Biến đổi hilbert trên không gian lp(t)

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❇Ò■ ❚❍➚ ◆❍⑨■

❇■➌◆ ✣✃■ ❍■▲❇❊❘❚ ❚❘➊◆ ❑❍➷◆● ●■❆◆ Lp(T)

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❍➔ ◆ë✐ ✲ ✷✵✶✽


❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❇Ò■ ❚❍➚ ◆❍⑨■

❇■➌◆ ✣✃■ ❍■▲❇❊❘❚ ❚❘➊◆ ❑❍➷◆● ●■❆◆ Lp(T)

❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤
▼➣ sè✿ ✽✹✻✵✶✵✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿
❚❙✳ ❇Ò■ ❑■➊◆ ❈×❮◆●

❍➔ ◆ë✐ ✲ ✷✵✶✽

▼❐❚ ❙➮ ❑Þ ❍■➏❯ ❚❖⑩◆ ❍➴❈

R

❚➟♣ t➜t ❝↔ ❝→ sè t❤ü❝

R+

❚➟♣ ❤ñ♣ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠

Rd

❑❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❊✉❝❧✐❞❡ d✲❝❤✐➲✉
❚➼❝❤ ✈æ ❤÷î♥❣ ❣✐ú❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt

·, ·
T = R/2π Z

❍➻♥❤ ①✉②➳♥ ♠ët ❝❤✐➲✉

C(T)

❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝✱
t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý 2π ✱ ♥❤➟♥ ❣✐→ trà ♣❤ù❝ tr➯♥ T

C α (T)

❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ ❍☎♦❧❞❡r✱
t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý 2π ✱ ♥❤➟♥ ❣✐→ trà ♣❤ù❝ tr➯♥ T

Lp (T)

❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❜➟❝ p ❦❤↔ t➼❝❤✱
t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý 2π ✱ ♥❤➟♥ ❣✐→ trà ♣❤ù❝ tr➯♥ T

f
T

f

Lp (T)

❈❤✉➞♥ ❝õ❛ ❤➔♠ f tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp (T)
❚➼❝❤ ♣❤➙♥ ❧➜② tr➯♥ ✤♦↕♥ ❜➜② ❦ý ❝â ✤ë ❞➔✐ 2π
▲✐➯♥ ❤ñ♣ ❝õ❛ f


▲❮■ ❈❷▼ ❒◆
✣➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ✈î✐ ✤➲ t➔✐ ✧❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr➯♥ ❦❤æ♥❣ ❣✐❛♥

Lp (T)✧✱ tr÷î❝ ❤➳t tæ✐ ①✐♥ ✤÷ñ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❝→❝ t❤➛② ❝æ
tr♦♥❣ ❦❤♦❛ ❚♦→♥ ❝ò♥❣ ❝→❝ t❤➛② ❝æ ♣❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷
♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï tæ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ q✉❛✳
❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t❤➛② ❣✐→♦ ❚❙✳ ❇ò✐ ❑✐➯♥ ❈÷í♥❣✱ ♥❣÷í✐ ✤➣
trü❝ t✐➳♣ ❤÷î♥❣ ❞➝♥✱ ❝❤➾ ❜↔♦ ✈➔ ✤â♥❣ ❣â♣ ♥❤✐➲✉ þ ❦✐➳♥ q✉þ ❜→✉ ✤➸ tæ✐ ❝â
t❤➸ ❤♦➔♥ t❤➔♥❤ ❜➔✐ ▲✉➟♥ ✈➠♥ ♥➔②✳
▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝
❝õ❛ ❜↔♥ t❤➙♥ ♥➯♥ ❝❤➢❝ ❝❤➢♥ ✤➲ t➔✐ ♥➔② ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳
❱➻ ✈➟②✱ tæ✐ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷ñ❝ sü ❝↔♠ t❤æ♥❣ ✈➔ ♥❤ú♥❣ þ ❦✐➳♥ ✤â♥❣ ❣â♣
❝õ❛ t❤➛② ❝æ✱ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ❝õ❛ tæ✐ ✤÷ñ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳

❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦

❍➔ ◆ë✐✱ t❤→♥❣ ✼ ♥➠♠ ✷✵✶✽
❍å❝ ✈✐➯♥ t❤ü❝ ❤✐➺♥

❇ò✐ ❚❤à ◆❤➔✐


▲❮■ ❈❆▼ ✣❖❆◆
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ❦➳t q✉↔ ❝õ❛ tæ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝
t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝ò♥❣ ✈î✐ sü ❣✐ó♣ ✤ï ❝õ❛ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ❝→❝ t❤➛② ❝æ
tr♦♥❣ ❦❤♦❛ t♦→♥✱ ✤➦❝ ❜✐➺t ❧➔ sü ❤÷î♥❣ ❞➝♥ t➟♥ t➻♥❤ ❝õ❛ t❤➛② ❣✐→♦ ❚❙✳ ❇ò✐
❑✐➯♥ ❈÷í♥❣✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ▲✉➟♥ ✈➠♥✱ tæ✐ ❝â t❤❛♠ ❦❤↔♦ ♥❤ú♥❣ t➔✐
❧✐➺✉ ❝â ❧✐➯♥ q✉❛♥ ✤➣ ✤÷ñ❝ ❤➺ t❤è♥❣ tr♦♥❣ ♠ö❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ▲✉➟♥
✈➠♥ ✧❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp (T)✧ ❦❤æ♥❣ ❝â trò♥❣ ❧➦♣ ✈î✐ ❝→❝
▲✉➟♥ ✈➠♥ ❦❤→❝✳

❍➔ ◆ë✐✱ t❤→♥❣ ✼ ♥➠♠ ✷✵✶✽
❍å❝ ✈✐➯♥

❇ò✐ ❚❤à ◆❤➔✐

✸


▼ö❝ ❧ö❝
▲❮■ ❈⑩▼ ❒◆
▲❮■ ❈❆▼ ✣❖❆◆

✸

▼Ð ✣❺❯

✻

✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✶✶

✶✳✶

❑❤æ♥❣ ❣✐❛♥ ✤ë ✤♦✱ ❤➔♠ ❝❤➾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✶✳✷

❑❤→✐ ♥✐➺♠ ❦❤æ♥❣ ❣✐❛♥ Lp (T)✱

2

(Z) ✈➔ W 1,1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷ ❈❍❯➱■ ❋❖❯❘■❊❘

✶✻

✷✳✶

❍➺ sè ❋♦✉r✐❡r

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷✳✷

❍➺ sè ❋♦✉r✐❡r tr♦♥❣ L1 (T) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✷✳✸

❍➺ sè ❋♦✉r✐❡r tr♦♥❣ L2 (T) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷✳✹

❍➔♠ ❝ü❝ ✤↕✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✸ ❇■➌◆ ✣✃■ ❍■▲❇❊❘❚ ❚❘➊◆ ❑❍➷◆● ●■❆◆ Lp(T)

✸✻

✸✳✶

❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr➯♥ L2 (T) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✸✳✷

P❤➙♥ t➼❝❤ ❈❛❧❞❡râ♥✲❩②❣♠✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✸✳✸

❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr➯♥ Lp (T) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✹

▼Ö❈ ▲Ö❈

❑➌❚ ▲❯❾◆

✺✵

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✺✶

✺


▼Ð ✣❺❯
✶✳ ▲þ ❞♦ ❝❤å♥ ✤➲ t➔✐
❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt t❤✉ë❝ ❧➽♥❤ ✈ü❝ ❣✐↔✐ t➼❝❤ ✤✐➲✉ ❤á❛✱ ①✉➜t ❤✐➺♥ ❧➛♥ ✤➛✉
t✐➯♥ ❦❤✐ ❉✳ ❍✐❧❜❡rt ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ❘✐❡♠❛♥♥✲❍✐❧❜❡rt✿ ❚➻♠ ❤➔♠ ❝❤➾♥❤
❤➻♥❤ ❦❤✐ ❜✐➳t ✏❜÷î❝ ♥❤↔②✑ ❝õ❛ ♥â ❦❤✐ ✤✐ q✉❛ ♠ët ✤÷í♥❣ ❝♦♥❣✳ ❙❛✉ ✤â✱ ♣❤➨♣
❜✐➳♥ ✤ê✐ ❍✐❧❜❡rt ✤➣ ✤÷ñ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠✱ ✤➦❝ ❜✐➺t✱ ♥â ❧➔
❦❤ð✐ ♥❣✉ç♥ ❝õ❛ ❧þ t❤✉②➳t t➼❝❤ ♣❤➙♥ ❦ý ❞à ❞♦ ❆✳ ❩②❣♠✉♥❞✱ ❆✳P✳ ❈❛❧❞❡râ♥
❝ô♥❣ ♥❤÷ ❙✳●✳ ▼✐❦❤❧✐♥ ✈➔ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ❦❤→❝ ①➙② ❞ü♥❣ ❧➯♥✳
❱➲ ♠➦t ✤à♥❤ ♥❣❤➽❛✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❍✐❧❜❡rt ✤ì♥ ❣✐↔♥ ❧➔ t➼❝❤ ♣❤➙♥ ✏s✉②
rë♥❣✑ ♥❤÷ s❛✉
∞

f (y)
dy,
−∞ x − y

1
H(f )(x) =
π
ð ✤â f : R → C ❧➔ ♠ët ❤➔♠ ✤õ tèt✳

❈â t❤➸ t❤➜② t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ♥➔② t❤❡♦ ❤❛✐ ♥❣❤➽❛✿ ❝â ✤✐➸♠ ❦➻ ❞à t↕✐ x
✈➔ ♠✐➲♥ ❧➜② t➼❝❤ ♣❤➙♥ ❧➔ ✈æ ❤↕♥✳
◆➳✉ ❤✐➸✉ t➼❝❤ ♣❤➙♥ tr➯♥ ♥❤÷ ❝→❝❤ ❧➜② ❣✐î✐ ❤↕♥ ❝õ❛ t➼❝❤ ♣❤➙♥ s✉② rë♥❣
t❤æ♥❣ t❤÷í♥❣ t❤➻ s➩ ❝â r➜t ➼t ❤➔♠ ✤➸ t➼❝❤ ♣❤➙♥ ♥➔② ❤ë✐ tö✳ ❈❤➥♥❣ ❤↕♥ ❤➔♠
2

f (x) = e−x ❧➔ ❤➔♠ ❦❤→ tèt ♥❤÷♥❣ ❦❤✐ t❤❛② ✈➔♦ t➼❝❤ ♣❤➙♥ tr➯♥ t❛ ❝❤➾ ✤÷ñ❝
♠ët t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ❦❤æ♥❣ ❤ë✐ tö t↕✐ ❜➜t ❝ù ✤✐➸♠ x ♥➔♦✳
◆➳✉ ❤✐➸✉ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❡♦ ❦✐➸✉ ❣✐→ trà ❝❤➼♥❤ ✭♣r✐♥❝✐♣❛❧ ✈❛❧✉❡✮✱ ♥❣❤➽❛

✻


▼Ð ✣❺❯
❧➔

H(f )(x) = lim+

→0
M →+∞

1
π

<|x−y|
f (y)
dy,
x−y

❦❤✐ ✤â✱ ✈î✐ f (·) t❤✉ë❝ ❧î♣ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤✱ ❣✐î✐ ❤↕♥ tr➯♥ ❤♦➔♥ t♦➔♥ ①→❝
✤à♥❤ t↕✐ ♠å✐ ✤✐➸♠ x✳
✣➸ t❤➜② ✤÷ñ❝ ✤✐➲✉ ♥➔② t❛ ❝â t❤➸ ♥❤➻♥ ❞÷î✐ ❣â❝ ✤ë ✏❤➔♠ s✉② rë♥❣✑ q✉❛
✈➼ ❞ö ❞÷î✐ ✤➙②✳
❳➨t ❤➔♠ s✉② rë♥❣ P V

1
x

♥❤÷ s❛✉✿

1 ∞ φ(x)
dx,
P V (x), φ =P V.
π −∞ x
1
φ(x)
= lim+
dx, ∀φ ∈ S(R).
π M→0
x
<|x|→+∞
❚ø ✤→♥❤ ❣✐→

| P V (x), φ | ≤
s✉② r❛ P V

1
x

2
π

sup |φ (x)| + sup |xφ(x)|
x∈R

x∈R

∈ S (R)✳

❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② ❜✐➳♥ ✤ê✐ ❍✐❧❜❡rt t❤ü❝ ❝❤➜t ❧➔ →♥❤ ①↕ t➼❝❤ ❝❤➟♣

f → H(f ) = P V

1
x

∗f

✈➔ ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tø S(R) ✈➔♦ E(R)✳
◆❤÷ ✈➟②✱ ❜✐➳♥ ✤ê✐ ❍✐❧❜❡rt ❧➔ t♦→♥ tû ❞↕♥❣ t➼❝❤ ❝❤➟♣✱ ❜➜t ❜✐➳♥ ✈î✐ ♣❤➨♣
❞à❝❤ ❝❤✉②➸♥✳ ✣➦❝ ❜✐➺t✱ t❛ ❝â t❤➸ ✈✐➳t ❜✐➳♥ ✤ê✐ ❍✐❧❜❡rt ♥❤÷ t♦→♥ tû ❤↕❝❤
❋♦✉r✐❡r ✈î✐ ❤↕❝❤ ❧➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r F P V

s❛✉✿

✼

1
x

∈ S (R) ✤÷ñ❝ t➼♥❤ ♥❤÷


é
S(R) t õ

1
x

F PV

,

=

1
lim+
M0
+

1
= lim+
M0

+

<|x|
(x)
dx
x

()
R

<|x|
e2ix
d
x

= i sign(), ()
s r F P V

1
x

() = i sign() õ ờ rt t

tỷ rr

H(f )(x) = F 1 (i sign()f())(x).
ợ ú t õ t ởt số t t ừ ờ
rt õ t t t tr L2 : |H(f )|L2 = |f |L2

H2 = I
t t q ờ rt tr ổ R
ữủ ự tr tt tr ữỡ tr t
t t R T t t q ỏ ữủ t
ỳ ổ
ợ ố ữủ t s ỹ ữủ sỹ ữợ
ừ ũ ữớ tổ ỹ ồ t ờ rt tr
ổ Lp (T) ự tỹ tốt

ử ự
tố õ tự ỡ ỵ tt ộ ỗ sỹ
ở tử ở tử t
r t t ỡ ừ ỵ tt ờ rt tr ổ



é
Lp (T) ợ T tr R

ử ự
ởt tờ q t ừ ử ự ở
ữỡ ự

ố tữủ ự
ố tữủ ự ộ rr sỹ t rõ
ờ rt
P ự t tr ữợ
q ố tữủ ự

Pữỡ ự
ỷ ử tự ữỡ ừ t ữỡ
ự ỵ tt t
t ự t õ q t t
tr ữợ õ q t
ỗ ữỡ
ữỡ tự
ữỡ ộ rr
ữỡ ờ rt tr ổ Lp (T)
ũ t ợ sỹ ố ừ t s tớ
õ ụ ợ ố ợ t tổ tr q
tr ổ tr ọ ỳ t sõt ổ
ữủ ỵ õ õ ừ t ổ ữủ
t ỡ



▼Ð ✣❺❯
❈✉è✐ ❝ò♥❣✱ tæ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝æ tr♦♥❣ ❦❤♦❛ ❚♦→♥
tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝æ ♣❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝✱ ✤➦❝
❜✐➺t ❧➔ t❤➛② ❚❙✳ ❇ò✐ ❑✐➯♥ ❈÷í♥❣ ✤➣ t➟♥ t➻♥❤ ❤÷î♥❣ ❞➝♥ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥
✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ▲✉➟♥ ✈➠♥✳

✶✵


❈❤÷ì♥❣ ✶
▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ✤ë ✤♦✱ ❤➔♠ ❝❤➾
❚r♦♥❣ ♠ö❝ ♥➔②✱ ❝❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ❧✐➯♥ q✉❛♥ tî✐ ❤➔♠

✤♦ ✤÷ñ❝✱ ❤➔♠ ❝❤➾✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ●✐↔ sû X ❧➔ ♠ët t➟♣ ❦❤→❝ ré♥❣✱ M ❧➔ ♠ët σ✲✤↕✐ sè
♥❤ú♥❣ t➟♣ ❝♦♥ ❝õ❛ X ✈➔ µ : M → [0, ∞] ❧➔ ♠ët ✤ë ✤♦ tr➯♥ X ✳ ❑❤✐ ✤â✱
❜ë ✭X, M, µ✮ ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦✳
❈→❝ t➟♣ t❤✉ë❝ M ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ✤♦ ✤÷ñ❝✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ●✐↔ sû f ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ t➟♣ ✤♦ ✤÷ñ❝
A ⊂ X ✳ ❑❤✐ ✤â✱ t❛ ❣å✐ f ❧➔ ❤➔♠ ✤♦ ✤÷ñ❝ ♥➳✉ ✈î✐ ♠å✐ sè a t❤➻ t➟♣
{x ∈ A : f (x) < a}
✤♦ ✤÷ñ❝✳
●✐↔ sû A ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ X ✱ ❦❤✐ ✤â ❤➔♠


 1, ♥➳✉ x ∈ A

1A (x) =


 0,

✶✶

♥➳✉ x ∈
/ A.


ữỡ
ữủ ồ ừ t A

ổ Lp(T)

2

(Z)

W 1,1

r ử ú tổ ởt số ổ ỡ ữủ sỷ
ử tr

sỷ H ổ t t tr trữớ số ự
C õ

ã, ã : H ì H C
ữủ ồ t ổ ữợ tọ
x, y = y, x ,

x, y H

ax + by, z = a x, z + b y, z , a, b C, x, y, z H
x, x 0, x H, x, x = 0 x = 0
ồ H ũ ợ t ổ ữợ tr ổ t rt
ồ ổ t rt H ổ rt õ ũ ợ


x =

x, x

t ổ

ổ rt H H A H ởt
số tỹ trữợ õ f : A H ữủ ồ tử
r tr A H tỗ t số K 0 s ợ ồ x1 , x2 A
t õ

f (x1 ) f (x2 ) K x1 x2




.


❈❤÷ì♥❣ ✶✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ ❍☎
♦❧❞❡r tr➯♥ A ❦➼ ❤✐➺✉ ❧➔ C α [A]✳
❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ♥➯✉ ❧↕✐ ❦❤→✐ ♥✐➺♠ ♠ët tr♦♥❣ ♥❤ú♥❣ ❦❤æ♥❣ ❣✐❛♥ q✉❛♥
trå♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❦❤æ♥❣ ❣✐❛♥ Lp ✳
❑➼ ❤✐➺✉ T = R/2π Z ❧➔ ❤➻♥❤ ①✉②➳♥ ♠ët ❝❤✐➲✉❀ C(T) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝
❤➔♠ ❧✐➯♥ tö❝✱ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý 2π ✱ ♥❤➟♥ ❣✐→ trà ♣❤ù❝ tr➯♥ T❀ f ❧➔ ❧✐➯♥ ❤ñ♣
♣❤ù❝ ❝õ❛ f ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❚❛ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ Lp(T)✱ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❤➔♠ sè
❣✐→ trà ♣❤ù❝✱ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý 2π ✱ ❝â ❧ô② t❤ø❛ ❜➟❝ p ❦❤↔ t➼❝❤ tr➯♥ ♠ët
✤♦↕♥ ❜➜t ❦ý ❝â ✤ë ❞➔✐ 2π ✈î✐ ❝❤✉➞♥

f

Lp (T)

1
2π

=

|f (t)|p dt

1/p

.

T

❈❤ó þ r➡♥❣ t❛ ❝â ❜❛♦ ❤➔♠ t❤ù❝ L∞ (T) ⊂ L2 (T) ⊂ L1 (T)✳

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ●✐↔ sû f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ T✳ ❑❤✐ ✤â✱ t➼❝❤
❝❤➟♣ ❝õ❛ ❤❛✐ ❤➔♠ ✤â ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔

(f ∗ g)(t) =

1
2π

f (t − τ )g(τ )dτ,

∀t ∈ T,

T

♥➳✉ t➼❝❤ ♣❤➙♥ tr➯♥ ❧➔ ❝â ♥❣❤➽❛✳
●✐↔ sû f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ Rd ✳ ❑❤✐ ✤â✱ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐
❤➔♠ ✤â ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔

(f ∗ g)(x) =

f (x − y)g(y)dy,
Rd

♥➳✉ t➼❝❤ ♣❤➙♥ tr➯♥ ❧➔ ❝â ♥❣❤➽❛✳

✶✸

∀x ∈ Rd ,


❈❤÷ì♥❣ ✶✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✣à♥❤ ❧þ ✶✳✼ ✭❇➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✱ ①❡♠ ❬✶❪✮✳ ❈❤♦ ❤❛✐ ❤➔♠ f ∈ Lp ✈➔
g ∈ L1 ✱ 0 ≤ p ≤ ∞✱

❦❤✐ ✤â
f ∗g
f ∗g

Lp (T)

≤ f

Lp (T)

Lp (Rd )

≤ f

Lp (Rd )

g

L1 (T)

g

L1 (Rd ) ,

✈î✐ ✈➳ ❜➯♥ ♣❤↔✐ ❧➔ ①→❝ ✤à♥❤ ❤ú✉ ❤↕♥✳

◆❤➟♥ ①➨t ✶✳✽✳
❬✶✳❪ ❚➼❝❤ ❝❤➟♣ ❝â t➼♥❤ ❝❤➜t ❣✐❛♦ ❤♦→♥✱ ♥❣❤➽❛ ❧➔

(f ∗ g)(x) = (g ∗ f )(x).
❬✷✳❪ ❚➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ t➼❝❤ ❝❤➟♣ tr♦♥❣ Lp (T)✿ ◆➳✉ fm → f tr♦♥❣

Lp (T), 1 ≤ p ≤ ∞ ✈➔ g ∈ L1 (T) t❤➻ fm ∗ g → f ∗ g tr♦♥❣ Lp (T)✳
❬✸✳❪ ❚➼❝❤ ❝❤➟♣ ✈î✐ ♠ët ✤❛ t❤ù❝ ❧÷ñ♥❣ ❣✐→❝ ❝❤♦ t❛ ♠ët ✤❛ t❤ù❝ ✤↕✐ sè✱
n
ijt
j=−n aj e

♥❣❤➽❛ ❧➔✱ ♥➳✉ f ∈ L1 (T) ✈➔ P (t) =

t❤➻

n

✭✶✳✶✮

aj f (j)eijt

(P ∗ f )(t) =
j=−n

✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ Lp(T)✱ ❦❤✐ ✤â t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❤➔♠
❝õ❛ Lp (T) s❛♦ ❝❤♦ ♥â ✈➔ ✤↕♦ ❤➔♠ ②➳✉ ❝➜♣ k ❝õ❛ ♥â ✤➲✉ t❤✉ë❝ Lp (T) ✤÷ñ❝
❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ W k,p (T)✳
❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ W k,p ❝ò♥❣ ✈î✐ ❝❤✉➞♥ tü ♥❤✐➯♥ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
1
p

f

W k,p

Dα f

=
α

❧➟♣ t❤➔♥❤ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳

✶✹

p
Lp (T)


❈❤÷ì♥❣ ✶✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✳ ❑❤æ♥❣ ❣✐❛♥

2

(Z) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② ❜➻♥❤ ♣❤÷ì♥❣

❦❤↔ tê♥❣ tr➯♥ Z✱ ❝â ♥❣❤➽❛ ❧➔ ♥➳✉ {un }n∈Z t❤➻
∞

|un |2 < +∞.
n=−∞

✣à♥❤ ♥❣❤➽❛ ✶✳✶✶✳ ❑❤æ♥❣ ❣✐❛♥ C α(T) ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ ❍☎♦❧❞❡r tr➯♥ T✱
❣ç♠ t➜t ❝↔ ❝→❝ ❤➔♠ f ∈ C(T) t❤ä❛ ♠➣♥ ❝â ❤➡♥❣ sè A > 0 s❛♦ ❝❤♦

|f (t) − f (τ )| ≤ A|t − τ |α ✈î✐ ♠å✐ t, τ ∈ T ♠✐➵♥ ❧➔ |t − τ | ≤ 2π ✳

✶✺


❈❤÷ì♥❣ ✷
❈❍❯➱■ ❋❖❯❘■❊❘

✷✳✶ ❍➺ sè ❋♦✉r✐❡r
◆ë✐ ❞✉♥❣ ♠ö❝ ♥➔② ✤÷ñ❝ tê♥❣ ❤ñ♣ tø ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ sè ❬✶❪✱ ❬✼❪✳
❈❤♦ L1 (T) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤✱ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý 2π ♥❤➟♥
❣✐→ trà ♣❤ù❝ tr➯♥ T ✤÷ñ❝ tr❛♥❣ ❜à ❝❤✉➞♥

f

L1 (T)

=

1
2π

|f (t)|dt
T

✈➔ ♣❤➨♣ tà♥❤ t✐➳♥ ❜✐➳♥ ❤➔♠ f ∈ L1 (T) t❤➔♥❤ ❤➔♠ fτ ∈ L1 (T) ✤÷ñ❝ ①→❝
✤à♥❤ ❜ð✐ fτ (t) = f (t − τ )✳ ❚ø ✤â✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❤➺ sè ❋♦✉r✐❡r ✈➔ ❝❤✉é✐
❋♦✉r✐❡r

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❈❤♦ f ∈ L1(T) ✈➔ n ∈ Z✱ t❛ ❣å✐
f (n) =

1
2π

f (t)e−int dt

✭✷✳✶✮

T

❧➔ ❤➺ sè ❋♦✉r✐❡r t❤ù n ❝õ❛ ❤➔♠ f ✳ ❑❤✐ ✤â✱ ❝❤✉é✐ ❤➻♥❤ t❤ù❝
∞

f (n)eint

S[f ] =
n=1

✤÷ñ❝ ❣å✐ ❧➔ ❝❤✉é✐ ❋♦✉r✐❡r ❝õ❛ f ✳

✶✻

✭✷✳✷✮


❈❤÷ì♥❣ ✷✳ ❈❍❯➱■ ❋❖❯❘■❊❘
❉♦ L2 (T) ⊂ L1 (T) ♥➯♥ ❤➺ sè ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f tr♦♥❣ L2 (T) ❤♦➔♥ t♦➔♥
①→❝ ✤à♥❤ ✈➔ ❝á♥ ❝â t❤➸ ✈✐➳t ❞÷î✐ ❞↕♥❣ f = f, eint ✱ tr♦♥❣ ✤â f, g =
1
2π

T f (t)g(t)dt

❧➔ t➼❝❤ ✈æ ❤÷î♥❣ tr♦♥❣ L2 (T)✳

✣à♥❤ ❧þ ✷✳✷ ❝❤♦ ❝❤ó♥❣ t❛ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➺ sè ❋♦✉r✐❡r

✣à♥❤ ❧þ ✷✳✷ ✭①❡♠❬✷❪✮✳ ❈❤♦ ❤❛✐ ❤➔♠ f, g ∈ L1(T), τ ∈ T ✈➔ ❝→❝ sè j, n ∈
Z, c ∈ C✳

❑❤✐ ✤â✿
✐✳ ❈ë♥❣ t➼♥❤ (f + g)(n) = f (n) + g(n)❀
✐✐✳ ❚➼♥❤ t❤✉➛♥ ♥❤➜t (cf )(n) = cf (n)❀
✐✐✐✳ ❍➔♠ ❧✐➯♥ ❤ñ♣ f (n) = f (−n)❀
✐✈✳ ✣❛ t❤ù❝ ❧÷ñ♥❣ ❣✐→❝ P (t) = Nn=−N aneint ❝â P (n) = an ✈î✐ |n| ≤ N
✈➔ P (n) = 0 ✈î✐ |n| > N ✳
✈✳ ❇✐➳♥ ♣❤➨♣ tà♥❤ t✐➳♥ t❤➔♥❤ ♣❤➨♣ ✤✐➲✉ ❜✐➳♥ fτ (n) = e−intf (n)❀
✈✐✳ ❇✐➳♥ ♣❤➨♣ ✤✐➲✉ ❜✐➳♥ t❤➔♥❤ ♣❤➨♣ tà♥❤ t✐➳♥ [f (t)eijt] (n) = f (n − j)❀
✈✐✐✳ ⑩♥❤ ①↕ ✿ L1(T) → l∞(Z) ❜à ❝❤➦♥ ✈î✐ |f (n)| ≤ f L (T)✳
❉♦ ✤â✱ ♥➳✉ {fm} tr♦♥❣ L1(T) ❤ë✐ tö ✈➲ f ✱ ❦❤✐ ✤â fm(n) ❤ë✐ tö ✤➲✉ t❤❡♦
n ✈➲ f (n) ❦❤✐ m → ∞✳
1

❇ê ✤➲ ✷✳✸ ✭❈æ♥❣ t❤ù❝ s❛✐ ♣❤➙♥✱ ❬✷❪✮✳ ❈❤♦ n = 0✱ ❦❤✐ ✤â
f (n) =

1
4π

f (t) − f t −
T

❈❤ù♥❣ ♠✐♥❤✳ ❇➡♥❣ ❝→❝❤ ✤ê✐ ❜✐➳♥ t → t − πn

π
n

e−int dt.

✈➔ ♥❤í t➼♥❤ ❝❤➜t t✉➛♥ ❤♦➔♥

t❤❡♦ ❝❤✉ ❦ý 2π ✱ ❝❤ó♥❣ t❛ t❤➜②

1
2π
1
=−
2π

f (t)e−in(t+ n ) dt ✈➻ e−iπ = −1
π

f (n) = −

T

f t−
T

✶✼

π −int
e dt.
n

✭✷✳✸✮

❈❤÷ì♥❣ ✷✳ ❈❍❯➱■ ❋❖❯❘■❊❘
❚ø ✭✷✳✸✮ ✈➔ ✭✷✳✶✮ t❛ s✉② r❛

1
1
1
f (n) = f (n) + f (n) =
2
2
4π

f (t) − f t −
T

π
n

e−int dt.

❇ê ✤➲ ✷✳✹ s❛✉ ❝❤♦ t❛ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ♣❤➨♣ tà♥❤ t✐➳♥

❇ê ✤➲ ✷✳✹✳ ❈è ✤à♥❤ f ∈ Lp(T), 1 ≤ p < ∞✳ ❑❤✐ ✤â →♥❤ ①↕
φ : T → Lp (T)
τ → fτ

❧➔ ❧✐➯♥ tö❝✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû τ0 ∈ T ✈➔ g ∈ C(T)✱ ❦❤✐ ✤â✱ t❛ ❝â ✤→♥❤ ❣✐→
fτ − fτ0

Lp (T)

≤ fτ − gτ
=2 f −g

Lp (T)
Lp (T)

+ gτ − gτ0
+ gτ − gτ0

Lp (T)

+ gτ0 − fτ0

Lp (T)

Lp (T) .

❉♦ g ❧✐➯♥ tö❝ ✤➲✉✱ t❛ ❝❤♦ τ → τ0 s✉② r❛ gτ − gτ0 → 0 ❦❤✐ τ → 0✳ ❉♦
t➼♥❤ trò ♠➟t ❝õ❛ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tr♦♥❣ Lp (T), 1 ≤ p < ∞✱ ❤✐➺✉ sè f − g
❝â t❤➸ ❜➨ tò② þ✳ ❱➻ ✈➟②

lim sup fτ − fτ0
τ →τ0

Lp (T)

→ 0.

❍➺ q✉↔ ✷✳✺✳ ❬❇ê ✤➲ ❘✐❡♠❛♥♥ ✲ ▲❡❜❡s❣✉❡❪ f (n) → 0 ❦❤✐ |n| → ∞✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✸✱ t❛ s✉② r❛
f (n) ≤ f − f nπ
✶✽

L1 (T) .


❈❤÷ì♥❣ ✷✳ ❈❍❯➱■ ❋❖❯❘■❊❘
❈❤♦ |n| → ∞ ❦➳t ❤ñ♣ ✈î✐ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ♣❤➨♣ tà♥❤ t✐➳♥ tr♦♥❣ ❇ê ✤➲ ✷✳✹
s✉② r❛ f − f nπ → f − f0 = 0✱ ✈➻ f0 = f ✳ ❙✉② r❛ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳

✣à♥❤ ❧þ ✷✳✻✳ ◆➳✉ f ∈ C α(T), 0 < α ≤ 1✱ t❤➻ f (n) = O

1
|n|α

✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ s❛✐ ♣❤➙♥ tr♦♥❣ ❇ê ✤➲ ✷✳✸ t❛ ❝â
1
4π

f (n) =

f (t) − f t −
T

π
n

e−int dt.

❉♦ ✤â

f (n) ≤

1
π
A
4π n

α

2π =

const
.
|n|α

✣à♥❤ ❧þ ✷✳✼✳ ◆➳✉ f ❧➔ ❤➔♠ ❧✐➯♥ tö❝ t✉②➺t ✤è✐ (f ∈ W 1,1(T))✱ t✉➛♥ ❤♦➔♥
1
t❤❡♦ ❝❤✉ ❦➻ 2π✱ t❤➻ f (n) = o( n1 ) ✈➔ |f (n)| ≤ |n|

f

L1 (T)

✳

❈❤ù♥❣ ♠✐♥❤✳ ❚➼♥❤ ❧✐➯♥ tö❝ t✉②➺t ✤è✐ ❝õ❛ f ❝â ♥❣❤➽❛ ❧➔ ❤➔♠ f ❦❤↔ ✈✐ ❤➛✉
❦❤➢♣ ♥ì✐ ✈➔ t❛ ❝â
t

f (t) = f (0) +

f (τ )dτ,
0

tr♦♥❣ ✤â f ∈ L1 (T)✳
❙û ❞ö♥❣ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥✱ t❛ ❝â

f (n) =

1
2π

f (t)e−int dt =
T

1
2π

e−int
f (t)dt.
in
T

⑩♣ ❞ö♥❣ ❜ê ✤➲ ❘✐❡♠❛♥♥ ✲ ▲❡❜❡s❣✉❡ ✭❍➺ q✉↔ ✷✳✺✮ ❝❤♦ f ✱ t❛ s✉② r❛

f (n) =

1
1
1
f (n) = o(1) = o
,
in
in
n

✈î✐

|f (n)| ≤

1
1
f (n) ≤
f
|n|
|n|

✶✾

L1 (T) .


❈❤÷ì♥❣ ✷✳ ❈❍❯➱■ ❋❖❯❘■❊❘
❚➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ k ❧➛♥ t❛ ✤÷ñ❝ ✣à♥❤ ❧þ ✷✳✽ ❞÷î✐ ✤➙②

✣à♥❤ ❧þ ✷✳✽✳ ●✐↔ sû f ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❜➟❝ k (f ∈ W k,1(T))✱ t✉➛♥ ❤♦➔♥ t❤❡♦
❝❤✉ ❦➻ 2π✳ ❑❤✐ ✤â f (n) = o( |n|1 ) ✈➔ |f (n)| ≤ |n|1 k
k

f (k)

L1 (T)

✳

❈❤ó þ ✷✳✾✳ ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ✈➔ ✤ë ❞♦
▲❡❜❡s❣✉❡✲❙t✐❡❧t❥❡s df (t) t❤❛② ❝❤♦ f (t)dt✱ ❝❤ó♥❣ t❛ ❝ô♥❣ ❝â ❝→❝ ❦➳t q✉↔
t÷ì♥❣ tü ❝❤♦ ❧î♣ ❤➔♠ ❜✐➳♥ ♣❤➙♥ ❜à ❝❤➦♥✳
❈→❝ ❦➳t q✉↔ s❛✉ ❝❤♦ ❝❤ó♥❣ t❛ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➺ sè ❋♦✉r✐❡r ❝õ❛
t➼❝❤ ❝❤➟♣ ✭①❡♠ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✮✿

✣à♥❤ ❧þ ✷✳✶✵✳ ◆➳✉ f ∈ Lp(T), 1 ≤ p ≤ ∞ ✈➔ g ∈ L1(T) t❤➻ f ∗ g ∈ Lp(T)
✈î✐

f ∗g

Lp (T)

≤ f

Lp (T)

g

L1 (T)

✭✷✳✹✮

✈➔

f ∗ g(n) = f (n)g(n),

n ∈ Z.

❍ì♥ ♥ú❛✱ ♥➳✉ f ∈ C(T) ✈➔ g ∈ L1 (T) t❤➻ f ∗ g ∈ C(T)✳

❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✱ ✤à♥❤ ❧➼ ✶✳✼✱ t❛ s✉② r❛ ✭✷✳✹✮✳ ❍ì♥
♥ú❛✱ t❤❡♦ ✤à♥❤ ❧þ ❋✉❜✐♥✐ s✉② r❛

1
2π
1
=
2π

(f ∗ g)(n) =

T

T

1
2π
1
2π

f (t − τ )g(τ )dτ e−int dt
T

f (t − τ )e−in(t−τ ) dt g(τ )e−inτ dτ
T

= f (n)g(n).
❈✉è✐ ❝ò♥❣✱ ♥➳✉ f ∈ C(T) ✈➔ g ∈ L1 (T) t❤➻ f ∗ g ❧✐➯♥ tö❝ ✈➻ t❤❡♦ t➼♥❤ ❝❤➜t
❧✐➯♥ tö❝ ✤➲✉ ❝õ❛ f ✱ (f ∗ g)(t + δ) → (f ∗ g)(t) ❦❤✐ δ → 0✳
✷✵


ữỡ

ỵ tờ tr Lp(T) C(T) {kn} ởt
tờ


n=1 kn

<

f Lp(T), 1 p < t

tr Lp(T), n .

kn f f

ữỡ tỹ f C(T) t kn f f tr C(T)

ỵ tờ t sỷ r {kn} ởt
tờ f L1(T) t0 T sỷ r {kn} tọ
tt L t tr limn sup<|t| |kn(t)| = 0 (0; )
f L(T) õ
f tử t t0 t (kn f )(t0) f (t0) n
t ố ợ {kn} tự kn(t) =
kn (t)
f (t0 + h) + f (t0 h)
h0
2

L = lim

tỗ t t
(kn f )(t0 ) L

n .

số rr tr L1(T)
ổ A(T) ừ ổ L1(T) ữủ


A(T) = {f L1 (T) :

|f (n)| < }
nZ

ợ

f

A(T)

= f

1 (Z)

|f (n)|.

=
nZ




ữỡ

ú ỵ


: A(T)

1

(Z) ởt s t t

A(T) ổ

ừ a, b

1

(Z) ữủ



a b(n) =

a(m)b(n m), n Z.
mZ

ụ tữỡ tỹ ố ợ trữớ ủ L1 (T) t ụ õ

ab

1 (Z)

a

1 (Z)

b

1 (Z)



f, g A(T) t f g A(T)

f g = f g,
ỗ tớ

fg

A(T)

f

A(T)

g

A(T) .

ỵ ỵ r f A(T) f (t) = 0 ợ ồ t T
t 1/f A(T).

số rr tr L2(T)
sỷ H ổ rt ợ t ổ ữợ ữủ u, v
u =

u, u

ởt {un}nZ tr H õ t tỷ tờ



❈❤÷ì♥❣ ✷✳ ❈❍❯➱■ ❋❖❯❘■❊❘
❤ñ♣ ✈➔ t♦→♥ tû ❣✐↔✐ t➼❝❤ ❝õ❛ ❞➣② {un } ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿

❚♦→♥ tû tê♥❣ ❤ñ♣ S :

2

(Z) → H

{cn }n∈Z →

cn un
n

✈➔

❚♦→♥ tû ♣❤➙♥ t➼❝❤ T : H →

2

(Z)

u → { u, un }n∈Z .

✣à♥❤ ❧þ ✷✳✶✽✳ ◆➳✉ t♦→♥ tû ♣❤➙♥ t➼❝❤ T ❜à ❝❤➦♥✱ ❝â ♥❣❤➽❛ ❧➔
| < u, un > |2 ≤ (const .) u

2

✈î✐ ♠å✐ u ∈ H

n

t❤➻ t♦→♥ tû tê♥❣ ❤ñ♣ S ❝ô♥❣ ❜à ❝❤➦♥ ✈➔ ❝❤✉é✐

S({cn }) =

cn un

❤ë✐ tö ❦❤æ♥❣ ✤✐➲✉ ❦✐➺♥✱ ❝â ♥❣❤➽❛ ❧➔✱ ✈î✐ A ⊂ Z t❤➻ S({cn }n∈Z )−S({cn }n∈A )
❤ë✐ tö ✈➲ 0 ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ ❝→❝❤ ♠ð rë♥❣ ❝õ❛ A r❛ t♦➔♥ Z✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❣✐↔ t❤✐➳t T
T∗ :

2

❜à ❝❤➦♥✱ s✉② r❛ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ ♥â

(Z) → H ❜à ❝❤➦♥✳ ▲➜② ❞➣② ❜➜t ❦ý {cn }✱ u ∈ H ✈➔ N ≥ 1✱ ❝❤ó♥❣ t❛

❝â
N
T ∗ {cn }N
n=−N , u = {cn }n=−N , T u

2

N

=

cn < u, un >
n=−N
N

=

cn un , u .
n=−N

✷✸