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
Chuyên ngành: Toán 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

▼ö❝ ❧ö❝
▼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➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❙t❡✐♥✲❲❛✐♥❣❡r

✶✹

✸ ×î❝ ❧÷ñ♥❣ ❝❤✉➞♥ ❝õ❛ t♦→♥ tû t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣

✷✻

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✵
✹✵

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

✶


ớ ỡ
rữợ tr ở ừ tổ ỷ ớ ỡ
t s s t ừ tợ

sỹ ú ù t

t ũ ỳ ớ ở ổ ũ ỵ ừ tr sốt q tr tổ
t tốt
ổ ụ t ỡ sỹ ú ù ừ t ổ tr
ỡ ồ trữớ ồ ồ ỹ ồ ố ở
ồ t t tr tử tự t ú ù tổ
t õ ồ
ổ ỷ ớ ỡ ổ ở ú
ù tổ rt tr sốt tớ ự ồ t
ũ ố rt tú tr q tr ự ữ
ợ q ợ ổ t ự ồ ỏ tớ tỹ
ổ t tr ọ ỳ t sõt
ữủ ỵ õ õ ừ t ổ ữủ t ỡ
ở

▼ð ✤➛✉
❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✤➣ t❤✉ ❤ót ♥❤✐➲✉ sü q✉❛♥ t➙♠ ❝õ❛ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ ✈➔ ❝→❝
♥❤➔ ❱➟t ❧þ tø ❦❤✐ ①✉➜t ❤✐➺♥ ❝æ♥❣ tr➻♥❤ ❚❤➨♦r✐❡ ❆♥❛❧②t✐q✉❡ ❞❡ ❧❛ ❈❤❛❧❡✉r ❝õ❛ ❏♦s❡♣❤
❋♦✉r✐❡r ✈➔♦ ♥➠♠ ✶✽✷✷✳ ◆❤✐➲✉ ❜➔✐ t♦→♥ ▲þ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ❤➻♥❤
❤å❝ ✤↕✐ sè✱ ❧þ t❤✉②➳t ①→❝ s✉➜t✱ ❧þ t❤✉②➳t sè❀ ❝→❝ ❜➔✐ t♦→♥ ✈➲ q✉❛♥❣ ❤å❝✱ ➙♠ ❤å❝✱ ❝ì
❤å❝ ❧÷ñ♥❣ tû✱✳✳✳ ✤➲✉ ❝â t❤➸ ✤÷❛ ✈➲ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣✳
❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✤➣ ✈➔ ✤❛♥❣ ✤÷ñ❝ sû ❞ö♥❣ tr♦♥❣ ♥❤✐➲✉ ù♥❣ ❞ö♥❣ ❦❤→❝ ♥❤❛✉
✈➔ t❤✉ ❤ót ✤÷ñ❝ ♥❤✐➲✉ sü q✉❛♥ t➙♠ tø ❝→❝ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ❬✸✲✻❪✳ ◆❤✐➲✉ ♥❤➔ ♥❣❤✐➯♥
❝ù✉ ✤➣ r➜t ♥é ❧ü❝ ✤➸ ÷î❝ t➼♥❤ trü❝ t✐➳♣ ❣✐→ trà t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✈➔ tè❝ ✤ë s✉② ❣✐↔♠
❝õ❛ ❝❤✉➞♥ ❝õ❛ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❋♦✉r✐❡r ✭①❡♠ ❬✸✱ ✺✱ ✻❪ ✮✳
◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ✤÷ñ❝ ❝❤✐❛ ❧➔♠ ❜❛
❝❤÷ì♥❣✿
✳ ❈❤÷ì♥❣ ♥➔② ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱
t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à✱ t➼❝❤ ❝❤➟♣ ✈➔ ♠ët sè ✤à♥❤ ❧➼ q✉❛♥ trå♥❣ ❝õ❛
♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (R ) ✈➔ L (R )✳
✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤
❜➔② ✈➲ ✈✐➺❝ ✤→♥❤ ❣✐→ ❝➟♥ tr➯♥ ✈➔ ❝➟♥ ❞÷î✐ ❝õ❛ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❦ý ❞à
I(λ) =

dx
,
x

e
R

✈➔ ÷î❝ ❧÷ñ♥❣ ❝→❝ ❝➟♥ tr➯♥ ✈➔ ❝➟♥ ❞÷î✐ ♥➔② t❤æ♥❣ q✉❛ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ P (x)✳ ◆ë✐ ❞✉♥❣
❝❤÷ì♥❣ ♥➔② ✤÷ñ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✹❪✳
✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣
t❛ s➩ t➻♠ ❤✐➸✉ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❋♦✉r✐❡r ❞↕♥❣✿
(T φ)(x) =

e

ψ(x, y)φ(y)dy,

R

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


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

Ω
Ω

{(Ω , ϕ )}

R
R

ϕ

R
Ω

{Ω }

Ω, Ω ⊂ U

0 ≤ ϕ (x) ≤ 1, x ∈ Ω, j = 1, 2, ...,

Ω

ϕ (x) = 1, x ∈ Ω

ϕ ∈ C (R ), supp ϕ ⊂ Ω , j = 1, 2, ...,
{ϕ }

{Ω }

Ω

❚❛ ❝â ✤à♥❤ ❧þ s❛✉ ✈➲ ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à✳

R

{U }
{ϕ }

{U }

❚r÷î❝ ❦❤✐ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ t❛ ①➨t ❤➔♠ ρ : R → R ❧➔ ❤➔♠ ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷
s❛✉✿
ρ(x) :=

Ce
0,

,

♥➳✉ x < 1
♥➳✉ x ≥ 1

tr♦♥❣ ✤â✱ C ❧➔ ❤➡♥❣ sè s❛♦ ❝❤♦
ρ(x)dx = 1.
R

✹


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

x ≤ 1 , ρ(x) ≥ 0,

ρ(x)dx = 1,
R

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

ρ

=

ρ

x

.

❍➔♠ ρ ❝ô♥❣ ❝â ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ρ, ❝ö t❤➸ ❧➔
ρ ∈ C (R ), s✉♣♣ρ = B[0, ] = x ∈ R

x ≤

, ρ (x) ≥ 0,

ρ (x)dx = 1,
R

✈➔ ρ ❧➔ ❤➔♠ ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ x ✳ ❱î✐ ♠é✐ ❤➔♠ f ∈ L (R )✱ ✤➦t
f (x) = (f ∗ ρ ) (x) =

f (y)ρ (x − y)dy
R

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

f (x − y)ρ (y)dy =

f (y)ρ (x − y)dy.

R

f ∈ L (R )
f ∈ C (R )
supp f = K ⊂ R

f ∈ C (R ) supp f ⊂ K

K = K + B[0, ] = x ∈ R d(x, K) ≤

.

f ∈ C(R ), lim sup |f (x) − f (x)| = 0, K ⊂ R

✭✐✮ ❉➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ tø ✤➥♥❣ t❤ù❝ s❛✉
f (y)ρ (x − y)dy

D

f (y)D ρ (x − y)dy.

=

R

R

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

f (x)
R

f (y)ρ (x − y)dy
R

❱î✐ ♠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
R

✺


(f (x − y) − f (x)) p(y)dy

=

♥➯♥
|f (x) − f (x)| ≤

sup |f (x − y) − f (x)| .

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

❈❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤
K⊂R
0 ≤ ϕ(x) ≤ 1

∀x ∈ R

supp ϕ ⊂ K

ϕ(x) = 1, ∀x ∈ K

❳➨t χ(x) ❧➔ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ t➟♣ K
χ(x) :=

1, ♥➳✉ x ∈ K
0, ♥➳✉ x ∈
/K

❈â χ ∈ L (R ) ⊂ L (R ), supp χ = K
χ∗ρ

ϕ ∈ C (R )

>0

∈ C (R ), supp(λ ∗ ρ

✱ tù❝ ❧➔
,
.
.

✱ ♥➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶ ❝â
) ⊂ K , 0 ≤ (χ ∗ ρ

)(x)

∀x ∈ R .

▼➔
(χ ∗ ρ

)(x) =

χ(x − y)ρ

(y)dy

∀x ∈ R ,

♥➯♥
(χ ∗ ρ

)(x) ≤

ρ

(y)dy = 1

(χ ∗ ρ

)(x) =

ρ

(y)dy = 1, x ∈ K

✈➔
.

◆❤÷ ✈➟② ❤➔♠ ❝➛♥ t➻♠ ❧➔ ϕ(x) = χ ∗ ρ (x) ✳
❈❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤
❚ø ❣✐↔ t❤✐➳t K ❧➔ t➟♣ ❝♦♠♣❛❝t✱ {U }
K t❛ ❝â
W := K \ ∪
✻

U

⊂U

❧➔ ♠ët ♣❤õ ♠ð ❝õ❛


♥➯♥ tç♥ t↕✐

> 0 s❛♦ ❝❤♦
W ⊂ W + B(0, ) ⊂ U .

❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷✱ ❝â ❤➔♠ ψ ∈ C (R ; [0; 1]) s❛♦ ❝❤♦
V := W + B(0,

▲↕✐ ❝â✱ W := K \ ∪

) ⊂ supp ψ ⊂ W + B(0, ) ⊂ U , ψ (x) = 1, x ∈ V .

2

⊂ V ♠➔ V ❧➔ t➟♣ ♠ð ♥➯♥

U

W := K \ V ∪ ∪

❉♦ ✤â✱ tç♥ t↕✐
ψ ∈C

U

⊂U .

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

(R ; [0; 1]) s❛♦ ❝❤♦
V := W + B(0,

2

) ⊂ supp ψ ⊂ W + B(0, ) ⊂ U , ψ (x) = 1, x ∈ V .

❈ù ♥❤÷ t❤➳ t❛ ①➙② ❞ü♥❣ ✤÷ñ❝ ❞➣② ❝→❝ ❤➔♠ {ψj}
ψ ∈C

(R ; [0; 1]) , V := W + B(0,
ψ (x) = 1, x ∈ V ,

2

✈➔ ❝→❝ t➟♣ {V , W }

) ⊂ supp ψ ⊂ W + B(0, ) ⊂ U

ψ (x) > 0, x ∈ ∪

V (⊃ K) ,

✈➔
ψ (x) < N + 1, x ∈ R .

❈â K ⊂ ∪

V ♥➯♥ tç♥ t↕✐ sè

> 0 s❛♦ ❝❤♦

K ⊂ K + B(0, ) ⊂ ∪

V.

❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷ ❝â ❤➔♠ ❦❤æ♥❣ ➙♠ φ t❤ä❛ ♠➣♥
φ ∈ C (R ), K ⊂ K + B(0, /2) ⊂ s✉♣♣φ ⊂ K + B(0, ) ⊂ ∪

V,

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

✣➦t
ψ (x)

ϕ (x) :=
φ(x)

ψ (x) + (1 − φ(x)) N + 1 −
✼

t❤ä❛ ♠➣♥

ψ (x)


❝â
0 ≤ ϕ (x) ≤ 1, x ∈ K, j = 1, 2, ..., N, ϕ ∈ C (R ), supp ϕ ⊂ U , j = 1, 2, ..., N,

✈➔
ϕ (x) = 1, x ∈ K.

❈❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤

◆➳✉ f, g ∈ L (R ) t❛ ✤à♥❤ ♥❣❤➽❛
f ∗ g(x) =

f (x − y)g(y)dy =
R

f (y)g(x − y)dy
R

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

f (x − y)g(y)dy dx

R

R

|f (x − y)| dx dy ≤ f

|g(y)|

≤

R

g

R

R

R

♥➯♥ f ∗ g ∈ L (R ) ✈➔
f ∗g

R

≤ f

R

g

R

.

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

≤ f

g .

S Rn
S (R )
S (R ) = {ϕ ∈ C

(R ) : sup x D ϕ (x) < ∞
R

✽

∀α, β ∈ Z }.


❈❤♦ ❤➔♠ ϕ ∈ S (R )✱ ❦❤✐ ✤â
lim x D ϕ (x) = 0

∀α, β ∈ Z .

C (R )
S (R )
ϕ (x) = e

, x∈R

ϕ

S (R )

S (Rn )

f ∈ S (R )

f

f (ξ)

F (f ) (ξ)
F (f ) (ξ) = f (ξ) = (2π)

e

f (x) dx

R

x = (x , x , ..., x ) ∈ R , ξ = (ξ , ξ , ..., ξ ) ∈ R .
f ∈ S (R )
F

(f ) (x) = f (x) = (2π)

e

f (ξ) dξ

R

x = (x , x , ..., x ) ∈ R , ξ = (ξ , ξ , ..., ξ ) ∈ R .

❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❛ ❞➵ ❞➔♥❣ s✉② r❛✿ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✭✈➔ ♥❣÷ñ❝ ❝õ❛ ♥â✮ ❧➔ t✉②➳♥

t➼♥❤✱ ♥❣❤➽❛ ❧➔✿
F[λ f + λ f ] = λ F[f ] + λ F[f ]

✈➔
F

[λ f + λ f ] = λ F

[f ] + λ F

[f ]

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

✾


S (R )

F, F

D F () = (i) F (x (x)) () ,

D F

F () = (i) F (D (x)) () ,

F

S (R )
() = i F
() = i F

(x (x)) () .
(D (x)) () .

ờ rr ừ tở ổ
S (R ) õ
(F) () = (2)

e

(x) dx.



R

ử ỵ t t ử tở t số t õ
D (F) () ợ ồ Z
D (F) () = D

(2)

e

(x) dx

R

= (2)

(ix) e

(x) dx

R

e

= (i) (2)

x (x)dx

= (i) F (x (x)) ()

S (R ) ,

t
e

x (x) dx

S (R )

R

ở tử tt ố t tr R ồ Z
e

x (x) |x| | (x)|

S (R ) .

S (R )
|x| | (x)| dx

Z

R

ở tử tt ố t tr R
õ tỗ t D (F) () F C (R )
t ộ R , , Z õ
lim D

e

(x) = 0

S (R ) .

ỷ ử t t tứ || t ữủ
D (F) () =

(2)

e
R

(iD )

(ix) (x) dx.




◆❤÷ ✈➟②✱ ✈î✐ ♠é✐ α, β ∈ Z ✱ ❝â
ξ D (Fϕ) (ξ) = (2π)

e

(−iD )

(−ix) ϕ (x) dx,

✭✶✳✸✮

R

♥❤➟♥ t❤➜② r➡♥❣
e

(−iD )

(−ix) ϕ (x) dx

R

≤ sup D

(−x) ϕ (x)

dx

(1 + x )

R

R

(1 + x )

. ✭✶✳✹✮

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

≤ (2π)

sup D

(−x) ϕ (x)

dx

(1 + x )

R

R

|D ϕ (x)|

≤ C sup 1 + x

(1 + x )

∀α, β ∈ Z .

R

❉♦ ϕ ∈ S (R ) ♥➯♥
|D ϕ (x)| < ∞

sup 1 + x

∀α, β ∈ Z .

R

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

(−iD ) e

ϕ (x) dx

R

e

= (2π)

(−iD ) ϕ (x) dx

R

= (−i) F D ϕ (x) (ξ)

∀ϕ ∈ S (R ) .

❱➟② ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠
❣✐↔♠ ♥❤❛♥❤ S (R )✳ ✣è✐ ✈î✐ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝ F
tü✳
❈❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤
ϕ ∈ S (R )
F

Fϕ = FF

✶✶

ϕ = ϕ.

t❛ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣


ϕ, ψ ∈ S (R )
ϕ (x) Fψ (x) dx =

ψ (ξ)Fϕ (ξ) dξ

R

R

|ϕ (x)| dx =
R

|Fϕ (ξ)| dξ.
R

❙û ❞ö♥❣ ✤à♥❤ ♥❣❤➽❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝❤♦ ❤➔♠ ψ (x) tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥
❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (R )✱ ❝â
Fψ (x) = (2π)

e

ψ (ξ) dξ,

R

❦❤✐ ✤â ϕ, ψ ∈ S (R )✱ t❛ ❝â
ϕ (x) (2π)

R

ϕ (x) Fψ (x) dx.

ψ (ξ) dξ dx =

e

✭✶✳✺✮

R

R

❚÷ì♥❣ tü✱ t❛ ♥❤➟♥ ✤÷ñ❝
Fϕ (ξ) = (2π)

e

ϕ (x) dx

∀ϕ ∈ S (R ) ,

R

✈î✐ ϕ, ψ ∈ S (R )✱ ♥➯♥
ψ (ξ) (2π)
R

e

ϕ (x) dx dξ =

R

ψ (x) (Fϕ) (ξ) dξ.

✭✶✳✻✮

R

▼➦t ❦❤→❝✱ ✈î✐ ❝→❝ ❤➔♠ ϕ, ψ ∈ S (R ) t❤❡♦ ✤à♥❤ ❧þ ❋✉❜✐♥✐✱ ❝â
ϕ (x) (2π)

e

R

ψ (ξ) dξ dx

R

ψ (ξ) (2π)

=
R

e

ϕ (x) dx dξ. ✭✶✳✼✮

R

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

ψ (ξ) (Fϕ) (ξ) dξ
R

❇➡♥❣ ❝→❝❤ ❝❤♦ ❤➔♠
ψ=F

ϕ

t❛ t❤➜② r➡♥❣
F

ϕ = Fϕ,
✶✷

ϕ = Fψ

∀ϕ, ψ ∈ S (R ) .

✭✶✳✽✮


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

|Fϕ (ξ)| dξ

R

∀ϕ ∈ S (R ) .

R

◆❤÷ ✈➟②✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r F ❧➔ ♠ët ✤➥♥❣ ❝➜✉ t✉②➳♥ t➼♥❤✱ tü ❧✐➯♥ ❤ñ♣✱ ✤➥♥❣
❝ü tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (R ) ✈î✐ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ L (R )✳
❈❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤
❉÷î✐ ✤➙② t❛ s➩ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✈➲ t➼❝❤ ❝❤➟♣
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (R )✳
ϕ, ψ ∈ S (R )
F (ϕ ∗ ψ) (ξ) = (2π)
F

Fϕ (ξ) Fψ (ξ) .

(ϕ ∗ ψ) (ξ) = (2π)

F

ϕ (ξ) F

ψ (ξ) .

(2π)

F (ϕ (x) ψ (x)) (ξ) = Fϕ (ξ) ∗ Fψ (ξ) .

(2π)

F

(ϕ (x) ψ (x)) (ξ) = F

ϕ (ξ) ∗ F

ψ (ξ) .

L1 (Rn )
f ∈ L (R )

f

f (ξ)

F (f ) (ξ)
F (f ) (ξ) = f (ξ) = (2π)

e

f (x) dx

R

x = (x , x , ..., x ) ∈ R , ξ = (ξ , ξ , ..., ξ ) ∈ R .

R
R
f (y) ≤ (2π)

|f (x)| dx

∀y ∈ R .

R

❚ø ✤à♥❤ ♥❣❤➽❛ t❛ s✉② r❛
|f (y)| ≤ (2π)

|e

| |f (x)| dx.

R

❑➳t ❤ñ♣ ✤✐➲✉ ♥➔② ✈î✐ e

= 1✱ s✉② r❛

f (y) ≤ (2π)

|f (x)| dx
R

❈❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤
✶✸

∀y ∈ R .


❈❤÷ì♥❣ ✷
✣→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣
❙t❡✐♥✲❲❛✐♥❣❡r
❈❤♦ P ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ ✈î✐ ❤➺ sè t❤ü❝ ❝â ❜➟❝ ❦❤æ♥❣ ✈÷ñt q✉→ d✳ ❈❤♦
P ∈ P t❛ ①➨t ❣✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ s❛✉
I (P ) = p.v.

dt
.
t

e
R

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

d∈N

c

c log d ≤ sup p.v.

e
R

d

dx
.
x

❚r÷î❝ ❦❤✐ ✤÷❛ r❛ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ tr➯♥✱ t❛ ♥❤➢❝ ❧↕✐ ❜ê ✤➸ ❱❛♥❞❡r ❈♦r♣✉t✳
φ : [a, b] → R
t ∈ [a, b]
1

k

k=1

φ

φ

ψ

λ ∈ R,

[a, b]
e

ψ(x)dx ≤

C

ψ

|λ|
C

a, b
✶✹

φ, ψ.

+ ψ

,

(t) ≥ 1


❚✐➳♣ t❤❡♦✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❜ê ✤➲ s❛✉✳
n≥3 f
f (t) = −1

−1 +

−1, −1 +

− ,

R

≤t≤ −

f (t) = 1

≤ t ≤ 1−

|t| ≥ 1

f (t) = 0

1− , 1

c

I (f ) = p.v.

n

dt
≥ c log n.
t

e
R

✭✷✳✶✮

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

t

I(f ) = 2

✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥
sin f (t)
dt − 2
t

I (f ) ≥ 2

❚❛ t❤➜②✱ ✈î✐

≤t≤1−

sinf (t)
dt − 2
t

sin 1
dt = sin 1 log(n − 1),
t

✭✷✳✸✮

t❤➻ f (t) = nt✱ s✉② r❛
sin f (t)
dt ≤
t

✈î✐ 1 −

✭✷✳✷✮

t❤➻ f (t) = 1✱ s✉② r❛
sin f (t)
dt =
t

✈î✐ 0 ≤ t ≤

sin f (t)
dt .
t

✭✷✳✹✮

ndt = 1,

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

n(1 − t)
n
dt = n log
− 1.
t
n−1

❑➳t ❤ñ♣ ✭✷✳✷✮✱ ✭✷✳✸✮✱ ✭✷✳✹✮ ✈➔ ✭✷✳✺✮✱ t❛ t❤✉ ✤÷ñ❝
f (t)
dt − 2
t
n
= 2 sin 1 log (n − 1) − 2 − 2n log
+ 2.
n−1

I (f ) ≥ 2 sin 1 log (n − 1) − 2

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

❈❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤
✶✺

f (t)
dt
t

✭✷✳✺✮


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

φ (x) = C

x
4

✭✷✳✻✮

,

tr♦♥❣ ✤â ❤➡♥❣ sè C ✤÷ñ❝ ❝❤å♥ t❤ä❛ ♠➣♥ ✤➥♥❣ t❤ù❝
✭✷✳✼✮

φ (x) dx = 1.

❈❤ó þ r➡♥❣
1−

1= C

x
4

1
(1 − x ) dx = 2C B( , k + 1),
2

dx = 4C

ð ✤➙② B(., .) ❧➔ ❤➔♠ ❜❡t❛✳ ❙û ❞ö♥❣ ❝→❝ t➼♥❤ ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❜❡t❛✱ t❛ s✉② r❛ C ∼ k ✳
❱î✐ ❤➔♠ f ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶✱ t❛ ①➙② ❞ü♥❣ ❤➔♠ P ①→❝ ✤à♥❤ tr➯♥ R ♥❤÷
s❛✉
✭✷✳✽✮

f (x)φ (t − x) dx.

P (t) =

❘ã r➔♥❣ ❤➔♠ P ❧➔ ✤❛ t❤ù❝ ❜➟❝ ❦❤æ♥❣ ✈÷ñt q✉→ 2k ✳ ❚❛ ❝â ❜ê ✤➲ s❛✉ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t
❝õ❛ ❝→❝ ✤❛ t❤ù❝ P ✳
✭✷✳✽✮

P

2k − 1

P

a = (−1)

2C k
4

1−

1
.
n

P
P (t) = a t

+ ...

∀t ∈ R
P

(

)

(t) ≥ C 2k − 1 !

k
.
4

t ∈ [−1; 1]
P (t) =

(f (t + x) + f (t − x))φ (x) dx.

✶✻


✭✐✮ ❙û ❞ö♥❣ ✭✷✳✽✮ t❛ ❝â
f (x) φ (−t − x) dx =

P (−t) =

f (x) φ (t + x) dx

f (−x) φ (t − x) dx = −P (t) .

=

❉♦ ✤â P ❧➔ ♠ët ❤➔♠ ❧➫✳ ❍ì♥ ♥ú❛

P (t) = C

f (x)
k (−1)
m
4

=C

t−x

k
m

−

dx

4

f (x)(t − x)

dx,

✤✐➲✉ ♥➔② ❞➝♥ ✤➳♥

P (t) =

C (−1)
4

f (x) (x − t)

k (−1)
m
4

dx + C

f (x) (t − x)

dx.

❱➻ ✈➟②
P (t) = C

(−1)
4

f (x)dxt

(−1) 2k
4

−C

f (x) xdxt

✭✷✳✾✮

f (x)dx = 0✱ ❦➳t ❤ñ♣ ✤✐➲✉ ♥➔② ✈î✐ ✭✷✳✾✮✱ t❛ ♥❤➟♥ ✤÷ñ❝ ✤➥♥❣

❉♦ f ❧➔ ❤➔♠ ❧➫✱ ♥➯♥
t❤ù❝ s❛✉

P (t)= (−1)

2C k
4

1−

1
t
n

+ ....

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

❍ì♥ ♥ú❛✱ ❞♦ P

+ ...

2C k

4

1−

1
t.
n

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

(

)

(t) ≥ C 2k − 1 !

✭✐✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳

✶✼

k
4

∀t ∈ R.


✭✐✐✮ ❈è ✤à♥❤ t ∈ [−1, 1]✳ ❑❤✐ ✤â
f (t − x) φ (x) dx =

f (t − x) φ (x) χ

(x) dx

R

=

f (x)φ (t − x)χ

(t − x) dx

=

f (x) φ (t − x) dx= P (t).

❑➳t ❤ñ♣ ✈î✐ φ ❧➔ ❤➔♠ ❝❤➤♥✱ t❛ ♥❤➟♥ ✤÷ñ❝
P (t) =

f (t − x) φ (x) dx =

(f (t + x) + f (t − x))φ (x) dx.

❈❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤
f
A (x, t) = |f (t + x) + f (t − x) − 2f (t)|
A (x, t)
dtφ (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 ❤ñ♣ ✤✐➲✉ ♥➔② ✈î✐ ✭✷✳✶✵✮✱ ✭✷✳✶✶✮✱ 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,

1
1
≤t−x≤t+x≤1− .
n
n

✭✷✳✶✸✮

φ (x) dtdx t❤➔♥❤ ❜↔② t➼❝❤ ♣❤➙♥ s❛✉

❚❛ t→❝❤ t➼❝❤ ♣❤➙♥
. . . dtdx +

. . . dtdx +

. . . dtdx +

. . . dtdx +

+

. . . dtdx

. . . dtdx +

. . . dtdx.

❚❛ ✤→♥❤ ❣✐→ r✐➯♥❣ tø♥❣ t➼❝❤ ♣❤➙♥ ♥❤÷ s❛✉
A (x, t)
dtφ (x) dx ≤ 4 log 2
t
A (x, t)
dtφ (x) dx ≤
t
A(x, t)
dtφ (x) dx ≤
t
=

4nt
dtφ (x) dx =
t

4nxφ (x) dx ≤ 2,

4nt
dtφ (x) dx
t
A(x, t)
dtφ (x) dx
t

4φ (x) dx ≤ 2.

≤

φ (x) dx = 2 log 2,

4nx
dtφ (x) dx =
t

4nxlog (1 +

1
)φ (x) dx ≤ 2.
nx

✣➸ ✤→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥
A(x, t)
dtφ (x) dx
t

t❛ ♥❤➟♥ t❤➜② ✈î✐ x ∈ [0, − ], t ∈ [x + , ] t❤➻

≤ t−x ≤ t+x ≤ 1−

❝æ♥❣ t❤ù❝ ✭✷✳✶✸✮✱ t❛ ✤÷ñ❝ A(x, t) = 0. ❉♦ ✤â
A(x, t)
dtφ (x) dx = 0.
t

✶✾

✈➔ →♣ ❞ö♥❣

❚❛ ❝ô♥❣ ❝â ✤→♥❤ ❣✐→ s❛✉
A(x, t)
dtφ (x) dx≤
t

4
dtφ (x) dx
t

≤4

log (nx + 1)φ (x) dx.

❈è ✤à♥❤ α ∈ (0, 1)✳ ❘ã r➔♥❣
log (nx + 1)φ (x) dx =

≤
≤

log (nx + 1)φ (x) dx +

log n

+1
2

log n

+1

2

✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥
lim sup

log (nx + 1)φ (x) dx

+ C log (n + 1)
+ C log (n + 1) e

1−

x
4

dx

.

log(nx + 1) φ (x) dx
1−α
≤
.
logn
2

❉♦ α ❧➔ tò② þ tr♦♥❣ (0, 1)✱ ♥➯♥
A(x, t)
dtφ (x) dx= o(logn).
t

❈✉è✐ ❝ò♥❣✱
A(x, t)
dtφ (x) dx ≤
t

4
dtφ (x) dx
t

n
≤ 4 log c
2

(1 −

x
) dx ≤ cn log ne
4

= o(1).

❈❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤
❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ❧þ ✷✳✶✳
❳➨t P ❧➔ ✤❛ t❤ù❝ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ✭✷✳✽✮✱ n ≥ 3✳ ❑❤✐
✤â P ❧➔ ♠ët ✤❛ t❤ù❝ ❜➟❝ d = 2n − 1 ✈➔
sup p.v.

dx
≥ I (P )

x

e
R

✷✵

✭✷✳✶✹✮


tr♦♥❣ ✤â
I (P ) = p.v

e
R

dt
.
t

❉♦ P ❧➔ ❤➔♠ ❧➫ ♥➯♥
sin P (t)
dt .
t

I (P ) = 2

❳➨t R ≥ 1. ❙û ❞ö♥❣ ♣❤➛♥ ✭✐✮ ❝õ❛ ❜ê ✤➲ ✷✳✷ ✈➔ →♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✷✳✶ t❛ t❤✉ ✤÷ñ❝
sin P (t)
dt ≤ c ,

t

∀R ≥ 1.

❉♦ ✤â
✭✷✳✶✺✮

I(P ) ≥ I (P ) − c

tr♦♥❣ ✤â
sin P (t)
dt .
t

I (P ) =

⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✶ t❤➻ tç♥ t↕✐ ❤➡♥❣ sè c s❛♦ ❝❤♦ I(f ) ≥ c log n✳ ❉♦ ✤â
I (P ) ≥ I(f ) − |I (P ) − I(f )| ≥ c log n − |I (P ) − I(f )|.

❍❛②
✭✷✳✶✻✮

I (P ) ≥ c log n − |I (P ) − I(f )|

❍ì♥ ♥ú❛
sin P (t) − sin f (t)
dt
t

|I (P ) − I(f )| =

|P (t) − f (t)|
dt.
t

≤

❙û ❞ö♥❣ ♣❤➛♥ ✭✐✐✮ ❝õ❛ ❇ê ✤➲ ✷✳✷ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✮✱ t❛ ❝â
|P (t) − f (t)| ≤

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

✈î✐ 0 ≤ t≤ 1.

❱➻ ✈➟②
|I (P ) (t) − f (t)| ≤

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

❑➳t ❤ñ♣ ✤✐➲✉ ♥➔② ✈î✐ ❇ê ✤➲ ✷✳✸✱ t❛ s✉② r❛
|I (P ) (t) − f (t)| = o(logn).

✭✷✳✶✼✮

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

d∈N

c

sup p.v.

d

dx
≤ c log d.
x

e
R

❚r÷î❝ ❦❤✐ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ tr➯♥✱ t❛ ❝➛♥ ❦➳t q✉↔ ✈➲ t➟♣ ♠ù❝ ❞÷î✐ ❝õ❛ ♠ët ✤❛
t❤ù❝ tr♦♥❣ ❜ê ✤➲ s❛✉ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ❱✐♥♦❣r❛❞♦✈ [5]✳
h(t) = b + b t + . . . + b t

n
α

|{t ∈ [1, 2] : |h (t)| ≤ α }| ≤ c

|b |

max

.

❚➟♣ ❤ñ♣ E = {t ∈ [1, 2] : |h (t)| ≤ α } ❧➔ ❤ñ♣ ❝õ❛ ❝→❝ ✤♦↕♥ rí✐ ♥❤❛✉✳ ❚❛
❞à❝❤ ❝❤✉②➸♥ ❝→❝ ✤♦↕♥ ♥➔② ✤➸ t↕♦ ♥➯♥ ♠ët ❦❤♦↔♥❣ ✤ì♥ I ❝❤✐➲✉ ❞➔✐ |E | ✈➔ ❝❤✐❛ ✤➲✉
✤♦↕♥ I ❜➡♥❣ n + 1 ✤✐➸♠ ❝❤å♥✳ ❚✐➳♣ t❤❡♦✱ t❛ ❞à❝❤ ❝❤✉②➸♥ ❝→❝ ✤♦↕♥ ♠î✐ trð ❧↕✐ ✈à tr➼
❜❛♥ ✤➛✉ ❝õ❛ ♥â✱ t❤➻ ✭♥✰✶✮ ✤✐➸♠ ✤➣ ❝❤å♥ ❝ô♥❣ ❞à❝❤ ❝❤✉②➸♥ t❤❡♦ ✈➔ s➩ ❦➳t t❤ó❝ t↕✐
n + 1 ❝→❝ ✤✐➸♠ x , x , x , . . . , x ∈ E t❤ä❛ ♠➣♥
|x − x | ≥ |E |

|j − k|
.
n

✭✷✳✶✽✮

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

h (x )

(x − x ) (x − x ) . . . (x − x
(x − x ) (x − x ) . . . (x − x

) (x − x
) . . . (x − x )
.
) (x − x
) . . . (x − x )

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

h(x )

(−1)
σ
(x , . . . , xˆ , . . . , x )
(x − x ) (x − x ) . . . (x − x
) (x − x
) . . . (x − x )

✈î✐ k = 0, 1, . . . , n.
❚r♦♥❣ ❝æ♥❣ t❤ù❝ tr➯♥✱ σ
❝õ❛ x , . . . , xˆ

(x , . . . , xˆ

. . . , x ) ❧➔ ❤➔♠ sè ✤è✐ ①ù♥❣ ❝ì ❜↔♥ t❤ù (n − k)

ð ✤➙② x ❜à ❧÷ñ❝ ❜ä✳ ❑➳t ❤ñ♣ ✭✷✳✶✽✮ ✈➔

..., x
σ

(x , . . . , xˆ

..., x ) ≤
✷✷

n
2

n−k


n
n−k

2
≤ c√ ,
n

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

=

n
2
n−k

α
|E |

n

1
j! (n − 1)!

α
8 n
α

n
≤ c√
.
n! |E |
n n! |E |

n
2
n−k

✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥
8
max |b | ≤ c √

n
α
,
n n! |E |

❞♦ ✈➟②
|E | ≤

α

.

|b |

max

❈❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤
❚❛ ✤➦t
K =

sup

dt
.
t

e

✭✷✳✶✾✮

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

✈➔ ❜✐➸✉ ❞✐➵♥

P (t) = a t + a t + . . . + a t + a

t

+ ... + a t

= Q (t) + R (t)

ð ✤➙② Q (t) = a t + a t + . . . + a t ✈➔ R (t) = a
✣➦t |a | = max

t

+ ... + a t ✳

|a | ✈î✐ k + 1 ≤ l ≤ d✳ ❚❛ ❝â t❤➸ ❣✐↔ sû |a | = 1 ✈➔ ✈➻ ✈➟② |a | ≤ 1

✈î✐ ∀k + 1 ≤ j ≤ d✳ ❇➙② ❣✐í t→❝❤ t➼❝❤ ♣❤➙♥ ð ✭✷✳✶✾✮ t❤➔♥❤ ❤❛✐ t❤➔♥❤ ♣❤➛♥ ♥❤÷ s❛✉✿
e

dt
≤
t

e

dt
+
t

e

dt
t

✭✷✳✷✵✮

=I +I .

✷✸