Keling, biz o'rgangan funktsiyani interpolyatsiya qilish usuliga oid ikkita muhim savolni ko'rib chiqaylik

Interpolatsiya xatosining yangi bahosi.
Keling, yuqorida keltirilgan savollarning birinchisiga javobni ko'rib chiqaylik. \[{{P}_{n-1}}\in {{\mathbb{P}}_{n-1}}\] algebraik ko'phad bo'lsin – va u \[f\left( x \right)\] funktsiyaning qandaydir usul bilan olingan yaqinlashuvi bo‘lsin. Shunday qilib, har bir usul o'zining \[{{P}_{n-1}}\] ko‘phadga ega. \[\left| f\left( x \right)-{{P}_{n-1}}\left( x \right) \right|\] qiymati $x\in \left[ a,b \right]$ nuqtadagi yaqinlashish xatosini va sonni aniqlaydi.
\[{{\left\| f-{{P}_{n-1}} \right\|}_{\infty }}=\underset{x\in \left[ a,b \right]}{\mathop{\max }}\,\left| f\left( x \right)-{{P}_{n-1}}\left( x \right) \right|\]
bu usulning maksimal xatosidir.
Ta'rif 2. \[{{f}_{n-1}}\in {{\mathbb{P}}_{n-1}}\] algebraik ko'phadni eng yaxshi yagona yaqinlikdagi ko'phad deb ataymiz, agar
\[{{\left\| f-{{f}_{n-1}} \right\|}_{\infty }}=\underset{{{P}_{n-1}}\in {{\mathbb{P}}_{n-1}}}{\mathop{\min }}\,{{\left\| f-{{P}_{n-1}} \right\|}_{\infty }}\] (2)
Keyinchalik (2) ning yechimi mavjudligini va o'ziga xos tarzda aniqlanganligini ko'ramiz.
\[{{E}_{n}}\left( f \right)={{\left\| f-{{f}_{n-1}} \right\|}_{\infty }}\]
Kattalik eng yaxshi yagona yaqinlashish xatoligi deyiladi.
Keling, quyidagi oddiy izohlarni qilaylik.
1. Agar \[{{F}_{n-1}}\in {{\mathbb{P}}_{n-1}}\] \[f\left( x \right)\] ning qandaydir usul bilan olingan taqribiy ko‘rsatkichi bo‘lsa (masalan, \[{{F}_{n-1}}\] interpolyatsion ko‘phad), u holda \[{{\left\| f-{{F}_{n-1}} \right\|}_{\infty }}\ge {{E}_{n}}\left( f \right)\] bo‘ladi.
2. $\left[ a,b \right]$ dagi har qanday uzluksiz \[f\left( x \right)\] funksiya uchun \[n\to \infty \] da \[{{E}_{n}}\left( f \right)\to \infty \]. Bu to‘g‘ridan-to‘g‘ri Veyershtras teoremasidan kelib chiqadi.
Teorema 1. Haqiqiy baholar
\[{{E}_{n}}\left( f \right)\,\,\le {{\left\| f-{{L}_{n-1}} \right\|}_{\infty }}\le \left( 1+{{\Lambda }_{n}} \right){{E}_{n}}\left( f \right).\] (3)
Isbot. \[{{f}_{n-1}}\in {{\mathbb{P}}_{n-1}}\] \[f\left( x \right)\] ga eng yaxshi yagona yaqinlikdagi ko‘phad bo‘lsin. Interpolyatsiya ko'phadning yagonaligi sababli \[{{f}_{n-1}}\left( x \right)=\sum\limits_{i=1}^{n}{{{f}_{n-1}}\left( {{x}_{i}} \right){{\ell }_{i}}\left( x \right)}\]. Shuning uchun
\[\left| f\left( x \right)-{{L}_{n-1}}\left( x \right) \right|=\left| \left( f\left( x \right)-{{f}_{n-1}}\left( x \right) \right)+\left( {{f}_{n-1}}\left( x \right)-{{L}_{n-1}}\left( x \right) \right) \right|\le \]
\[\le \left| \left( f\left( x \right)-{{f}_{n-1}}\left( x \right) \right) \right|+\sum\limits_{i=1}^{n}{\left| {{f}_{n-1}}\left( x \right)-{{f}_{n-1}}\left( {{x}_{i}} \right) \right|}\left| {{\ell }_{i}}\left( x \right) \right|\le \]
\[\le \left( 1+{{\Lambda }_{n}} \right){{E}_{n}}\left( f \right){{\left\| f-{{f}_{n-1}} \right\|}_{\infty }}\le \left( 1+{{\Lambda }_{n}} \right){{E}_{n}}\left( f \right)\]
Quyidan baho \[{{E}_{n}}\left( f \right)\] ta'rifi bo'yicha amal qiladi.
Yuqori bahodan (3) ko'rinib turibdiki, interpolyatsion ko'phad \[{{L}_{n-1}}\left( x \right)\] eng yaxshi yagona yaqinlikdan ko'pi bilan \[1+{{\Lambda }_{n}}\] marta kam aniq.