(Luận văn thạc sĩ) một số kết quả về tính bị chặn của tích phân dao động

ĐẠI HỌC QUỐC GIA HÀ NỘI
TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN
---------------------

Nguyễn Thị Xâm

MỘT SỐ KẾT QUẢ VỀ TÍNH BỊ CHẶN CỦA
TÍCH PHÂN DAO ĐỘNG

LUẬN VĂN THẠC SĨ KHOA HỌC

Hà Nội - Năm 2019


ĐẠI HỌC QUỐC GIA HÀ NỘI
TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN
---------------------

Nguyễn Thị Xâm

MỘT SỐ KẾT QUẢ VỀ TÍNH BỊ CHẶN CỦA
TÍCH PHÂN DAO ĐỘNG
Chun ngành: Tốn Giải Tích
Mã số: 8460101.02
LUẬN VĂN THẠC SĨ KHOA HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC:TS. VŨ NHẬT HUY

Hà Nội - Năm 2019

▼ư❝ ❧ư❝
▼ð ✤➛✉
✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶
✶✳✷
✶✳✸
✶✳✹

✸
✹

P❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
❚➼❝❤ ❝❤➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✶✳✹✳✶ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠
♥❤❛♥❤ S (Rn) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✶✳✹✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ L1(Rn) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✷ ✣→♥❤ ❣✐→ t➼❝❤ ở tr



ìợ ữủ ừ t tỷ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣

✷✻

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✵
✹✵

✷✳✶ ✣→♥❤ ❣✐→ ữợ ừ t ở ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✷ ✣→♥❤ ❣✐→ ❝➟♥ tr➯♥ ❝õ❛ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✸✳✶ ❇ê ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✸✳✷ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✈ỵ✐ ❤➔♠ ♣❤❛ ❧❛✐ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✶


ớ ỡ
rữợ tr ở ừ ❧✉➟♥ ✈➠♥✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥
t❤➔♥❤ ✈➔ s➙✉ s➢❝ ♥❤➜t ❝õ❛ ♠➻♥❤ tỵ✐

❚❙✳ ❱ơ ◆❤➟t ❍✉②✱ ✈➻ sü ❣✐ó♣ ✤ï✱ ❝❤➾ ❜↔♦ t➟♥

t➻♥❤✱ ❝ị♥❣ ♥❤ú♥❣ ❧í✐ ✤ë♥❣ ổ ũ ỵ ừ tr sốt q tr➻♥❤ tỉ✐
❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣✳
❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝→♠ ì♥ sü ❣✐ó♣ ✤ï ❝õ❛ ❝→❝ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛
❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐
✈➔ ❑❤♦❛ ❙❛✉ ✤↕✐ ❤å❝✱ ✤➣ ♥❤✐➺t t➻♥❤ tr✉②➲♥ t❤ư ❦✐➳♥ t❤ù❝ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï tỉ✐
❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❈❛♦ ❤å❝✳
❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉æ♥ ✤ë♥❣ ✈✐➯♥✱ ❦❤✉②➳♥ ❦❤➼❝❤✱ ❣✐ó♣
✤ï tỉ✐ r➜t ♥❤✐➲✉ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤å❝ t➟♣✳
▼➦❝ ❞ò ✤➣ ❝è ❣➢♥❣ r➜t ♥❤✐➲✉ tú tr q tr ự ữ
ợ ❧➔♠ q✉❡♥ ✈ỵ✐ ❝ỉ♥❣ t→❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈➔ ❝á♥ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ t❤ü❝
❤✐➺♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ ❦➼♥❤
ữủ ỵ õ õ ừ t ổ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳
❍➔ ◆ë✐✱ ♥➠♠ ✷✵✶✾

◆❣✉②➵♥ ❚❤à ❳➙♠

✷


▼ð ✤➛✉
❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✤➣ t❤✉ ❤ót ♥❤✐➲✉ sü q✉❛♥ t➙♠ ❝õ❛ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ ✈➔ ❝→❝
♥❤➔ ❱➟t ỵ tứ t ổ tr r tq ❧❛ ❈❤❛❧❡✉r ❝õ❛ ❏♦s❡♣❤
❋♦✉r✐❡r ✈➔♦ ♥➠♠ ✶✽✷✷✳ ◆❤✐➲✉ ❜➔✐ t♦→♥ ỵ tt ữỡ tr r
ồ số ỵ tt st ỵ tt số t ✈➲ q✉❛♥❣ ❤å❝✱ ➙♠ ❤å❝✱ ❝ì
❤å❝ ❧÷đ♥❣ tû✱✳✳✳ ✤➲✉ ❝â t❤➸ ✤÷❛ ✈➲ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣✳
❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✤➣ ✈➔ ✤❛♥❣ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ♥❤✐➲✉ ù♥❣ ❞ư♥❣ ❦❤→❝ ♥❤❛✉
✈➔ t❤✉ ❤ót ✤÷đ❝ ♥❤✐➲✉ sü q✉❛♥ t➙♠ tø ❝→❝ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ❬✸✲✻❪✳ ◆❤✐➲✉
ự rt ộ ỹ ữợ t trỹ t✐➳♣ ❣✐→ trà t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✈➔ tè❝ ✤ë s✉② ❣✐↔♠
❝õ❛ ❝❤✉➞♥ ❝õ❛ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❋♦✉r✐❡r ✭①❡♠ ❬✸✱ ✺✱ ✻❪ ✮✳
◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❜❛
❝❤÷ì♥❣✿

❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❈❤÷ì♥❣ ♥➔② ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱

t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à✱ t➼❝❤ ❝❤➟♣ ✈➔ ♠ët sè ✤à♥❤ ❧➼ q✉❛♥ trå♥❣ ❝õ❛
♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn ) ✈➔ L1 (Rn )✳

❈❤÷ì♥❣ ✷✿ ✣→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❙t❡✐♥✲❲❛✐♥❣❡r✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤

❜➔② tr ữợ ừ t ở ý
eiP (x)

I() =

R

dx
,
x

ữợ ữủ tr ữợ tổ q ❝õ❛ ✤❛ t❤ù❝ P (x)✳ ◆ë✐ ❞✉♥❣
❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✹❪✳

❈❤÷ì♥❣ ✸✿ ✣→♥❤ ❣✐→ ❝❤✉➞♥ ❝õ❛ t♦→♥ tû ❞❛♦ ✤ë♥❣✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣

t❛ s➩ t➻♠ ❤✐➸✉ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❋♦✉r✐❡r ❞↕♥❣✿
eiλS(x,y) ψ(x, y)φ(y)dy,

(Tλ φ)(x) =
R

tr♦♥❣ ✤â S(x, y) ❧➔ ♠ët ❤➔♠ ♣❤❛ ♥❤➟♥ ❣✐→ trà t❤ü❝✱ ψ(x, y) ❧➔ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ❝â
❣✐→ ❝♦♠♣❛❝t ✈➔ λ ❧➔ ♠ët t❤❛♠ sè✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✸❪✳
✸


❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤
✤ì♥ ✈à✱ t➼❝❤ ❝❤➟♣ ✈➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦
❝❤➼♥❤ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✳

✶✳✶ P❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ Ω ❧➔ ♠ët t➟♣ ❤đ♣ tr♦♥❣ Rn✳ ▼ët ❤å ✤➳♠ ✤÷đ❝ ❝→❝ ❝➦♣ {(Ωj , ϕj )}∞j=1✱

tr♦♥❣ ✤â Ωj ❧➔ t➟♣ ♠ð tr♦♥❣ Rn✱ ϕj ❧➔ ❤➔♠ t❤✉ë❝ ❧ỵ♣ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ✈ỉ ❤↕♥ tr➯♥ Rn✱
✤÷đ❝ ❣å✐ ❧➔ ♠ët ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à ❝õ❛ t➟♣ Ω ♥➳✉ ❝→❝ t➼❝❤ ❝❤➜t s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿
∞
{Ωj }∞
j=1 ❧➔ ♠ët ♣❤õ ♠ð ❝õ❛ Ω, Ω ⊂ Uj=1 Ωj ✱ 0 ≤ ϕj (x) ≤ 1, x ∈ Ω, j = 1, 2, ...,
∞
ϕj ∈ C0∞ (Rn ), supp ϕj ⊂ Ωj , j = 1, 2, ...,
j=1 ϕj (x) = 1, x ∈ Ω✳
❚❛ ❝á♥ ❣å✐ {ϕj }∞j=1 ❧➔ ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à ù♥❣ ✈ỵ✐ ♣❤õ ♠ð {Ωj }∞j=1 ❝õ❛ t➟♣ Ω✳
❚❛ ❝â ỵ s ỡ

ỵ ❈❤♦ ❑ ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t tr♦♥❣ Rn✱ ❤å ❤ú✉ ❤↕♥ {Uj }Nj=1 ❧➔ ♠ët ♣❤õ

♠ð ❝õ❛ ❑✳ ❑❤✐ ✤â✱ tỗ t ởt ồ ỳ ừ ✈ỉ ❤↕♥ {ϕj }Nj=1 ①→❝ ✤à♥❤
♠ët ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à ự ợ ừ {Uj }Nj=1 ừ t
rữợ ự ỵ t t : Rn → R ❧➔ ❤➔♠ ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷
s❛✉✿
ρ(x) :=

Ce
0,

1
x 2 −1 ,

♥➳✉ x < 1
♥➳✉ x ≥ 1

tr♦♥❣ ✤â✱ C ❧➔ ❤➡♥❣ sè s❛♦ ❝❤♦

ρ(x)dx = 1.
Rn

✹


❍➔♠ ρ ❝â ❝→❝ t➼♥❤ ❝❤➜t ✿
ρ ∈ C0∞ (Rn ), s✉♣♣ρ = B[0, 1] = x ∈ Rn

x ≤ 1 , ρ(x) ≥ 0,

ρ(x)dx = 1,
Rn

✈➔ ρ ❧➔ ❤➔♠ ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ x ✳ ❱ỵ✐ ♠é✐ > 0✱ t❛ ①➨t ❤➔♠ ρ ♥❤÷ s❛✉
ρ

(x)

=

−n

x

ρ

.

❍➔♠ ρ ❝ơ♥❣ ❝â ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ρ, ❝ö t❤➸ ❧➔

ρ ∈ C0∞ (Rn ), s✉♣♣ρ = B[0, ] = x ∈ Rn

x ≤

, ρ (x) ≥ 0,

ρ (x)dx = 1,
Rn

✈➔ ρ ❧➔ ❤➔♠ ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ x ✳ ❱ỵ✐ ♠é✐ ❤➔♠ f ∈ L1loc (Rn )✱ ✤➦t
f (x) = (f ∗ ρ ) (x) =

f (y)ρ (x − y)dy
Rn

❱✐➺❝ ✤➦t ♥➔② ❝â ♥❣❤➽❛ ✈➻
f (y)ρ (x − y)dy =
Rn

f (x − y)ρ (y)dy =
Rn

f (y)ρ (x − y)dy.
B[x, ]

▼➺♥❤ ✤➲ ✶✳✶✳ ❈❤♦ f ∈ L1loc(Rn)✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦➳t ❧✉➟♥ s❛✉✳
✭✐✮ f ∈ C ∞(Rn)✳
✭✐✐✮ ◆➳✉ supp f = K ⊂ Rn t❤➻ f

∈ C0∞ (Rn )✱ supp f ⊂ K

tr♦♥❣ ✤â

K = K + B[0, ] = x ∈ Rn d(x, K) ≤

✭✐✐✐✮ ◆➳✉ f ∈ C(Rn), lim
→0

+

❈❤ù♥❣ ♠✐♥❤✳

.

sup |f (x) − f (x)| = 0, K ⊂ Rn ✳

x∈K

✭✐✮ ❉➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ tø ✤➥♥❣ t❤ù❝ s❛✉
Dxα

f (y)ρ (x − y)dy

f (y)Dxα ρ (x − y)dy.

=
Rn

Rn

✭✐✐✮ ❉♦ supp f = K ♥➯♥
f (y)ρ (x − y)dy =

f (x)
Rn

f (y)ρ (x − y)dy
Rn

❱ỵ✐ ♠é✐ x ∈
/ K ❝â x − y > , ∀y ∈ K ✳ ▼➔ supp ρ = B[0, 1] ♥➯♥ ρ (x − y) = 0, ∀y ∈ K ✳
❉♦ ✤â✱ f (x) = 0 ❦❤✐ x ∈
/ K ❤❛② supp f ⊂ K ✳
✭✐✐✐✮ ❉➵ t❤➜②
f (x) − f (x) =

(f (x − y) − f (x)) p(y)dy
Rn

✺


(f (x − y) − f (x)) p(y)dy

=
B(0,1)

♥➯♥
|f (x) − f (x)| ≤

sup |f (x − y) − f (x)| .
y∈B[0,1]

▼➔ f ∈ C(Rn ) ♥➯♥ f ❧✐➯♥ tö❝ ✤➲✉ tr➯♥ tø♥❣ t➟♣ ❝♦♠♣❛❝t K ⊂ Rn ✳ ❉♦ ✤â
lim sup |f (x) − f (x)| = 0, K ⊂ Rn .
→0+ x∈K

❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤

▼➺♥❤ ✤➲ ✶✳✷✳ ❈❤♦ t➟♣ K ⊂ Rn✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐

❝â ❤➔♠ ϕ ∈ C0∞(Rn) t❤ä❛ ♠➣♥
✈➔ ϕ(x) = 1, ∀x ∈ K /2✳
>0

0 ≤ ϕ(x) ≤ 1

∀x ∈ Rn ✱ supp ϕ ⊂ K

❈❤ù♥❣ ♠✐♥❤✳

❳➨t χ(x) ❧➔ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ t➟♣ K3 /4 ✱ tù❝ ❧➔
1, ♥➳✉ x ∈ K3
0, ♥➳✉ x ∈
/ K3

χ(x) :=

/4 ,
/4 .

.

❈â χ ∈ L1 (Rn ) ⊂ L1loc (Rn ), supp χ = K3 /4 ✱ ♥➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶ ❝â
χ∗ρ

/4

∈ C0∞ (Rn ), supp(λ ∗ ρ

/4 )

⊂ K , 0 ≤ (χ ∗ ρ

/4 )(x)

∀x ∈ Rn .

▼➔
(χ ∗ ρ

/4 )(x)

χ(x − y)ρ

=
B

/4 (y)dy

/4(0)

♥➯♥
(χ ∗ ρ

/4 )(x)

≤

ρ
B

/4 (y)dy

=1

∀x ∈ Rn ,

/4(0)

✈➔
(χ ∗ ρ

/4 )(x)

=

ρ
B

/4 (y)dy

= 1, x ∈ K

/2 .

/4(0)

◆❤÷ ✈➟② ❤➔♠ ❝➛♥ t➻♠ ❧➔ ϕ(x) = χ ∗ ρ (x) ✳
❈❤ù♥❣ ♠✐♥❤ ữủ t

ự ỵ

4

ứ tt K ❧➔ t➟♣ ❝♦♠♣❛❝t✱ {Uj }N
j=1 ❧➔ ♠ët ♣❤õ ♠ð ❝õ❛

K t❛ ❝â
W1 := K \ ∪N
J=2 Uj ⊂ U1
✻


tỗ t

1

> 0 s
W1 W1 + B(0, 1 ) ⊂ U1 .

❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷✱ ❝â ❤➔♠ ψ1 ∈ C0∞ (Rn ; [0; 1]) s❛♦ ❝❤♦
V1 := W1 + B(0,

1

2

) ⊂ supp ψ1 ⊂ W1 + B(0, 1 ) ⊂ U1 , ψ1 (x) = 1, x ∈ V1 .

▲↕✐ ❝â✱ W1 := K \ ∪N
J=2 Uj ⊂ V1 ♠➔ V1 ❧➔ t➟♣ ♠ð ♥➯♥
W2 := K \ V1 N
J=3 Uj

õ tỗ t

2

U2 .

> 0 s❛♦ ❝❤♦ W2 ⊂ W2 + B(0, 2 ) ⊂ U2 . ❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷✱ ❝â ♠ët ❤➔♠

ψ2 ∈ C0∞ (Rn ; [0; 1]) s❛♦ ❝❤♦
V2 := W2 + B(0,

2

2

) ⊂ supp ψ2 ⊂ W2 + B(0, 2 ) ⊂ U2 , ψ2 (x) = 1, x ∈ V2 .

N
❈ù ♥❤÷ t❤➳ t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ❞➣② ❝→❝ ❤➔♠ {ψj}N
j=1 ✈➔ ❝→❝ t➟♣ {Vj , Wj }j=1 t❤ä❛ ♠➣♥

ψj ∈ C0∞ (Rn ; [0; 1]) , Vj := Wj + B(0,

j

2

) ⊂ supp ψj ⊂ Wj + B(0, j ) ⊂ Uj

N

ψj (x) > 0, x ∈ ∪N
j=1 Vj (⊃ K) ,

ψj (x) = 1, x ∈ Vj ,
j=1

✈➔

N

ψj (x) < N + 1, x ∈ Rn .
j=1

❈â K N

j=1 Vj tỗ t số > 0 s❛♦ ❝❤♦
K ⊂ K + B(0, ) ⊂ ∪N
j=1 Vj .

❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷ ❝â ❤➔♠ ❦❤æ♥❣ ➙♠ φ t❤ä❛ ♠➣♥
φ ∈ C0∞ (Rn ), K ⊂ K + B(0, /2) ⊂ s✉♣♣φ ⊂ K + B(0, ) ⊂ ∪N
j=1 Vj ,

✈➔
0 ≤ φ(x) ≤ 1, x ∈ Rn , φ(x) = 1, x ∈ K + B(0, /2).

✣➦t
ψj (x)

ϕj (x) :=
φ(x)

N
k=1 ψk (x)

+ (1 − φ(x)) N + 1 −
✼

N
k=1 ψk (x)


❝â
0 ≤ ϕj (x) ≤ 1, x ∈ K, j = 1, 2, ..., N, ϕj ∈ C0∞ (Rn ), supp ϕj ⊂ Uj , j = 1, 2, ..., N,

✈➔

N

ϕj (x) = 1, x ∈ K.
j=1

❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤

✶✳✷ ❚➼❝❤ ❝❤➟♣
◆➳✉ f, g ∈ L1 (Rn ) t❛ ✤à♥❤ ♥❣❤➽❛
f ∗ g(x) =

f (x − y)g(y)dy =

f (y)g(x − y)dy

Rn

Rn

①→❝ ✤à♥❤ ✈ỵ✐ ♠å✐ x ∈ Rn ✳ ❚❛ ❣å✐ f ∗ g ❧➔ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤➔♠ f t❤❡♦ ❤➔♠ g ✳ ❘ã r➔♥❣✱
tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t➼❝❤ ❝❤➟♣ ❝õ❛ ❤➔♠ f t❤❡♦ ❤➔♠ g ✈➔ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤➔♠ g t❤❡♦
❤➔♠ f ❧➔ ♥❤÷ ♥❤❛✉✳ ❚ø ✣à♥❤ ỵ õ
|f g(x)| d(x) =

f (x y)g(y)dy dx

Rn

Rn

|f (x − y)| dx dy ≤ f

|g(y)|

≤
Rn

Rn

L1 (Rn )

g

L1 (Rn )

♥➯♥ f ∗ g ∈ L1 (Rn ) ✈➔
f ∗g

L1 (Rn )

≤ f

L1 (Rn )

g

L1 (Rn ) .

❚ê♥❣ q✉→t✱ ✈ỵ✐ f ∈ L1 (Rn ), g ∈ Lp (Rn )(1 ≤ p ≤ ∞) t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣ ♥❤÷
s❛✉
f ∗g

p

≤ f

p

g 1.

✶✳✸ ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn)
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❑❤æ♥❣ ❣✐❛♥ S (Rn) ❧➔ t➟♣ ❤ñ♣
S (Rn ) = {ϕ ∈ C ∞ (Rn ) : sup xα Dβ ϕ (x) < ∞
x∈Rn

✽

∀α, β ∈ Zn+ }.


❈❤♦ ❤➔♠ ϕ ∈ S (Rn )✱ ❦❤✐ ✤â
lim xα Dβ ϕ (x) = 0

x →∞

∀α, β ∈ Zn+ .

❱➼ ❞ư ✶✳✶✳ ❑❤ỉ♥❣ ❣✐❛♥ C0∞(Rn) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠

♥❤❛♥❤ S (Rn)✳

❱➼ ❞ư ✶✳✷✳ ❈❤♦ ❤➔♠ sè ϕ (x) = e− x
❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn)✳

2

, x ∈ Rn ✳

❑❤✐ ✤â ϕ ❧➔ ❤➔♠ sè t❤✉ë❝ ❦❤æ♥❣ ❣✐❛♥

✶✳✹ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r
✶✳✹✳✶ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤
S (Rn )

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❈❤♦ ❤➔♠ f ∈ S (Rn)✳ ❇✐➳♥ ờ rr ừ f ỵ f ()
F (f ) (ξ)✱ ❧➔ ❤➔♠ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

F (f ) (ξ) = f (ξ) = (2π)−n/2

e−i x,ξ f (x) dx
Rn

tr♦♥❣ ✤â x = (x1, x2, ..., xn) ∈ Rn, ξ = (ξ1, ξ2, ..., ξn) ∈ Rn.

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷đ❝ ❝õ❛ ❤➔♠ f ∈ S (Rn) ❧➔ ❤➔♠ ✤÷đ❝ ①→❝ ✤à♥❤

❜ð✐

∨

F −1 (f ) (x) = f (x) = (2π)−n/2

ei x,ξ f (ξ) dξ
Rn

tr♦♥❣ ✤â x = (x1, x2, ..., xn) ∈ Rn, ξ = (ξ1, ξ2, ..., ξn) ∈ Rn.
❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❛ ❞➵ ❞➔♥❣ s✉② r❛✿ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✭✈➔ ♥❣÷đ❝ ❝õ❛ ♥â✮ ❧➔ t✉②➳♥
t➼♥❤✱ ♥❣❤➽❛ ❧➔✿
F[λ1 f1 + λ2 f2 ] = λ1 F[f1 ] + λ2 F[f2 ]

✈➔
F −1 [λ1 f1 + λ2 f2 ] = λ1 F −1 [f1 ] + λ2 F −1 [f2 ]

❇➙② ❣✐í t❛ ①➨t ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✱ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷đ❝ ❝õ❛ ❤➔♠
t❤✉ë❝ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn )✳

✾


ỵ S (Rn) ✤â Fϕ, F −1ϕ ∈ S (Rn) ✈➔
• Dα Fϕ (ξ) = (−i)|α| F (xα ϕ (x)) (ξ) ,

Dα F −1 ϕ (ξ) = i|α| F −1 (xα ϕ (x)) (ξ) .

• ξ α Fϕ (ξ) = (−i)|α| F (Dα ϕ (x)) (ξ) ,

ξ α F −1 ϕ (ξ) = i|α| F −1 (Dα ϕ (x)) (ξ) .

❈❤ù♥❣ ♠✐♥❤✳

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ ϕ t❤✉ë❝ ❦❤æ♥❣ ❣✐❛♥

❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn ) ❝â
(Fϕ) (ξ) = (2)n/2

ei x, (x) dx.



Rn

ử ỵ t ❦❤↔ ✈✐ ❝→❝ t➼❝❤ ♣❤➙♥ ♣❤ö t❤✉ë❝ t❤❛♠ sè✱ t❛ ❝â ✤↕♦ ❤➔♠
Dξα (Fϕ) (ξ) ✈ỵ✐ ♠å✐ α ∈ Zn+ ✈➔
Dξα (Fϕ) (ξ) = Dξα

(2π)−n/2

e−i x,ξ ϕ (x) dx
Rn

= (2π)−n/2

(−ix)α e−i x,ξ ϕ (x) dx
Rn

= (−i)|α| (2π)−n/2

e−i x,ξ xα ϕ (x)dx
Rn

|α|

α

= (−i) F (x ϕ (x)) (ξ)

∀ϕ ∈ S (Rn ) ,

❞♦ t➼❝❤ ♣❤➙♥
e−i x,ξ xα ϕ (x) dx

∀ϕ ∈ S (Rn )

Rn

❤ë✐ tö t✉②➺t ✤è✐ ✈➔ ✤➲✉ t❤❡♦ ξ tr♦♥❣ Rn ✈➔ ♠å✐ α ∈ Zn+ ✳❱➻
e−i x,ξ xα ϕ (x) ≤ |x|α |ϕ (x)|

∀ϕ ∈ S (Rn ) .

❉♦ ❤➔♠ ϕ ∈ S (Rn ) ♥➯♥
|x|α |ϕ (x)| dx

∀α ∈ Zn+

Rn

❤ë✐ tö t✉②➺t ✤è✐ ✈➔ ✤➲✉ t❤❡♦ ξ tr Rn
õ tỗ t D (F) (ξ)✱ ❞➝♥ ✤➳♥ Fϕ ∈ C ∞ (Rn )✳

❱➻ t❤➳ ♠é✐ ξ ∈ Rn , β, γ ∈ Zn+ ✱ ❝â
lim ξ β Dxγ e−i x,ξ ϕ (x) = 0

x →∞

∀ϕ ∈ S (Rn ) .

❙û ❞ö♥❣ ♣❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ |β| ❧➛♥ ❝❤♦ ✭✶✳✷✮✱ t❛ ✤÷đ❝
Dξα (Fϕ) (ξ) = ξ −β (2π)−n/2

e−i x,ξ (−iDx )β (−ix)α ϕ (x) dx.
Rn

✶✵

✭✶✳✷✮


ữ ợ ộ , Zn+ õ
β Dξα (Fϕ) (ξ) = (2π)−n/2

✭✶✳✸✮

e−i x,ξ (−iDx )β (−ix)α ϕ (x) dx,
Rn

♥❤➟♥ t❤➜② r➡♥❣
e−i x,ξ (−iDx )β (−ix)α ϕ (x) dx
Rn

≤ sup Dxβ (−x)α ϕ (x)

dx

(1 + x )n+1

x∈Rn

Rn

(1 + x )n+1

. ✭✶✳✹✮

❑➳t ❤ñ♣ ✭✶✳✸✮ ✈➔ ✭✶✳✹✮✱ t❛ ♥❤➟♥ ✤÷đ❝
sup ξ β Dξα Fϕ (ξ)

ξ∈Rn

≤ (2π)−n/2 sup Dxβ (−x)α ϕ (x)
x∈Rn

≤ C sup 1 + x

dx

(1 + x )n+1
Rn

2 n+1+|α|

x∈Rn

|Dγ ϕ (x)|

(1 + x )n+1

∀α, β ∈ Zn+ .

γ≤β

❉♦ ϕ ∈ S (Rn ) ♥➯♥
sup 1 + x

2 n+1+|α|

x∈Rn

|Dγ ϕ (x)| < ∞

∀α, β ∈ Zn+ .

γ≤β

✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ Fϕ ∈ S (Rn )✳
❚ø ❝æ♥❣ t❤ù❝ ✭✶✳✸✮✱ ❝❤♦ α = 0, β ∈ Zn+ t❛ ♥❤➟♥ ✤÷đ❝
ξ β Fϕ (ξ) = (2π)−n/2

(−iDx )β e−i x,ξ ϕ (x) dx
Rn

= (2π)−n/2
|β|

e−i x,ξ (−iDx )β ϕ (x) dx
Rn
β

= (−i) F D ϕ (x) (ξ)

∀ϕ ∈ S (Rn ) .

❱➟② ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠
❣✐↔♠ ♥❤❛♥❤ S (Rn )✳ ✣è✐ ✈ỵ✐ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷đ❝ F −1 t❛ ❝❤ù♥❣ ♠✐♥❤ tữỡ
tỹ
ự ữủ t

ỵ ∈ S (Rn)✳ ❑❤✐ ✤â
F −1 Fϕ = FF −1 ϕ = ϕ.

❚ø ✤â s✉② r❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✭❝ơ♥❣ ♥❤÷ ♥❣÷đ❝ ❝õ❛ ♥â✮ ❧➔ ♣❤➨♣ ù♥❣ ✶✲✶✳
✶✶


▼➺♥❤ ✤➲ ✶✳✸✳ ❈❤♦ ❝→❝ ❤➔♠ ϕ, ψ ∈ S (Rn)✳ ❑❤✐ ✤â✱
ϕ (x) Fψ (x) dx =
Rn

ψ (ξ)Fϕ (ξ) dξ
Rn

✈➔
|ϕ (x)|2 dx =
Rn

❈❤ù♥❣ ♠✐♥❤✳

|Fϕ (ξ)|2 dξ.
Rn

❙û ❞ö♥❣ ✤à♥❤ ♥❣❤➽❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝❤♦ ❤➔♠ ψ (x) tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥

❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn )✱ ❝â
Fψ (x) = (2π)−n/2

e−i x,ξ ψ (ξ) dξ,
Rn

❦❤✐ ✤â ϕ, ψ ∈ S (Rn )✱ t❛ ❝â
ϕ (x) (2π)−n/2
Rn

e−i x,ξ ψ (ξ) dξ dx =

ϕ (x) Fψ (x) dx.

✭✶✳✺✮

Rn

Rn

❚÷ì♥❣ tü✱ t❛ ♥❤➟♥ ✤÷đ❝
Fϕ (ξ) = (2π)−n/2

e−i x,ξ ϕ (x) dx

∀ϕ ∈ S (Rn ) ,

Rn

✈ỵ✐ ϕ, ψ ∈ S (Rn )✱ ♥➯♥
ψ (ξ) (2π)−n/2
Rn

e−i x,ξ ϕ (x) dx dξ =
Rn

ψ (x) (Fϕ) (ξ) dξ.

✭✶✳✻✮

Rn

▼➦t ❦❤→❝✱ ✈ỵ✐ ❝→❝ ❤➔♠ ϕ, ψ ∈ S (Rn ) t ỵ õ
(x) (2)n/2
Rn

ei x, () dξ dx
Rn

ψ (ξ) (2π)−n/2

=
Rn

e−i x,ξ ϕ (x) dx dξ. ✭✶✳✼✮
Rn

❑➳t ❤đ♣ ✭✶✳✺✮✱ ✭✶✳✻✮ ✈➔ ✭✶✳✼✮✱ t❛ ✤↕t ✤÷đ❝
ϕ (x) Fψ (x) dx =
Rn

ψ (ξ) (Fϕ) (ξ) dξ
Rn

❇➡♥❣ ❝→❝❤ ❝❤♦ ❤➔♠
ψ = F −1 ϕ

t❛ t❤➜② r➡♥❣
F −1 ϕ = Fϕ,
✶✷

ϕ = Fψ

∀ϕ, ψ ∈ S (Rn ) .

✭✶✳✽✮

✈➔ sû ❞ư♥❣ ✭✶✳✽✮✱ t❛ ♥❤➟♥ ✤÷đ❝
|ϕ (x)|2 dx =
Rn

|Fϕ (ξ)|2 dξ

∀ϕ ∈ S (Rn ) .

Rn

◆❤÷ ✈➟②✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r F ❧➔ ♠ët ✤➥♥❣ ❝➜✉ t✉②➳♥ t➼♥❤✱ tü ❧✐➯♥ ❤đ♣✱ ✤➥♥❣
❝ü tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn ) ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ L2 (Rn )✳
❈❤ù♥❣ ữủ t
ữợ t s tr ởt sè t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✈➲ t➼❝❤ ❝❤➟♣
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn )✳

▼➺♥❤ ✤➲ ✶✳✹✳ ❈❤♦ ❝→❝ ❤➔♠ ϕ, ψ ∈ S (Rn)✳ ❑❤✐ ✤â✱
F (ϕ ∗ ψ) (ξ) = (2π)n/2 Fϕ (ξ) Fψ (ξ) .
F −1 (ϕ ∗ ψ) (ξ) = (2π)n/2 F −1 ϕ (ξ) F −1 ψ (ξ) .
(2π)n/2 F (ϕ (x) ψ (x)) (ξ) = Fϕ (ξ) ∗ Fψ (ξ) .
(2π)n/2 F −1 (ϕ (x) ψ (x)) (ξ) = F −1 ϕ (ξ) ∗ F −1 ψ (ξ) .

✶✳✹✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ L1(Rn)

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❈❤♦ ❤➔♠ f ∈ L1 (Rn)✳ ❷♥❤ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f ỵ f ()

F (f ) ()

✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

F (f ) (ξ) = f (ξ) = (2π)−n/2

e−i x,ξ f (x) dx
Rn

tr♦♥❣ ✤â x = (x1, x2, ..., xn) ∈ Rn, ξ = (ξ1, ξ2, ..., ξn) ∈ Rn.
▼➺♥❤ ✤➲ ✶✳✺✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ♠ët ❤➔♠ ❦❤↔ t➼❝❤ t✉②➺t ✤è✐ tr➯♥ Rn ❧➔ ♠ët ❤➔♠
❜à ❝❤➦♥ tr➯♥ Rn✳ ❍ì♥ ♥ú❛
f (y) ≤ (2π)−n/2

❈❤ù♥❣ ♠✐♥❤✳

|f (x)| dx

∀y ∈ Rn .

Rn

❚ø ✤à♥❤ ♥❣❤➽❛ t❛ s✉② r❛
|e−ixy | |f (x)| dx.

|f (y)| ≤ (2π)−n/2
Rn

❑➳t ❤đ♣ ✤✐➲✉ ♥➔② ✈ỵ✐ e−ixy = 1✱ s✉② r❛
f (y) ≤ (2π)−n/2

|f (x)| dx
Rn

❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤
✶✸

∀y ∈ Rn .


❈❤÷ì♥❣ ✷
✣→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣
❙t❡✐♥✲❲❛✐♥❣❡r
❈❤♦ Pd ❧➔ t➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ ✈ỵ✐ ❤➺ sè t❤ü❝ ❝â ❜➟❝ ❦❤ỉ♥❣ ✈÷đt q✉→ d✳ ❈❤♦
P ∈ Pd t❛ ①➨t ❣✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ s❛✉
eiP (t)

I (P ) = p.v.
R

dt
.
t

▼ư❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ ✤→♥❤ ❣✐→ tr ữợ ừ I (P ) ❝→❝ ❤➡♥❣
sè ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ ❜➟❝ d ❝õ❛ ✤❛ t❤ù❝ P (x)✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✷ ♥➔② ❞ü❛ tr➯♥ t
số

ữợ ừ t ở
ỵ d N õ tỗ t số ữỡ c1 ổ ử tở d
s

eiP (x)

c1 log d sup p.v.
P Pd

R

dx
.
x

rữợ ữ r ự ỵ tr t ❜ê ✤➸ ❱❛♥❞❡r ❈♦r♣✉t✳

▼➺♥❤ ✤➲ ✷✳✶✳ ❈❤♦ φ : [a, b] → R ❧➔ ❤➔♠ sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝➜♣ k t❤ä❛ ♠➣♥

φ(k) (t) ≥ 1

✈ỵ✐ ♠å✐ t ∈ [a, b] ✭♥➳✉ k = 1 t❛ ❣✐↔ sû t❤➯♠ φ ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉✮ ✈➔ ❝❤♦ ψ ❧➔ ❤➔♠ ❦❤↔
✈✐ ❝➜♣ 1 tr➯♥ [a, b]✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ λ ∈ R, t❛ ❧✉æ♥ ❝â
b

eiλφ(x) ψ(x)dx ≤
a

Ck
1

|λ| k

ψ

∞

tr♦♥❣ ✤â ❤➡♥❣ sè Ck ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ a, b ✈➔ φ, ψ.
✶✹

+ ψ

1

,


❚✐➳♣ t❤❡♦✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❜ê ✤➲ s❛✉✳

❇ê ✤➲ ✷✳✶✳ ❈❤♦ n ≥ 3✱ f ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝ tr➯♥ R t❤ä❛ ♠➣♥ f (t) = 1 ♥➳✉ n1 ≤ t ≤ 1− n1 ✱

♥➳✉ −1 + n1 ≤ t ≤ − n1 ✱ f (t) = 0 ♥➳✉ |t| ≥ 1 ✈➔ t✉②➳♥ t➼♥❤ tr♦♥❣ ♠é✐ ❦❤♦↔♥❣
−1, −1 + n1 ✱ − n1 , n1 ✈➔ 1 − n1 , 1 õ tỗ t số c ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ n
s❛♦ ❝❤♦

f (t) = −1

eif (t)

I (f ) = p.v.
R

❈❤ù♥❣ ♠✐♥❤✳

dt
≥ c log n.

t

✭✷✳✶✮

❚ø ❣✐↔ t❤✐➳t✱ t❛ s✉② r❛ f ❧➔ ❤➔♠ ❧➫ ✈➔ f (t) = 0 ∀ |t| ≥ 1✱ ❞♦ ✤â
1

sin f (t)
dt .
t

I(f ) = 2
0

✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥
1− n1

I (f ) ≥ 2
1
n

❚❛ t❤➜②✱ ✈ỵ✐

1
n

≤t≤1−
1− n1
1
n

✈ỵ✐ 0 ≤ t ≤

1
n

sin f (t)
dt − 2
t
1
n

1− n1

sin f (t)
dt =
t

1− n1

sin f (t)
dt .
t

✭✷✳✷✮

1
n

sin 1

dt = sin 1 log(n − 1),
t

✭✷✳✸✮

t❤➻ f (t) = nt✱ s✉② r❛

0
1
n

0

1

sinf (t)
dt − 2
t

t❤➻ f (t) = 1✱ s✉② r❛

1
n

✈ỵ✐ 1 −

1
n

sin f (t)

dt ≤
t

1
n

✭✷✳✹✮

ndt = 1,
0

≤ t ≤ 1 t❤➻ f (t) = n(1 − t)✱ s✉② r❛
1
1− n1

sin f (t)
dt ≤
t

1
1− n1

n(1 − t)
n
dt = n log
− 1.
t
n−1

❑➳t ❤ñ♣ ✭✷✳✷✮✱ ✭✷✳✸✮✱ ✭✷✳✹✮ ✈➔ ✭✷✳✺✮✱ t❛ t❤✉ ✤÷đ❝

1
n

1

f (t)
f (t)
I (f ) ≥ 2 sin 1 log (n − 1) − 2
dt − 2
dt
t
t
0
1− n1
n
= 2 sin 1 log (n − 1) − 2 − 2n log
+ 2.
n−1

✣✐➲✉ ♥➔② ❝❤♦ t❛
I (f ) ≥ 2 sin 1 log (n − 1) − 4 ≥ c log n.

❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤
✶✺

✭✷✳✺✮


❱ỵ✐ ♠é✐ k ∈ N✱ t❛ ①→❝ ✤à♥❤ ❤➔♠ φk : R → R ♥❤÷ s❛✉✿
x2

1−
4

φk (x) = Ck

k2

,

✭✷✳✻✮

tr♦♥❣ ✤â ❤➡♥❣ sè Ck ✤÷đ❝ ❝❤å♥ t❤ä❛ ♠➣♥ ✤➥♥❣ t❤ù❝
2

φk (x) dx = 1.



2

ú ỵ r
2

1 = Ck
2

k2

x2
1

4

1

dx = 4Ck
0

1
k2
(1 x2 ) dx = 2Ck B( , k 2 + 1),
2

ð ✤➙② B(., .) ❧➔ ❤➔♠ ❜❡t❛✳ ❙û ❞ö♥❣ ❝→❝ t➼♥❤ ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❜❡t❛✱ t❛ s✉② r❛ Ck k
ợ f ữ tr ờ ✤➲ ✷✳✶✱ t❛ ①➙② ❞ü♥❣ ❤➔♠ Pk ①→❝ ✤à♥❤ tr➯♥ R ♥❤÷
s❛✉

1

Pk (t) =
−1

f (x)φk (t − x) dx.

✭✷✳✽✮

❘ã r➔♥❣ ❤➔♠ Pk ❧➔ ✤❛ t❤ù❝ ❜➟❝ ❦❤ỉ♥❣ ✈÷đt q✉→ 2k 2 ✳ ❚❛ ❝â ❜ê ✤➲ s❛✉ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t
❝õ❛ ❝→❝ ✤❛ t❤ù❝ Pk ✳

❇ê ✤➲ ✷✳✷✳ ❈❤♦ Pk ❧➔ ❤➔♠ sè ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ❝ỉ♥❣ t❤ù❝ ✭✷✳✽✮ ð tr➯♥✳ ❑❤✐

✤â t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉
✭✐✮ Pk ❧➔ ♠ët ✤❛ t❤ù❝ ❧➫ ❜➟❝ 2k2 − 1 ợ số t ữủ t t ổ tự
ak = (−1)k

2

+1

2Ck k 2
1
1−
.
2
k
n
4

❚ù❝ ❧➔✱ Pk ❝â ❞↕♥❣ s❛✉
Pk (t) = ak t2k

2

−1

+ ...

◆â✐ r✐➯♥❣✱ ∀t ∈ R t❛ ❧✉æ♥ ❝â
(2k2 −1)

Pk

k3
(t) ≥ C 2k − 1 ! k2 .
4
2

✭✐✐✮ ❱ỵ✐ t ∈ [−1; 1] t❛ ❝â
2

(f (t + x) + f (t − x))φk (x) dx.

Pk (t) =
0

✶✻


❈❤ù♥❣ ♠✐♥❤✳

✭✐✮ ❙û ❞ö♥❣ ✭✷✳✽✮ t❛ ❝â
1

1

f (x) φk (−t − x) dx =

Pk (−t) =

f (x) φk (t + x) dx

−1
1

−1

f (−x) φk (t − x) dx = −P k (t) .

=
−1

❉♦ ✤â Pk ❧➔ ♠ët ❤➔♠ ❧➫✳ ❍ì♥ ♥ú❛

Pk (t) = Ck

2

k2

1

k2
m

f (x)
−1

m=0

k2

k 2 (−1)m
m
4m

= Ck
m=0

t−x
−

m

4

dx

1

f (x)(t − x)2m dx,
−1

✤✐➲✉ ♥➔② ❞➝♥ ✤➳♥
Ck (−1)k
Pk (t) =
2
4k

2

k2 −1

1

2

f (x) (x − t)2k dx + Ck
−1

m=0

k 2 (−1)m
m
4m

1

f (x) (t − x)2m dx.
−1

❱➻ ✈➟②
(−1)3k
Pk (t) = Ck
2
4k

❉♦ f ❧➔ ❤➔♠ ❧➫✱ ♥➯♥

2

2

1
2k

f (x)dxt
−1
1
f (x)dx
−1

2

(−1)k 2k 2
− Ck
2
4k

1

f (x) xdxt2k

2

+1 2Ck k
2
4k

2

1−

1 2k2 −1
t
+ ....
n

❳➨t sè ❤↕♥❣ ❣➢♥ ✈ỵ✐ ❜➟❝ ❝❛♦ ♥❤➜t tr♦♥❣ ❝æ♥❣ t❤ù❝ tr➯♥ t❛ ❝â
ak = = (−1)k

2

+1 2Ck k
2
4k

2

1−

1
t.
n

(t) = (2k 2 − 1)!ak ✈➔ ak ∼ k ✱ t❛ s✉② r❛
(2k2 −1)

Pk

+ ...

✭✷✳✾✮

= 0✱ ❦➳t ❤ñ♣ ợ t ữủ

Pk (t)= (1)k

(2k2 −1)

−1

−1

t❤ù❝ s❛✉

❍ì♥ ♥ú❛✱ ❞♦ Pk

2

k3
(t) ≥ C 2k − 1 ! k2
4
2

✭✐✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳

✶✼

∀t ∈ R.

✭✐✐✮ ❈è ✤à♥❤ t ∈ [−1, 1]✳ ❑❤✐ ✤â
2

f (t − x) φk (x) dx =
−2

f (t − x) φk (x) χ[−2,2] (x) dx
R
1

f (x)φk (t − x)χ[−2,2] (t − x) dx

=
−1
1

f (x) φk (t − x) dx= P k (t).

=
1

t ủ ợ k t ữủ
2

2

f (t − x) φk (x) dx =

Pk (t) =

(f (t + x) + f (t − x))φk (x) dx.

−2

0

❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤

❇ê ✤➲ ✷✳✸✳ ❈❤♦ ❤➔♠ f ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶✳ ❚❛ ①→❝ ✤à♥❤

A (x, t) = |f (t + x) + f (t − x) − 2f (t)|✳
2
0

❈❤ù♥❣ ♠✐♥❤✳

1
0

❑❤✐ ✤â

A (x, t)
dtφn (x) dx = o(logn).
t

❚ø ❣✐↔ t❤✐➳t t❛ s✉② r❛
|f (x) − f (y)| ≤ n|x − y|

∀x, y ∈ R.

✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥
A (x, t) ≤ |f (t + x) − f (t)| + |f (t − x) − f (t)|

✭✷✳✶✵✮

≤ nx + nx ≤ 2nx.

▼➦t ❦❤→❝✱ ❞♦ f (t) ≤ nt✱ ♥➯♥
A(x, t) = |f (t + x) − f (x) + f (t − x) − f (−x) − 2f (t)|
≤ |f (t + x) − f (x)| + |f (t − x) − f (−x)| + 2 |f (t)|
≤ nt + nt + 2nt = 4nt.

❍ì♥ ♥ú❛✱ ❞♦ |f | ❜à ❝❤➦♥ ❜ð✐ 1✱ ♥➯♥
A(x, t) ≤ 4.
✶✽

✭✷✳✶✶✮


❑➳t ❤đ♣ ✤✐➲✉ ♥➔② ✈ỵ✐ ✭✷✳✶✵✮✱ ✭✷✳✶✶✮✱ t❛ s✉② r❛
✭✷✳✶✷✮

A (x, t) ≤ 4 min (nx, nt, 1) .

❍ì♥ ♥ú❛✱ tø ❣✐↔ t❤✐➳t t❛ s✉② r❛
❦❤✐

A (x, t) = 0,

❚❛ t→❝❤ t➼❝❤ ♣❤➙♥

2

2 1 A(x,t)
t φn (x) dtdx
0 0
1
n

1

1
− n1
2

1
n

2

. . . dtdx +
0

1
− n1
2

0

x+ n1

x
1
2

2

. . . dtdx +

x+ n1

0

x+ n1

. . . dtdx

0

. . . dtdx +

+

1
n

. . . dtdx +
1
n

0

1
2

✭✷✳✶✸✮

t❤➔♥❤ ❜↔② t➼❝❤ ♣❤➙♥ s❛✉

x

. . . dtdx +
1
2

0

1
1
≤t−x≤t+x≤1− .
n
n

1
n

1
n

. . . dtdx.
1

− n1
2

1
n

❚❛ ✤→♥❤ ❣✐→ r✐➯♥❣ tø♥❣ t➼❝❤ ♣❤➙♥ ♥❤÷ s❛✉
2

1
1
2

0
1
n

x

0

0

1
n

0

1
n

A (x, t)
dtφn (x) dx ≤
t
1
n

2

x

0

1
n

1
n

0

1
n

x+ n1

4φn (x) dx ≤ 2.
0
1
n

x+ n1

≤
0

0x

x

4nx
dtφn (x) dx =
t

✣➸ ✤→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥

1
− n1
2

1
2

x+ n1

0

A(x, t)
dtφn (x) dx
t

4nxlog (1 +

0

0

x+ n1

1
)φ (x) dx ≤ 2.
nx n

A(x, t)
dtφn (x) dx
t
1
n

≤ t−x ≤ t+x ≤ 1−

❝ỉ♥❣ t❤ù❝ ✭✷✳✶✸✮✱ t❛ ✤÷đ❝ A(x, t) = 0. ❉♦ ✤â
1
2

4nxφn (x) dx ≤ 2,
0

1
n

t❛ ♥❤➟♥ t❤➜② ✈ỵ✐ x ∈ [0, 12 − n1 ], t ∈ [x + n1 , 12 ] t❤➻
1
− n1
2

1
n

4nt
dtφn (x) dx
t

1
n

0

φn (x) dx = 2 log 2,
0

4nt
dtφn (x) dx =
t

0
2

A(x, t)
dtφn (x) dx ≤
t

=

2

A (x, t)
dtφn (x) dx ≤ 4 log 2
t

A(x, t)
dtφn (x) dx = 0.
t

✶✾

1
n

✈➔ →♣ ❞ö♥❣


❚❛ ❝ô♥❣ ❝â ✤→♥❤ ❣✐→ s❛✉
1
− n1
2
1
n

x+ n1
1
n

1
− n1
2

A(x, t)
dtφn (x) dx≤
t

1
n

x+ n1
1
n

4
dtφn (x) dx
t

1

≤4
1
n

log (nx + 1)φn (x) dx.

❈è ✤à♥❤ α ∈ (0, 1)✳ ❘ã r➔♥❣
1

nα

1
1
n

log (nx + 1)φn (x) dx =

1

log (nx + 1)φn (x) dx +

1
n

log n1−α + 1
+ Cn log (n + 1)
≤
2
≤

1
nα

log (nx + 1)φn (x) dx
1

1
nα

x2
1−
4

n2

dx

log n1−α + 1
1 2(1−α)
+ Cn log (n + 1) e− 4 n
.
2

✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥
lim sup
n→∞

log(nx + 1) n (x) dx
1

.
logn
2

tũ ỵ tr (0, 1) ♥➯♥
1
− n1
2

x+ n1

1
n

1
n

A(x, t)
dtφn (x) dx= o(logn).
t

❈✉è✐ ❝ò♥❣✱
1
2

2
1
− n1
2

1
n

A(x, t)
dtφn (x) dx ≤
t

1
2

2
1
− n1
2

1
n

n
≤ 4 log cn
2

4
dtφn (x) dx
t
n2

2
1
− n1
2

1 2
x2
(1 − ) dx ≤ cn log ne− 16 n = o(1).
4

❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤
❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ỵ

ự ỵ

t Pn t❤ù❝ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ✭✷✳✽✮✱ n ≥ 3✳ ❑❤✐

✤â Pn ❧➔ ♠ët ✤❛ t❤ù❝ ❜➟❝ d = 2n2 − 1 ✈➔
eiP (x)

sup p.v.
P ∈Pd

R

✷✵

dx
≥ I (Pn )
x

✭✷✳✶✹✮


tr♦♥❣ ✤â
eiPn (t)

I (Pn ) = p.v
R

❉♦ Pn ❧➔ ❤➔♠ ❧➫ ♥➯♥

+∞

I (Pn ) = 2
0

dt
.
t

sin Pn (t)
dt .
t

❳➨t R ≥ 1. ❙û ❞ö♥❣ ♣❤➛♥ ✭✐✮ ❝õ❛ ❜ê ✤➲ ✷✳✷ ✈➔ →♣ ❞ư♥❣ ▼➺♥❤ ✤➲ ✷✳✶ t❛ t❤✉ ✤÷đ❝
R
1

sin Pn (t)
dt ≤ c1 ,
t

∀R ≥ 1.

❉♦ ✤â
✭✷✳✶✺✮

I(Pn ) ≥ I1 (Pn ) − c1

tr♦♥❣ ✤â

1

sin Pn (t)
dt .
t

I1 (Pn ) =
0

ử ờ t tỗ t số c2 s❛♦ ❝❤♦ I(f ) ≥ c2 log n✳ ❉♦ ✤â
I1 (Pn ) ≥ I(f ) − |I1 (Pn ) − I(f )| ≥ c2 log n − |I1 (Pn ) − I(f )|.

❍❛②
✭✷✳✶✻✮

I1 (Pn ) ≥ c log n − |I1 (Pn ) − I(f )|

❍ì♥ ♥ú❛
1

|I1 (Pn ) − I(f )| =
0
1

≤
0

sin Pn (t) − sin f (t)
dt
t

|Pn (t) − f (t)|
dt.
t

❙û ❞ö♥❣ ♣❤➛♥ ✭✐✐✮ ❝õ❛ ❇ê ✤➲ ✷✳✷ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✮✱ t❛ ❝â
2

|Pn (t) − f (t)| ≤

|f (t + x) + f (t − x) − 2f (t)| φn (x) dx

✈ỵ✐ 0 ≤ t≤ 1.

0

❱➻ ✈➟②
2

1

|I1 (Pn ) (t) − f (t)| ≤
0

0

|f (t + x) + f (t − x) − 2f (t)|
dtφn (x) dx.
t

❑➳t ❤đ♣ ✤✐➲✉ ♥➔② ✈ỵ✐ ❇ê ✤➲ ✷✳✸✱ t❛ s✉② r❛
|I1 (Pn ) (t) − f (t)| = o(logn).

✭✷✳✶✼✮

❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❜➡♥❣ ❝→❝❤ →♣ ❞ư♥❣ ✭✷✳✶✹✮✱ ✭✷✳✶✺✮ ✭✷✳✶✻✮ ✱ ✈➔ ✭✷✳✶✼✮✳
✷✶


✷✳✷ ✣→♥❤ ❣✐→ ❝➟♥ tr➯♥ ❝õ❛ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣
✣à♥❤ ỵ d N õ tỗ t ❤➡♥❣ sè ❞÷ì♥❣ c2 ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ d

s❛♦ ❝❤♦

eiP (x)

sup p.v.
P Pd

R

dx
c2 log d.
x

rữợ ự t q tr t t q t ự ữợ ❝õ❛ ♠ët ✤❛
t❤ù❝ tr♦♥❣ ❜ê ✤➲ s❛✉ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ❱✐♥♦❣r❛❞♦✈ [5]✳

❇ê ✤➲ ✷✳✹✳ ❈❤♦ h(t) = b0 + b1t + . . . + b1tn ❧➔ ✤❛ t❤ù❝ ❜➟❝ n✳ ❑❤✐ ✤â
α

|{t ∈ [1, 2] : |h (t)| ≤ α }| ≤ c

❈❤ù♥❣ ♠✐♥❤✳

max0≤k≤n |bk |

1
n

.

❚➟♣ ❤ñ♣ Eα = {t ∈ [1, 2] : |h (t)| ≤ α } ❧➔ ❤đ♣ ❝õ❛ ❝→❝ ✤♦↕♥ rí✐ ♥❤❛✉✳ ❚❛

❞à❝❤ ❝❤✉②➸♥ ❝→❝ ✤♦↕♥ ♥➔② ✤➸ t↕♦ ♥➯♥ ♠ët ❦❤♦↔♥❣ ✤ì♥ I ❝❤✐➲✉ ❞➔✐ |Eα | ✈➔ ❝❤✐❛ ✤➲✉
✤♦↕♥ I ❜➡♥❣ n + 1 ✤✐➸♠ ❝❤å♥✳ ❚✐➳♣ t❤❡♦✱ t❛ ❞à❝❤ ❝❤✉②➸♥ ❝→❝ ✤♦↕♥ ♠ỵ✐ trð ❧↕✐ ✈à tr➼
❜❛♥ ✤➛✉ ❝õ❛ ♥â✱ t❤➻ ✭♥✰✶✮ ✤✐➸♠ ✤➣ ❝❤å♥ ❝ô♥❣ ❞à❝❤ ❝❤✉②➸♥ t❤❡♦ ✈➔ s➩ ❦➳t t❤ó❝ t↕✐
n + 1 ❝→❝ ✤✐➸♠ x0 , x1 , x2 , . . . , xn ∈ Eα t❤ä❛ ♠➣♥
|xj − xk | ≥ |Eα |

|j − k|
.
n

✭✷✳✶✽✮

✣❛ t❤ù❝ ▲❛❣r❛♥❣❡ ✈ỵ✐ ❝→❝ ❣✐→ trà ♥ë✐ s✉② h (x0 ) , h (x1 ) , . . . , h(xn ) ❝❤➼♥❤ ❧➔ ✤❛ t❤ù❝
h(x) :
n

h (x) =

h (xj )
j=0

(x − x0 ) (x − x1 ) . . . (x − xj−1 ) (x − xj+1 ) . . . (x − xn )
.
(xj − x0 ) (xj − x1 ) . . . (xj − xj−1 ) (xj − xj+1 ) . . . (xj − xn )

❱➻ ✈➟②✱ t❛ ❝â ❝æ♥❣ t❤ù❝ t➼♥❤ ❝→❝ ❤➺ sè ❝õ❛ ✤❛ t❤ù❝ h(x) ♥❤÷ s❛✉✿
n

bk =
j=0

(−1)n−k σn−k (x0 , . . . , xˆj , . . . , xn )
h(xj )
(xj − x0 ) (xj − x1 ) . . . (xj − xj−1 ) (xj − xj+1 ) . . . (xj − xn )

✈ỵ✐ k = 0, 1, . . . , n.
❚r♦♥❣ ❝æ♥❣ t❤ù❝ tr➯♥✱ σn−k (x0 , . . . , xˆj , . . . , xn ) ❧➔ ❤➔♠ sè ✤è✐ ①ù♥❣ ❝ì ❜↔♥ t❤ù (n − k)
❝õ❛ x0 , . . . , xˆj , . . . , xn ð ✤➙② xj ❜à ❧÷đ❝ ❜ä✳ ❑➳t ❤đ♣ ✭✷✳✶✽✮ ✈➔
σn−k (x0 , . . . , xˆj , . . . , xn ) ≤
✷✷

n
2n−k
n−k

2n
≤ c√ ,
n

n
n−k

t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ❝❤♦ ♠å✐ k = 0, 1, . . . , n,
|bk | ≤

=

n
α
2n−k nn
n−k
|Eα |n

n

j=0

1
j! (n − 1)!

8n nn α
n
nn α
.
22n−k

n ≤ c√
n! |Eα |
n−k
n n! |Eα |n

✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥

8n nn α
max |bk | ≤ c √
,
0≤k≤n
n n! |Eα |n

❞♦ ✈➟②

1
n

α

|Eα | ≤

.

max0≤k≤n |bk |

❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤

❈❤ù♥❣ ♠✐♥❤ ✣à♥❤ ỵ

t
Kd =

eiP (t)

sup
P Pd , ,R

|t|R

dt
.
t



t❤ù❝ ❜➜t ❦ý P ❜➟❝ ❦❤ỉ♥❣ ✈÷đt q✉→ d✱ ❣✐↔ ✤à♥❤ r➡♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔②
❦❤æ♥❣ ❝â ❤➺ sè tü ❞♦✱ tù❝ ❧➔ P ✭✵✮ = ✵✳ ❚❛ ✤➦t k =

d
2

✈➔ ❜✐➸✉ ❞✐➵♥

P (t) = a1 t + a2 t2 + . . . + ak tk + ak+1 tk+1 + . . . + a2d td
= Q (t) + R (t)

ð ✤➙② Q (t) = a1 t + a2 t2 + . . . + ak tk ✈➔ R (t) = ak+1 tk+1 + . . . + a2d td ✳
✣➦t |al | = maxk+1≤j≤d |aj | ✈ỵ✐ k + 1 ≤ l ≤ d✳ ❚❛ ❝â t❤➸ ❣✐↔ sû |al | = 1 ✈➔ ✈➻ ✈➟② |aj | ≤ 1
✈ỵ✐ ∀k + 1 ≤ j ≤ d✳ ❇➙② ❣✐í t→❝❤ t➼❝❤ ♣❤➙♥ ð ✭✷✳✶✾✮ t❤➔♥❤ ❤❛✐ t❤➔♥❤ ♣❤➛♥ ♥❤÷ s❛✉✿

eiP (t)
≤|t|≤R

dt
≤
t

eiP (t)
≤|t|≤1

dt
+
t

eiP (t)
1≤|t|≤R

dt
t

✭✷✳✷✵✮

= I1 + I2 .

✷✸