Giải gần đúng hệ phương trình tích phân kì dị của một hệ phương trình cặp tích phân fourier (LV thạc sĩ)

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

◆●➷ ❚❍➚ ❚❍❆◆❍

●■❷■ ●❺◆ ✣Ó◆●
❍➏ P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆ ❑➐ ❉➚
❈Õ❆ ▼❐❚ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❈➄P
❚➑❈❍ P❍❹◆ ❋❖❯❘■❊❘
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

◆●➷ ❚❍➚ ❚❍❆◆❍

●■❷■ ●❺◆ ✣Ó◆●
❍➏ P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆ ❑➐ ❉➚
❈Õ❆ ▼❐❚ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❈➄P
❚➑❈❍ P❍❹◆ ❋❖❯❘■❊❘
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
❍÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝
❚❙✳ ◆●❯❨➍◆ ❚❍➚ ◆●❹◆

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


✐

▲í✐ ❝❛♠ ✤♦❛♥
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣
t❤ü❝ ✈➔ ❦❤æ♥❣ trò♥❣ ❧➦♣ ✈î✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ❚æ✐ ❝ô♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣
♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷ñ❝ ❝↔♠ ì♥ ✈➔ ❝→❝
t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤÷ñ❝ ❝❤➾ rã ♥❣✉ç♥ ❣è❝✳
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✺
◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

◆❣æ ❚❤à ❚❤❛♥❤




ớ ỡ
t ữủ ởt tổ ổ ữủ
sỹ ữợ ú ù t t ừ ổ
t tọ ỏ t ỡ s s ổ ỷ ớ tr
t ừ tổ ố ợ ỳ ổ tổ
ổ t ỡ trữớ ồ ữ
ồ ũ Pỏ ự ừ trữớ ồ
ữ ồ trữớ ồ ữ
ỵ ổ ợ ồ trữớ
ồ ữ ồ t t tr t ỳ
tự qỵ ụ ữ t tổ t õ ồ
ổ ỷ ớ ỡ tợ trữớ r ồ ờ tổ P ổ

t ỡ ỡ tổ ổ t t tổ t õ
ồ ổ ỡ ỳ ữớ t ổ ở
ộ trủ t ồ tổ tr sốt q tr ồ t
tỹ tr trồ ỡ
t
ữớ t

ổ


✐✐✐

▼ö❝ ❧ö❝
▲í✐ ❝❛♠ ✤♦❛♥

✐

▲í✐ ❝↔♠ ì♥

✐✐

▼ö❝ ❧ö❝

✐✐✐

▼ð ✤➛✉

✶

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✸

✶✳✶ ▲î♣ ❤➔♠ ❍♦❧❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ ●✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✶ ●✐→ trà ❝❤➼♥❤ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✷ ●✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✳ ✳ ✳
✶✳✸ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ❦ý ❞à tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ L2ρ
✶✳✸✳✶ ❑❤æ♥❣ ❣✐❛♥ L2ρ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✷ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ❦ý ❞à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✹ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ❧♦↕✐ ♠ët ✳ ✳ ✳ ✳
✶✳✺ ❈→❝ ✤❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✺✳✶ ✣❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ❧♦↕✐ ♠ët ✳ ✳ ✳ ✳ ✳
✶✳✺✳✷ ✣❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ❧♦↕✐ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻ ❍➺ ✈æ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤

✳
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✳ ✸
✳ ✺
✳ ✺
✳ ✺
✳ ✻
✳ ✻
✳ ✼
✳ ✼
✳ ✽
✳ ✽
✳ ✶✵
✳ ✶✷


✐✈

✶✳✼ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤ ✳ ✳ ✳ ✳ ✳
✶✳✼✳✶ ❑❤æ♥❣ ❣✐❛♥ S ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤ ✳
✶✳✼✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✽ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠ ✳ ✳ ✳ ✳ ✳
✶✳✽✳✶ ❑❤æ♥❣ ❣✐❛♥ S ❝õ❛ ❝→❝ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠
✶✳✽✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠ ✳
✶✳✽✳✸ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ t➼❝❤ ❝❤➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✾ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✾✳✶ ❑❤æ♥❣ ❣✐❛♥ H s(R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✾✳✷ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ Hos(Ω), Ho,os (Ω), H s(Ω) ✳ ✳ ✳ ✳ ✳

✶✳✾✳✸ ✣à♥❤ ❧þ ♥❤ó♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✵ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈❡❝tì ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✵✳✶ ❑❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✶ P❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✷ ❚♦→♥ tû ❣✐↔ ✈✐ ♣❤➙♥ ✈❡❝tì ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✶✹
✶✹
✶✹
✶✺
✶✺
✶✻
✶✼
✶✼
✶✼
✶✽
✶✾
✶✾
✶✾
✷✶
✷✷

✷ ●✐↔✐ ❣➛♥ ✤ó♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ❝õ❛ ♠ët ❤➺
♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r
✷✹
✷✳✶ ❚➼♥❤ ❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r
✷✳✶✳✶ P❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✷ ✣÷❛ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✳ ✳ ✳

✷✳✶✳✸ ❚➼♥❤ ❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥
✭✷✳✶✵✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✹ ✣÷❛ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ❤➺ ♣❤÷ì♥❣
tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ♥❤➙♥ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✺ ✣÷❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ♥❤➙♥ ❈❛✉❝❤②
✈➲ ❤➺ ✈æ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✳ ✳

✷✹
✷✹
✷✺
✷✻
✷✾
✸✸


✈

✷✳✷ ●✐↔✐ ❣➛♥ ✤ó♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ❝õ❛ ♠ët ❤➺
♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✷✳✷✳✶ ✣÷❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✈➲ ❞↕♥❣ ❦❤æ♥❣
t❤ù ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✷✳✷✳✷ ❚➼♥❤ ❣➛♥ ✤ó♥❣ ♥❣❤✐➺♠ ❝õ❛ ♠ët ❤➺ ♣❤÷ì♥❣ tr➻♥❤
t➼❝❤ ♣❤➙♥ ❦ý ❞à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✻✵

ỵ tt ữỡ tr t ữủ
t ỷ t r t t
ồ q t ú ữỡ tr t
b
a

(t)
dt +
xt

b

(t)K(x, t)dt = f (x),
a



tr õ f (x) K(x, t) ỳ t (t) t
K(x, t) tữớ tử tr ỳ t
S = {(x, t) : (x, t) [a, b] ì [a, b]}.

Pữỡ tr t t tr t ộ
ủ ừ t t ố ợ ổ trỡ ữ t
t ựt t r t t ú ừ tt ỗ
ữỡ ú ữỡ tr t
ỗ ữỡ ữỡ trỹ t ữỡ ở s
ữỡ r ữỡ s tự tỹ ữỡ
tự trỹ ởt số ữỡ tr t ữủ

tỹ tữỡ tỹ ữỡ tr t ữỡ tr
t ữủ ờ tứ ữỡ tr t
ồ q t ự t
ữủ ừ ởt số ữỡ tr t rr t
t ộ ủ ừ ữỡ tr ỏ ữỡ tr
s ỏ ợ ố ữủ t ữỡ tr t
ú ữỡ tr t ú tổ ồ
t ú ữỡ tr t ừ ởt ữỡ
tr t t rr t




t ỗ ữỡ ở
ữỡ ởt tr tờ q ởt số tự ỡ ợ
r t tr ừ t t tỷ t
tr ổ L2 ữỡ tr t ổ
ữỡ tr số t t tự s ờ rr
ừ ỡ ờ rr ừ s rở
t ổ ổ tỡ
t t tử t tỷ tỡ
ữỡ tr t q ừ ử tr
t ữủ ừ ữỡ tr t t
t ộ ủ ừ ữỡ tr ỏ
tr t tỗ t t ừ ữỡ
tr t rr ữ ữỡ tr t rr
ữỡ tr t s õ ữ ữỡ
tr t ổ ữỡ tr số
t t ử ú tổ tỹ ú ữỡ tr
t ừ ữỡ tr t rr ợ

ữợ ữ ữỡ tr t ổ tự
t ú tr ừ ữỡ tr t tỹ
ú ổ ữỡ tr số t t ữủ
t ửt s õ t ú ừ ữỡ tr
t
ữủ t t trữớ ồ ữ
ữợ sỹ ữợ ồ ừ ữủ
tọ ỏ t ỡ t s s t tợ ổ ữợ
trữớ ồ ữ ồ t ồ
t ủ t t ữủ ồ ừ


✸

❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ▲î♣ ❤➔♠ ❍♦❧❞❡r
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❬✸❪✳ ●✐↔ sû L ❧➔ ✤÷í♥❣ ❝♦♥❣ trì♥ ✈➔ ϕ(ξ) ❧➔ ❤➔♠ ❝→❝

✤✐➸♠ ♣❤ù❝ ξ ∈ L. ◆â✐ r➡♥❣ ❤➔♠ ϕ(ξ) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ✭✤✐➲✉
❦✐➺♥ Hλ✮ tr➯♥ ✤÷í♥❣ ❝♦♥❣ L ♥➳✉ ✈î✐ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý ξ1, ξ2 ∈ L t❛ ❝â ❜➜t
✤➥♥❣ t❤ù❝
λ
|ϕ(ξ2 ) − ϕ(ξ1 )| < A |ξ2 − ξ1 | ,
✭✶✳✶✮
tr♦♥❣ ✤â A, λ ❧➔ ❝→❝ ❤➡♥❣ sè ❞÷ì♥❣✳
◆➳✉ λ > 1 t❤➻ tø ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✮ s✉② r❛ ϕ (ξ) ≡ 0 tr➯♥ L ✈➔ ❞♦ ✤â
ϕ(ξ) ≡ const, ξ ∈ L. ❱➻ ✈➟② t❛ ❧✉æ♥ ❧✉æ♥ ❝❤♦ r➡♥❣ 0 < λ ≤ 1. ◆➳✉ λ = 1
t❤➻ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r trð t❤➔♥❤ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③✳ ❘ã r➡♥❣ λ ❝➔♥❣ ♥❤ä t❤➻
❧î♣ ❤➔♠ Hλ ❝➔♥❣ rë♥❣✳ ▲î♣ ❤➔♠ ❍♦❧❞❡r ❤➭♣ ♥❤➜t ❧➔ ❧î♣ ❤➔♠ ▲✐♣s❝❤✐t③✳

❉➵ t❤➜② r➡♥❣✱ ♥➳✉ ❝→❝ ❤➔♠ ϕ1(ξ), ϕ2(ξ) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r
t÷ì♥❣ ù♥❣ ✈î✐ ❝→❝ ❝❤➾ sè λ1, λ2✱ t❤➻ tê♥❣✱ t➼❝❤ ✈➔ ❝↔ t❤÷ì♥❣ ✭✈î✐ ✤✐➲✉
❦✐➺♥ ♠➝✉ t❤ù❝ ❦❤→❝ ❦❤æ♥❣✮ ❝ô♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ✈î✐ ❝❤➾ sè
λ = min(λ1 , λ2 )✳
◆➳✉ ❤➔♠ ϕ(ξ) ❝â ✤↕♦ ❤➔♠ ❤ú✉ ❤↕♥ tr➯♥ L t❤➻ ♥â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
▲✐♣s❝❤✐t③✳ ✣✐➲✉ ♥➔② ✤÷ñ❝ s✉② r❛ tø ✣à♥❤ ❧þ ✈➲ sè ❣✐❛ ❤ú✉ ❤↕♥✳ ◆❣÷ñ❝ ❧↕✐
♥â✐ ❝❤✉♥❣ ❦❤æ♥❣ ✤ó♥❣✳ ❚❤➼ ❞ö✱ ❤➔♠
ϕ(ξ) = |ξ|, ξ ∈ R,

t❤✉ë❝ ❧î♣ ❤➔♠ ❍♦❧❞❡r tr➯♥ R✱ ♥❤÷♥❣ ❦❤æ♥❣ ❝â ✤↕♦ ❤➔♠ t↕✐ ξ = 0✳


✹

❱➼ ❞ö ✶✳✶✳✷✳ ❍➔♠ sè ϕ(x)

√

t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ✈➔ ❝❤➾
sè λ = 1/2 tr➯♥ ♠å✐ ❦❤♦↔♥❣ ❝õ❛ trö❝ t❤ü❝✳ ◆➳✉ ♥❤÷ ❦❤♦↔♥❣ ✤â ❦❤æ♥❣
❝❤ù❛ ❣è❝ tå❛ ✤ë t❤➻ ϕ(x) ❝á♥ ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤✱ ❞♦ ✤â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
▲✐♣s❝❤✐t③✳
❱➼ ❞ö ✶✳✶✳✸✳ ❳➨t ❤➔♠ sè
ϕ(x) =

=




x

1
,
lnx

0 < x ≤ 21 ,

ϕ(0) = 0.

❉➵ t❤➜② r➡♥❣ ❤➔♠ sè ϕ(x) ❧➔ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ 0 ≤ x ≤ 12 ✳ ◆❤÷♥❣ ✈➻
limx→0+ xλ lnx = 0, ∀λ > 0,

♥➯♥ ✈î✐ ♠å✐ A ✈➔ λ ❝â t❤➸ t➻♠ ✤÷ñ❝ ❣✐→ trà ❝õ❛ x s❛♦ ❝❤♦
|ϕ(x) − ϕ(0)| =

1
> Axλ .
lnx

◆❤÷ ✈➟②✱ ❤➔♠ ϕ(x) tr➯♥ ✤♦↕♥ ♥â✐ tr➯♥ ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ❬✸❪✳ ❑➼ ❤✐➺✉ Hα(r), 0 < α ≤ 1, r ≥ 0 ❧➔ ❧î♣ ❤➔♠ ①→❝
✤à♥❤ tr➯♥ ✤♦↕♥ [a, b] ❝â ✤↕♦ ❤➔♠ ❝➜♣ r t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ✈î✐ sè
♠ô α✳
❑❤→✐ ♥✐➺♠ ✈➲ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ❝â t❤➸ ♠ð rë♥❣ ❝❤♦ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ✈î✐
sè ❜✐➳♥ ❤ú✉ ❤↕♥ ❜➜t ❦ý✳ ✣➸ ✤ì♥ ❣✐↔♥ t❛ ①➨t tr÷í♥❣ ❤ñ♣ ❤➔♠ ❤❛✐ ❜✐➳♥✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ❬✸❪✳ ❍➔♠ ❤❛✐ ❜✐➳♥ ϕ(ξ, τ ) tr➯♥ D t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
❍♦❧❞❡r ♥➳✉ ✈î✐ ♠å✐ ξ1, ξ2, τ1, τ2 ∈ D ❝â ❜➜t ✤➥♥❣ t❤ù❝
|ϕ(ξ2 , τ2 ) − ϕ(ξ1 , τ1 )|

µ

ν

A |ξ2 − ξ1 | + B |τ2 − τ1 | ,

tr♦♥❣ ✤â A, B, µ, ν ❧➔ ❝→❝ ❤➔♥❣ sè ❞÷ì♥❣❀ µ, ν

1✳

◆➳✉ λ = min(µ, ν) ✈➔ C = max(A, B)✱ t❤➻
|ϕ(ξ2 , τ2 ) − ϕ(ξ1 , τ1 )|

λ

λ

C[|ξ2 − ξ1 | + |τ2 − τ1 | ].

❘ã r➔♥❣ ❧➔✱ ♥➳✉ ϕ(ξ, τ ) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r t❤❡♦ ❤é♥ ❤ñ♣ (ξ, τ )
t❤➻ ♥â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r t❤❡♦ ξ ✤➲✉ t❤❡♦ τ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉
❦✐➺♥ ❍♦❧❞❡r t❤❡♦ τ ✤➲✉ t❤❡♦ ξ ✳


✺

✶✳✷ ●✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦ý ❞à
✶✳✷✳✶ ●✐→ trà ❝❤➼♥❤ ❈❛✉❝❤②

●✐↔ sû a ✈➔ b ❧➔ ❤❛✐ ✤✐➸♠ ❤ú✉ ❤↕♥✳ ❳➨t t➼❝❤ ♣❤➙♥

b

dx
(a < c < b).
x−c

a

❈❤ó♥❣ t❛ ❤➣② t➼♥❤ t➼❝❤ ♣❤➙♥ tr➯♥ ✤➙② ♥❤÷ ❧➔ t➼♥❤ t➼❝❤ ♣❤➙♥ s✉② rë♥❣✱ t❛
❝â
b

 c−

dx
= lim 1 →0, 2 →0 
x−c

a

1

dx
+
x−c

a

= ln

b

c+

b−c
+ lim 1 →0, 2 →0 ln
c−a

1



dx 
x−c

2

.

2

✭✶✳✷✮

●✐î✐ ❤↕♥ ❝õ❛ ❜✐➸✉ t❤ù❝ ❝✉è✐ ❝ò♥❣ tr♦♥❣ ✭✶✳✷✮ rã r➔♥❣ ❧➔ ♣❤ö t❤✉ë❝ ✈➔♦
❝→❝❤ t✐➳♥ ✤➳♥ ✵ ❝õ❛ 1 ✈➔ 2✳ ❱➻ ✈➟② t➼❝❤ ♣❤➙♥ ✭✶✳✷✮ ❦❤æ♥❣ tç♥ t↕✐ ♥➳✉ ①➨t
♥â ♥❤÷ ♠ët t➼❝❤ ♣❤➙♥ s✉② rë♥❣✳ ❚➼❝❤ ♣❤➙♥ tr➯♥ ✤÷ñ❝ ❣å✐ ❧➔ t➼❝❤ ♣❤➙♥ ❦ý
❞à✳ ❚✉② ♥❤✐➯♥✱ ♥➳✉ 1 = 2 t❤➻ tø ✭✶✳✷✮ t❛ ❝â ❦❤→✐ ♥✐➺♠ ✈➲ ❣✐→ trà ❝❤➼♥❤ ❝õ❛
t➼❝❤ ♣❤➙♥ ❦ý ❞à s❛✉✿
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❬✸❪✳ ●✐→ trà ❝❤➼♥❤ t❤❡♦ ❈❛✉❝❤② ❝õ❛ t➼❝❤ ♣❤➙♥ ❦ý ❞à
b

dx
(a < c < b),
x−c

a

✤÷ñ❝ ❤✐➸✉ ❧➔
b

 c−

dx
= lim →0 
x−c

a

b

dx
+
x−c

a

dx 
b−c
= ln
.

x−c
c−a

c+

✶✳✷✳✷ ●✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦ý ❞à

❳➨t t➼❝❤ ♣❤➙♥

b

a



ϕ(x)dx
(a < c < b),
x−c




tr õ (x) tọ r tr [a, b] ú t
ờ t tr ữ s
b

b

(x)dx
=

xc

a

b

(x) (y)
dx + (c)
xc

a

dx
.
xc



a

(x) tọ r
(x) (c)
A
,
<
1
xc
|x c|

õ t tự t tr tỗ t ữ t s rở ỏ

t tự tỗ t t ữ t õ
b

b

(x)dx
=
xc

a

bc
(x) (y)
dx + (c) ln
.
xc
ca

a

tỷ t ý tr ổ L2
ổ L

2


ợ a < x < b t trồ
(x) = (x a) (b x) , , > 1.

ỵ L2(a, b) t ừ tt u(x) ữỡ t ợ

trồ


21

b
2

(x) |u(x)| dx < .

u :=



a

ổ ữợ tr L2(a, b) ữủ ổ tự
b

(u, v) :=

(x)u(x)v(x)dx.



a

ó r ợ t ổ ữợ t L2(a, b) ởt ổ
rt

✼

✶✳✸✳✷ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ❦ý ❞à

❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ L2ρ(a, b)✱ ①➨t t♦→♥ tû
b

1
SJ [u] (x) =
iπ

u(y)dy
, x ∈ J := (a, b),
y−x

✭✶✳✻✮

a

tr♦♥❣ ✤â t➼❝❤ ♣❤➙♥ ✤÷ñ❝ ❤✐➸✉ t❤❡♦ ❣✐→ trà ❝❤➼♥❤ ❈❛✉❝❤②✳
✣à♥❤ ❧➼ ✶✳✸✳✷✳ ❬✸❪✳ ❱î✐ ρ(x) = (x − a)α(b − x)β , 1 < α, β < 1, −∞ <
a < b < ∞ t❤➻ t♦→♥ tû SJ ❜à ❝❤➦♥✱ ❞♦ ✤â ❧➔ ❧✐➯♥ tö❝ tr♦♥❣ L2ρ (a, b)✳

✶✳✹ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ❧♦↕✐ ♠ët
❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à s❛✉
1
π

b

a

ϕ(τ )
dτ = f (ξ), a < ξ < b.
τ −ξ

✭✶✳✼✮

P❤÷ì♥❣ tr➻♥❤ ✭✶✳✼✮ ❧➔ ♠ët tr÷í♥❣ ❤ñ♣ r✐➯♥❣ q✉❛♥ trå♥❣ ❝õ❛ ❝→❝ ♣❤÷ì♥❣
tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à t❤÷í♥❣ ❣➦♣ tr♦♥❣ ♥❤✐➲✉ ❜➔✐ t♦→♥ ❝ì ❤å❝ ✈➔ ❱➟t ❧þ
t♦→♥✳ ❚r♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❛ ❣✐↔ t❤✐➳t r➡♥❣ ❤➔♠ f (ξ) t❤ä❛ ♠➣♥ ✤✐➲✉
❦✐➺♥ ❍♦❧❞❡r✳ ❚ò② t❤✉ë❝ ✈➔♦ ❞→♥❣ ✤✐➺✉ ❝õ❛ ➞♥ ❤➔♠ ð ❝→❝ ✤➛✉ ♠ót ❝õ❛ ✤♦↕♥
[a, b], t❛ ❝â ❝→❝ ❝æ♥❣ t❤ù❝ ♥❣❤✐➺♠ s➙✉ ✤➙② ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳
✶✳ ◆❣❤✐➺♠ ❦❤æ♥❣ ❜à ❝❤➦♥ ð ❤❛✐ ✤➛✉ ♠ót✿


ϕ(ξ) = −

1
1
(ξ − a)(b − ξ) π



b

(τ − a)(b − τ )f (τ )
dτ + a0  ,
τ −ξ

a

✭✶✳✽✮

a < ξ < b,

tr♦♥❣ ✤â a0 ❧➔ ❤➡♥❣ sè tò② þ✳
✷✳ ◆❣❤✐➺♠ ❜à ❝❤➦♥ t↕✐ ✤➛✉ ♠ót ξ = a ✈➔ ❦❤æ♥❣ ❜à ❝❤➦♥ t↕✐ ✤➛✉ ♠ót ξ = b
✿
ϕ(ξ) = −

b

ξ−a1
b−ξπ
a

b − τ f (τ )
dτ.
τ − aτ − ξ

✭✶✳✾✮


✽

✸✳ ◆❣❤✐➺♠ ❦❤æ♥❣ ❜à ❝❤➦♥ t↕✐ t = a ✈➔ ❜à ❝❤➦♥ t↕✐ t = b ✿
ϕ(ξ) = −

b

b−ξ 1
ξ − aπ

τ − a f (τ )
dτ.
b−τ τ −ξ

✭✶✳✶✵✮

a

✹✳ ◆❣❤✐➺♠ ❜à ❝❤➦♥ t↕✐ ❤❛✐ ✤➛✉ ♠ót ✿
b

1
ϕ(ξ) = − (ξ − a)(b − ξ)
π
a

f (τ )
dτ
,
(τ − a)(b − τ ) τ − ξ

✭✶✳✶✶✮

✈î✐ ✤✐➲✉ ❦✐➺♥
b

a

f (τ )dτ
= 0.
(τ − a)(b − τ )

✭✶✳✶✷✮

✶✳✺ ❈→❝ ✤❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈
✶✳✺✳✶ ✣❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ❧♦↕✐ ♠ët

✶✳ ✣à♥❤ ♥❣❤➽❛✳❬✶✷❪✳ ✣❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ❜➟❝ n ❧♦↕✐ ♠ët Tn(x) ✤÷ñ❝ ①→❝
✤à♥❤ ♥❤÷ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥
Tn+1 (x) − 2xTn (x) + Tn−1 (x) = 0,
T0 (x) = 1,
T1 (x) = x.

◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ tr➯♥ ❧➔
Tn (x) = cos(n arccos x), Tn (cos θ) = cos(nθ), (n = 0, 1, 2, · · · ).

✷✳ ❇✐➸✉ t❤ù❝ ❤✐➸♥
[ n2 ]
n
(−1)m (n − m − 1)!
Tn (x) =
(2x)n−m .
2 m=0
m!(n − 2m)!

✾

✸✳ ❈→❝ ✤❛ t❤ù❝ ❜➟❝ t❤➜♣
T0 (x) = 1,
T1 (x) = x,
T2 (x) = 2x2 − 1,
T3 (x) = 4x3 − 3x,
T4 (x) = 8x4 − 8x2 + 1,
T5 (x) = 16x5 − 20x3 + 5x.

✹✳ ▼ët sè ❤➺ t❤ù❝
Tn (−x) = (−1)n Tn (x), Tn (1) = 1, Tn (−1) = (−1)n ,
Tn+m (x) + Tn−m (x)
,
Tn (x)Tm (x) =
2
Tn (Tm (x)) = Tnm (x).

✺✳ ❚rü❝ ❣✐❛♦
1

−1

✻✳ ❈→❝ ❤➺ t❤ù❝ ♣❤ê
1

−1

1
π




0,



m = n,
Tm (x)Tn (x)
√
dx = π, m = n = 0,

1 − x2


 π , m = n = 0.
2

Tn (y)dy
√
= πUn−1 (x),
(y − x) 1 − y 2
1

ln
−1

1 Tn (y)dy
√

= σn Tk (x),
|x − y| 1 − y 2

(n = 0, 1, 2, ...).

tr♦♥❣ ✤â Un(x) ❧➔ ✤❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ❧♦↕✐ ❤❛✐✱ ❝á♥
σn =


ln 2,

n=0

1,

n = 1, 2, ...

n

✼✳ ◆❣❤✐➺♠ ❝õ❛ Tn(x)


✶✵

❚➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ Tn(x) ✤➲✉ t❤✉ë❝ ✤♦↕♥ [−1, 1] ✈➔ ✤÷ñ❝ ①→❝ ✤à♥❤
t❤❡♦ ❝æ♥❣ t❤ù❝
xk = cos θk = cos

(2k − 1)π
, k = 1, 2, ....

2n

✽✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
(1 − x)y − xy + n2 y = 0, y = Tn (x).

✶✳✺✳✷ ✣❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ❧♦↕✐ ❤❛✐

✶✳ ✣à♥❤ ♥❣❤➽❛✳❬✶✷❪✳ ✣❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ❜➟❝ n ❧♦↕✐ ❤❛✐ Un(x) ✤÷ñ❝ ①→❝
✤à♥❤ ♥❤÷ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥
Un+1 (x) − 2xUn (x) + Un−1 (x) = 0,
U0 (x) = 1,
U1 (x) = 2x.

◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ tr➯♥ ❧➔
x = cos θ, Un (cos θ) =

sin[(n + 1)θ]
.
sin θ

✷✳ ❇✐➸✉ t❤ù❝ ❤✐➸♥
[ n2 ]

(−1)m (n − m)!
Un (x) =
(2x)n−m , n = 1, 2, ....
m!(n − 2m)!
m=0

✸✳ ❈→❝ ✤❛ t❤ù❝ ❜➟❝ t❤➜♣

U0 (x) = 1,
U1 (x) = 2x,
U2 (x) = 4x2 − 1,
U3 (x) = 8x3 − 4x,
U4 (x) = 16x4 − 12x2 + 1,
U5 (x) = 32x5 − 32x3 + 6x.


✶✶

✹✳ ▼ët sè ❤➺ t❤ù❝ ❣✐ú❛ Tn(x) ✈➔ Un(x)
Un (−x) = (−1)n Un (x),
Un (1) = n + 1, Un (−1) = (−1)n (n + 1),
Tn−m (x) − Tn+m+2 (x)
,
Un (x)Um (x) =
2(1 − x2 )
1
Tm Un (x) = [Un−m (x) + Un+m (x)],
2
d
Tn (x) = nUn−1 (x),
dx
xTn (x) − Tn+1 (x) = (1 − x2 )Un−1 (x),
Tn (x) = Un (x) − xUn−1 (x).

✺✳ ❚rü❝ ❣✐❛♦
1

√

Um (x)Un (x) 1 − x2 dx =


0,

m = n,

π,

m = n.

2

−1

✻✳ ❈→❝ ❤➺ t❤ù❝ ♣❤ê
1

√

1 − y 2 Un−1 (y)dy
= −πTn (x), (n = 1, 2, ...)
(y − x)

−1

tr♦♥❣ ✤â Tn(x) ❧➔ ✤❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ❧♦↕✐ ♠ët✳
✼✳ ◆❣❤✐➺♠ ❝õ❛ Un(x)
❚➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ Un(x) ✤➲✉ t❤✉ë❝ ✤♦↕♥ [−1, 1] ✈➔ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
❝æ♥❣ t❤ù❝ s❛✉

xk = cos θk = cos

✽✳P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥

kπ
,
n+1

k = 1, 2, ..., n.

(1 − x)y − 3xy + n(n + 2)y = 0, y = Un (x).

❚❛ ❝â ♠ët sè ❝æ♥❣ t❤ù❝ s❛✉ ❬✶✷❪✿
Tn (cos θ) = cos(nθ), Un (cos θ) =
b
a

sin(n + 1)θ
,
sin θ

Tk [η(x)] Tj [η(x)]
dx = αk δkj ,
ρ(x)

✭✶✳✶✸✮
✭✶✳✶✹✮


✶✷

b

Uk [η(x)] Uj [η(x)] ρ(x)dx = βδkj ,

✭✶✳✶✺✮

Tk [η(y)] dy
−2π
dx =
Um−1 [η(x)], k = 0, 1, ...,
(x − y)ρ(y)
b−a

✭✶✳✶✻✮

a
b
a
b
a

ρ(y)Uk−1 [η(y)] dy
π(b − a)
=
Tk [η(x)] , k = 1, 2, ...
x−y
2

π

✈î✐ δkj ❧➔ ❦➼ ❤✐➺✉ ❑r♦♥❡❝❦❡r ✈➔ αk =  π
2

✭✶✳✶✼✮

k=0
k = 1, 2, ...

π(b − a)2
2x − (a − b)
β=
, η(x) =
.
8
b−a

✶✳✻ ❍➺ ✈æ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤
❳➨t ❤➺ ✈æ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ s❛✉
∞

xi =

ci,k xk + bi , (i = 1, 2, ...),

✭✶✳✶✽✮

k=1

tr♦♥❣ ✤â xi ❧➔ ❝→❝ sè ❝➛♥ ①→❝ ✤à♥❤✱ ci,k ✈➔ bi ❧➔ ❝→❝ ❤➺ sè ✤➣ ❜✐➳t✳
✣à♥❤ ♥❣❤➽❛ ✶✳✻✳✶✳ ❬✺❪✳ ❚➟♣ ❤ñ♣ ♥❤ú♥❣ sè x1, x2, ... ✤÷ñ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠

❝õ❛ ❤➺ ✭✶✳✶✽✮ ♥➳✉ ❦❤✐ t❤❛② ♥❤ú♥❣ sè ✤â ✈➔♦ ✈➳ ♣❤↔✐ ❝õ❛ ✭✶✳✶✽✮ t❛ ❝â ❝→❝
❝❤✉é✐ ❤ë✐ tö ✈➔ t➜t ❝↔ ♥❤ú♥❣ ✤➥♥❣ t❤ù❝ ✤÷ñ❝ t❤ä❛ ♠➣♥✳ ◆❣❤✐➺♠ ✤÷ñ❝ ❣å✐
❧➔ ❝❤➼♥❤ ♥➳✉ ♥â t➻♠ ✤÷ñ❝ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ✈î✐ ❣✐→ trà
❜❛♥ ✤➛✉ ❜➡♥❣ ❦❤æ♥❣✳
✣à♥❤ ♥❣❤➽❛ ✶✳✻✳✷✳ ❬✺❪✳ ❍➺ ✈æ ❤↕♥ ✭✶✳✶✽✮ ✤÷ñ❝ ❣å✐ ❧➔ ❤➺ ❝❤➼♥❤ q✉② ♥➳✉
∞

|ci,k | < 1, (i = 1, 2, ...).

◆➳✉ ❝â t❤➯♠ ✤✐➲✉ ❦✐➺♥

k=1

∞

|ci,k |
k=1

✭✶✳✶✾✮

1 − θ < 1, 0 < θ < 1, (i = 1, 2, ...),

✭✶✳✷✵✮

t❤➻ ❤➺ ♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ❤♦➔♥ t♦➔♥ ❝❤➼♥❤ q✉②✳ ◆➳✉ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝✭✶✳✶✾✮
✭t÷ì♥❣ ù♥❣ ✱✭✭✶✳✷✵✮✮✮ ✤ó♥❣ ✈î✐ i = N + 1, N + 2, ..., t❤➻ ❤➺ ✭✶✳✶✽✮ ✤÷ñ❝ ❣å✐
❧➔ tü❛ ❝❤➼♥❤ q✉② ✭t÷ì♥❣ ù♥❣✱ tü❛ ❤♦➔♥ t♦➔♥ ❝❤➼♥❤ q✉②✮✳

ỵ



i = 1

|ci,k | , (i = 1, 2, ...).
k=1

q i > 0 t q i > 0.
sỷ q số tỹ bi tọ

|bi | Ki , (K = const > 0).

ỹ tỗ t ừ số tỹ
ừ ổ q tọ t õ õ
|xi| K õ t t ữủ ữỡ
t
ỹ t ửt x ừ q


xi =

cik xk + bi , (i = 1, 2, 3, ...),
k=1

ũ ợ số tỹ tọ |bi| Ki õ t t ữủ
ữỡ t ửt xNi ừ ỳ
N

xi =

cik xk + bi , (i = 1, 2, 3, ..., N ),
k=1

t
xi = limN xN
i .

r P q õ t õ ổ q
ởt t ổ

limi xi = 0.


✶✹

✶✳✼ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤
✶✳✼✳✶ ❑❤æ♥❣ ❣✐❛♥ S ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤

✣à♥❤ ♥❣❤➽❛ ✶✳✼✳✶✳ ❬✹✱ ✶✸✱ ✶✹❪✳ ❑➼ ❤✐➺✉ S = S(R) ❧➔ t➟♣ ❤ñ♣ ❝õ❛ ❝→❝
❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ϕ ∈ C ∞(R), t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
p
p

|Dk ϕ| < ∞, p = 0, 1, 2, ..., m,

|[ϕ]|p = sup(1 + |x|)
x∈R

k=0

tr♦♥❣ ✤â ❦➼ ❤✐➺✉ D = dxd ✳ ❉➣② {|[ϕ]|p}k ❧➔ ♠ët ❤å ❝→❝ ♥û❛ ❝❤✉➞♥✳ ❉➣②
{ϕk } ⊂ S ✤÷ñ❝ ❣å✐ ❧➔ ❤ë✐ tö ✤➳♥ ❤➔♠ ϕ ∈ S ✱ ♥➳✉ |[ϕk − ϕ]|p → 0, ❦❤✐
k → ∞; p = 0, 1, 2, ..., m. ❚➟♣ ❤ñ♣ S ✈î✐ ❤ë✐ tö tr➯♥ ✤➙② ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣
❣✐❛♥ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤✳
❱➼ ❞ö✿ ❤➔♠ ϕ(x) = e−x ∈ C ∞(R) ❧➔ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤✳
✣à♥❤ ❧➼ ✶✳✼✳✷✳ ❬✹✱ ✶✸✱ ✶✹❪✳ ❚➟♣ ❤ñ♣ Co∞(R) ❝õ❛ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ❝â
❣✐→ ❝♦♠♣❛❝t tr♦♥❣ R ❧➔ trò ♠➟t tr♦♥❣ S t❤❡♦ tæ♣æ ❝õ❛ S ✳
2

✶✳✼✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥

❱➻ ❤➔♠ ❝ì ❜↔♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ S ❧➔ ♥❤ú♥❣ ❤➔♠ ❦❤↔ tê♥❣ tr♦♥❣ R ♥➯♥
❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝
+∞

ϕ(x)eix.ξ dx, ϕ ∈ S.

F [ϕ](ξ) =
−∞

❙❛✉ ✤➙② ❧➔ ❝→❝ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥
S✳
✶✳ ✣↕♦ ❤➔♠ sè ❧➛♥ tò② þ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r
Dα F [ϕ](ξ) = F [(ix)α ϕ](ξ).

✷✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ✤↕♦ ❤➔♠
F [Dα ϕ](ξ) = (−iξ)α F [ϕ](ξ).

✶✺

✸✳ ✣➥♥❣ t❤ù❝ P❛rs❡✈❛❧
●✐↔ sû f ∈ L1(R). ❑❤✐ ✤â t❛ ❝â ✤➥♥❣ t❤ù❝
+∞

+∞

f (x)F [ϕ](x)dx, ϕ ∈ S.

F [f ](ξ)ϕ(ξ)dξ =
−∞

−∞

✭✶✳✷✷✮

✹✳ ❈æ♥❣ t❤ù❝ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝
ϕ = F −1 [F [ϕ]] = F [F −1 [ϕ]], F −1 [ϕ(ξ)](x) =

1
F [ϕ(−ξ)](x).
2π

✣à♥❤ ❧➼ ✶✳✼✳✸✳ ❬✹✱ ✶✸✱ ✶✹❪✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r F tø S ✈➔♦ S ❧➔ →♥❤ ①↕ t÷ì♥❣
ù♥❣ ♠ët ✲ ♠ët ✈➔ ❧✐➯♥ tö❝ ✈➔♦ ❝❤➼♥❤ ♥â✱ ♥❣❤➽❛ ❧➔ ♠ët ✤➥♥❣ ❝➜✉ t✉②➳♥ t➼♥❤✳

✶✳✽ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠

✶✳✽✳✶ ❑❤æ♥❣ ❣✐❛♥ S ❝õ❛ ❝→❝ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠

✣à♥❤ ♥❣❤➽❛ ✶✳✽✳✶✳ ❬✹✱ ✶✸✱ ✶✹❪✳ ▼å✐ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr➯♥

✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠✳ ❚➟♣ ❤ñ♣ ❝→❝ ❤➔♠ s✉② rë♥❣ t➠♥❣
❝❤➟♠ ❦þ ❤✐➺✉ ❧➔ S ✳ ●✐→ trà ❝õ❛ f ∈ S tr➯♥ ♣❤➛♥ tû ϕ ∈ S ✤÷ñ❝ ❦➼ ❤✐➺✉
❧➔ < f, ϕ >✱ ❝á♥ tr➯♥ ♣❤➞♥ tû ❧✐➯♥ ❤ñ♣ ♣❤ù❝ ϕ✱ ❦➼ ❤✐➺✉ ❧➔ (f, ϕ)✳ ❉➣②
{fk } ∈ S ❤ë✐ tö ✤➳♥ f ∈ S ✱ ♥➳✉ < fk , ϕ >→< f, ϕ >, ϕ ∈ S ✳
S

●✐↔ sû f ❧➔ ❤➔♠ ❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣✱ ♥❣♦➔✐ r❛ ✤è✐ ✈î✐ N > 0 ♥➔♦ ✤â✿
+∞

|f (x)|(1 + |x|)−N dx < ∞.
−∞

❑❤✐ ✤â ❤➔♠ f t÷ì♥❣ ù♥❣ ✈î✐ ♠ët ♣❤✐➳♠ ❤➔♠ tr➯♥ S t❤❡♦ ❝æ♥❣ t❤ù❝✿
+∞

(f, ϕ) =

f (x)ϕ(x)dx.
−∞

P❤✐➳♠ ❤➔♠ tr➯♥ ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ s✉② rë♥❣ ❝❤➼♥❤ q✉②✳ ❉➵ t❤➜② r➡♥❣ ♣❤✐➳♠
❤➔♠ tr➯♥ ✤➙② ❧➔ t✉②➳♥ t➼♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ S ✳
✣à♥❤ ❧➼ ✶✳✽✳✷✳ ❬✶✸❪✳ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠ S ❧➔ ❦❤æ♥❣
❣✐❛♥ ✤➛② ✤õ✳

✶✻

✶✳✽✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠

❈æ♥❣ t❤ù❝ ✭✶✳✷✷✮ ❝â t❤➸ ✈✐➳t ❧↕✐ ð ❞↕♥❣
F [f ], ϕ >=< f, F [ϕ] , ϕ ∈ S.

❈æ♥❣ t❤ù❝ ♥➔② ❧➔ ❝ì sð ❝õ❛ ✤à♥❤ ♥❣❤➽❛ s❛✉ ✤➙②✳
✣à♥❤ ♥❣❤➽❛ ✶✳✽✳✸✳ ❬✶✸✱ ✶✹❪✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ s✉② rë♥❣ t➠♥❣
❝❤➟♠ f ❧➔ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠ F [f ] ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝
< F [f ], ϕ >=< f, F [f ] >, f ∈ S , ϕ ∈ S.
✭✶✳✷✸✮
❱➻ ♣❤➨♣ t♦→♥ ϕ → F [ϕ] ❧➔ ✤➥♥❣ ❝➜✉ ✈➔ ❧✐➯♥ tö❝ tø S ✈➔ S ✱ ♥➯♥ ♣❤✐➳♠ ❤➔♠
F [f ] ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝ ✭✶✳✷✸✮ ✤÷ñ❝ ❤✐➸✉ t❤❡♦ ♥❣❤➽❛ S ✱ ❤ì♥ ♥ú❛✱
♣❤➨♣ t♦→♥ f → F [f ] ❧➔ t✉②➳♥ t➼♥❤ ✈➔ ❧✐➯♥ tö❝ tø S ✈➔♦ S ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✽✳✹✳ ❬✶✸✱ ✶✹❪✳ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r F −1 ✤÷ñ❝ ①→❝ ✤à♥❤
tr♦♥❣ S t❤❡♦ ❝æ♥❣ t❤ù❝
1
F −1 [f ] =
F [f (−x)], f ∈ S .
✭✶✳✷✹✮
2π
tr♦♥❣ ✤â f (−x) ❧➔ ❤➔♠ s✉② rë♥❣ ♣❤↔♥ ①↕ ❝õ❛ ❤➔♠ s✉② rë♥❣ f (x) :
< f (−x), ϕ(x) >=< f, ϕ(−x) >, ϕ ∈ S.

❘ã r➔♥❣ ❧➔ F −1 ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tø S ✈➔♦ S ✳ ❚❛ s➩ ❝❤ù♥❣ tä
r➡♥❣✱ t♦→♥ tû F −1 ❧➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝ ❝õ❛ F ✱ ♥❣❤➽❛ ❧➔
F −1 [F [f ]] = f, F [F −1 [f ]] = f, f ∈ S .
✭✶✳✷✺✮
❚❤➟t ✈➟②✱ t❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ S ✱ t❤➻ ❝→❝ ❝æ♥❣ t❤ù❝

tr♦♥❣ ✭✶✳✷✺✮ ✤ó♥❣ tr♦♥❣ S trò ♠➟t tr♦♥❣ S ✱ ❞♦ ✤â ✭✶✳✷✺✮ ❝ô♥❣ ✤ó♥❣ tr♦♥❣
S✳

❈→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r
✶✳ ✣↕♦ ❤➔♠ ❝õ❛ ❝→❝ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r
Dα F [f ] = F [(ix)α f ], f ∈ S .

✭✶✳✷✻✮

✷✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ✤↕♦ ❤➔♠
Dα F [f ] = (−iξ)α F [f ], f ∈ S .

✭✶✳✷✼✮


✶✼

✸✳ ✣➥♥❣ t❤ù❝ P❛rs❡✈❛❧
< F [f ], F [ϕ] >= 2π < f (−x), ϕ(x) >,

f ∈ S , ϕ ∈ S.

✭✶✳✷✽✮

✹✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❞à❝❤ ❝❤✉②➸♥

✭✶✳✷✾✮
✺✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ s✉② rë♥❣ ❝â ❣✐→ ❝♦♠♣❛❝t✳ ◆➳✉ f ∈ S ✈➔ ❝â
❣✐→ ❝♦♠♣❛❝t✱ t❤➻ F [f ] ∈ C ∞ ✈➔ t➠♥❣ ❝❤➟♠ ð ✈æ ❝ò♥❣✱ ♥❣❤➽❛ ❧➔
F [f (x − x0 )] = eiξx0 F [f ], f ∈ S .

|Dα F [f ](ξ)| ≤ Cmα (1 + |ξ|2 )mα /2 .

✶✳✽✳✸ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ t➼❝❤ ❝❤➟♣

✣à♥❤ ♥❣❤➽❛ ✶✳✽✳✺✳ ❬✹✱ ✶✸✱ ✶✹❪✳ ✐✮ ◆➳✉ f ∈ S , η ∈ S ✱ t❤➻ f ∗ η ✤÷ñ❝ ①→❝
✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝

f ∗ η =< f (y), η(x − y) > .

❑❤✐ ✤â
F [f ∗ η](ξ) = F [f ](ξ)F [η](ξ).

✐✐✮ ◆➳✉ f, g ∈ S ✱ s✉♣♣ g ❧➔ t➟♣ ❝♦♠♣❛❝t t❤➻ f ∗ g ∈ S ✈➔ ✤÷ñ❝ ①→❝ ✤à♥❤
t❤❡♦ ❝æ♥❣ t❤ù❝
< f ∗ g, ϕ >=< f (y), < g(x), ϕ(x + y) >> .

❑❤✐ ✤â
F [f ∗ g] = F [f ].F [g].

✶✳✾ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈
✶✳✾✳✶ ❑❤æ♥❣ ❣✐❛♥ H (R)
s

✣à♥❤ ♥❣❤➽❛ ✶✳✾✳✶✳ ❬✹✱ ✶✸✱ ✶✹❪✳ ●✐↔ sû s ∈ R✳ ❑➼ ❤✐➺✉ H (R) ❧➔ ❦❤æ♥❣
s

❣✐❛♥ ❝õ❛ ❝→❝ ❤➔♠ s✉② rë♥❣ u ∈ S ✱ ❝â ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r u(ξ) t❤ä❛ ♠➣♥ ✤✐➲✉
❦✐➺♥✿
+∞

2
||u||s =
|u(ξ)|2 (1 + |ξ|2s )dξ < ∞.
✭✶✳✸✵✮
−∞




ỵ H s ổ ừ u = F [u] ợ u H s(R). ổ
tự tr H s tr H s t r H s
H s ổ rt ợ t ổ ữợ
+



(1 + ||)2s u()v()d.

(u, v)s =


ố ừ ổ H (R) ổ
s

H s(R). r t ủ Co(R) ừ ổ õ
t trũ t tr H s(R), s R.

ổ H (), H
s
o

s
s
o,o (), H ()

sỷ ởt

ổ tr R Hos() ổ ừ ổ
H s(R) ữủ ữ õ ừ Co() t ừ
H s (R) ủ ừ tr H s (R) õ tr ữủ
s
()
Ho,o

tr Hos() ụ ữủ ổ tự ồ
u Hos() õ su t sỷ u Hos(). õ tỗ t
{uk } Co() ở tử u t ừ H s(R) = R/
ữ t õ < uk , >= 0 ợ ồ Co( ) t tử s
r < u, >= 0 ợ ồ Co( ) õ ự tọ su ữ
Hos() Ho,os ().
sỷ f H s(R) f ừ f
tr
< f , >=< f, > ợ ồ Co ().
p, l tữỡ ự t tỷ tr t tỷ t
tr r R ủ ừ tở H s(R) tr
H s () tr H s () ữủ t ổ tự
||f ||H () = inf ||lf ||s ,

l
s

tr õ inf t tt t tr lf H s(R) ừ f H s()