormal ketma-ketlikka nisbatan Gilbert fazosida x{\displaystyle x} elementning koeffitsientlari haqidagi jumladir. Tengsizlik 1828-yilda nemis olimi Friedrich Bessel (astronomiya, matematika, fizika va geodeziya fanlari olimi) tomonidan keltirib chiqarilgan.[1]

H{\displaystyle H} Gilbert fazosi va e1,e2,...{\displaystyle e_1, e_2,...} H{\displaystyle H} dagi ortonormal ketma-ketlik bo'lsin. U holda, H{\displaystyle H} dagi har qanday vektor x{\displaystyle x} uchun

∑k=1∞|⟨x,ek⟩|2≤‖x‖2,{\displaystyle \sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2,}

munosabat o'rinli bo'ladi, bu yerda ⟨·,·⟩ belgi H{\displaystyle H} Gilbert fazosidagi skalyar ko'paytmani ifodalaydi.[2][3][4] Agar ek{\displaystyle e_k} yo'nalishdagi x{\displaystyle x} vektor proyeksiyaning "cheksiz yig'indi" sidan tuzilgan

x′=∑k=1∞⟨x,ek⟩ek,{\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 e1,e2,…{\displaystyle e_1, e_2, \dots} orqali ifodalanishi mumkin bo'lgan x′∈H{\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 x′{\displaystyle x'} ham x{\displaystyle x} bilan almashtirilishi kerak bo'ladi).

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

0≤‖x−∑k=1n⟨x,ek⟩ek‖2=‖x‖2−2∑k=1nRe⁡⟨x,⟨x,ek⟩ek⟩+∑k=1n|⟨x,ek⟩|2=‖x‖2−2∑k=1n|⟨x,ek⟩|2+∑k=1n|⟨x,ek⟩|2=‖x‖2−∑k=1n|⟨x,ek⟩|2.{\displaystyle \begin{align} 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{align}}

Yana qarang

  • Koshi-Shvarz tengsizligi
  • Parseval teoremasi

Manbalar

  1. "Bessel inequality (Wayback Machine saytida 2022-06-05 sanasida arxivlangan) - Encyclopedia of Mathematics (Wayback Machine saytida 2022-06-05 sanasida arxivlangan)".
  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.

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“. 2013-yil 4-mayda asl nusxadan arxivlangan. Qaraldi: 2022-yil 5-iyun.
  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.  
"https://new.ziyosfera.uz/ziyo/Bessel_tengsizligi" dan olindi