Peano yadrosi teoremasi - Peano kernel theorem
Raqamli tahlilda foydalaniladigan matematik teorema
Yilda raqamli tahlil, Peano yadrosi teoremasi raqamli yaqinlashuvning keng klassi uchun xato chegaralarida umumiy natijadir (masalan sonli kvadratchalar ) bilan belgilanadi chiziqli funktsiyalar. Bunga bog'liq Juzeppe Peano.[1]
Bayonot
Ruxsat bering
barchaning makoni bo'ling farqlanadigan funktsiyalar
uchun belgilangan
ular chegaralangan o'zgarish kuni
va ruxsat bering
bo'lishi a chiziqli funktsional kuni
. Buni taxmin qiling
bu
marta doimiy ravishda farqlanadigan va bu
yo'q qiladi darajadagi barcha polinomlar
, ya'ni
![{ displaystyle Lp = 0, qquad forall p in mathbb {P} _ { nu} [x].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1088ed1492b09db29a571202ae5a889c0aa1f66)
Bundan tashqari, har qanday kishi uchun
ikki tomonlama funktsiya ![g (x, theta)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad353faedd3077be2daf2387d1dccef6a2b80972)
bilan
![{ displaystyle g (x, cdot), , g ( cdot, theta) in C ^ { nu +1} [a, b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60f59d00ed362cd5d387eef0d55afbbd5f563d02)
, quyidagilar amal qiladi:
![{ displaystyle L int _ {a} ^ {b} g (x, theta) , d theta = int _ {a} ^ {b} Lg (x, theta) , d theta, }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f952cea37dd090b83563098d7e17305408e8f85f)
va ni aniqlang
Peano yadrosi ning
![L](https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8)
kabi
![{ displaystyle k ( theta) = L [(x- theta) _ {+} ^ { nu}], qquad theta in [a, b],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/608a1f43f7369fd3ad50ab9a72466a517e1c040e)
yozuvlarni kiritish
![{ displaystyle (x- theta) _ {+} ^ { nu} = { begin {case} (x- theta) ^ { nu}, & x geq theta, 0, & x leq theta. end {case}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92c4eda4e8c602397deb636aac3811569028f80c)
The
Peano yadrosi teoremasi keyin buni ta'kidlaydi
![{ displaystyle Lf = { frac {1} { nu!}} int _ {a} ^ {b} k ( theta) f ^ {( nu +1)} ( theta) , d teta,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10d8d0d8365b2e95ec4b50f5d32eb89b3c7e8a71)
taqdim etilgan
![{ displaystyle k in { mathcal {V}} [a, b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a64be74ceee3b987b20a49de1fa54912caf17e97)
.
[1][2]Chegaralar
Qiymatining bir necha chegaralari
ushbu natijadan foydalaning:
![{ displaystyle { begin {aligned} | Lf | & leq { frac {1} { nu!}} | k | _ {1} | f ^ {( nu +1)}} | _ { infty} [5pt] | Lf | & leq { frac {1} { nu!}} | k | _ { infty} | f ^ {( nu +1)} | _ {1} [5pt] | Lf | & leq { frac {1} { nu!}} | K | _ {2} | f ^ {( nu +1)} | _ {2} end {hizalangan}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2764ece246dfc561f08d3443aa1a664a80db8e57)
qayerda
,
va
ular taksik, Evklid va maksimal normalar navbati bilan.[2]
Ilova
Amalda Peano yadrosi teoremasining asosiy qo'llanilishi hamma uchun aniq bo'lgan taxminiy xatolarni bog'lashdir.
. Yuqoridagi teorema Teylor polinomi uchun
ajralmas qoldiq bilan:
![{ displaystyle { begin {aligned} f (x) = f (a) + {} & (xa) f '(a) + { frac {(xa) ^ {2}} {2}} f' ' (a) + cdots [6pt] & cdots + { frac {(xa) ^ { nu}} { nu!}} f ^ { nu} (a) + { frac {1} { nu!}} int _ {a} ^ {x} (xa) ^ { nu} f ^ {( nu +1)} ( theta) , d theta, end {hizalangan}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd5171f23168a23b4ca528d885fcad6e538b4a4)
belgilaydigan
dan foydalanib, taxminiy xato sifatida chiziqlilik ning
aniqligi bilan birga
tugmachasini o'ng tomonida yo'q qilish va
o'chirish uchun yozuv
-integral integral chegaralaridan bog'liqlik.[3]
Shuningdek qarang
Adabiyotlar