Phương pháp xây dựng khung sóng nhỏ dựa trên các nguyên lý mở rộng

✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ĩ ò Pì PP ❳❹❨ ❉Ü◆● ❑❍❯◆● ❙➶◆● ◆❍➘ ❉Ü❆ ❚❘➊◆ ❈⑩❈ ◆●❯❨➊◆ ▲Þ ▼Ð ❘❐◆● ❚♦→♥ ù♥❣ ❞ö♥❣ ▼➣ sè✿ ✽✹✻✵✶✶✷✳✵✶ ❈❤✉②➯♥ ♥❣➔♥❤✿ ữợ s P❤❛♥ ❇ị✐ ❚❤à P❤÷đ♥❣ ❍⑨ ◆❐■✱ ✽✴✷✵✷✵ ▼ư❝ ❧ư❝ ▲í✐ õ ỡ ợ t ỵ t❤✉②➳t ❦❤✉♥❣ ✷ ✹ ✺ ✶✳✶ ❙ì ❧÷đ❝ ✈➲ ❝ì sð ✈➔ ♠ët ✈➔✐ ✤✐➸♠ ②➳✉ ❝õ❛ ❝ì sð ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ợ t ỵ tt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ▼ët ✈➔✐ ✈➼ ❞ö ❦❤✉♥❣ ❦❤➢❝ ♣❤ư❝ ✤✐➸♠ ②➳✉ ❝õ❛ ❝ì sð ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷ ●✐ỵ✐ t❤✐➺✉ ❦❤✉♥❣ sâ♥❣ ♥❤ä ✷✳✶ ●✐ỵ✐ t❤✐➺✉ ❦❤✉♥❣ sâ♥❣ ♥❤ä ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ✤❛ ♣❤➙♥ ❣✐↔✐ ✲ ▼❘❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✾ ✶✾ ✷✹ ✸ P❤÷ì♥❣ ♣❤→♣ ①➙② ỹ sõ ọ ỹ tr ỵ rở ✸✶ ✸✳✶ ▼ët sè ❤↕♥ ❝❤➳ ❦❤✐ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ▼❘❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ P❤÷ì♥❣ ♣❤→♣ t❤→❝ tr✐➸♥ ✤ì♥ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✸ P❤÷ì♥❣ ♣❤→♣ t❤→❝ tr✐➸♥ ♥❣❤✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✾ ✹✾ ✶ ▲í✐ ♥â✐ ✤➛✉ ❚r♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì✱ ♠ët tr♦♥❣ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ q✉❛♥ trå♥❣ ♥❤➜t ❧➔ ❦❤→✐ ♥✐➺♠ ❝ì sð✱ ♥❤í ✤â ♠é✐ ✈➨❝ tì tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❝â t❤➸ ✈✐➳t ♥❤÷ tê ❤đ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❝ì sð✳ ❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ❦✐➺♥ ✤➸ trð t❤➔♥❤ ❝ì sð ❦❤→ ❝❤➦t ❝❤➩✱ ❦❤ỉ♥❣ ❝❤♦ ♣❤➨♣ sü ♣❤ư t❤✉ë❝ t✉②➳♥ t➼♥❤ ❣✐ú❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❝ì sð✳ ✣✐➲✉ ♥➔② ❧➔♠ ❝❤♦ ✈✐➺❝ t➻♠ ❤♦➦❝ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ❝→❝ ❝ì sð t❤ä❛ ♠➣♥ ♠ët sè ✤✐➲✉ ❦✐➺♥ ❜ê s✉♥❣✳ ✣➙② ỵ t t ởt ổ ❧✐♥❤ ❤♦↕t ❤ì♥✱ ❦❤✉♥❣ ❝❤➼♥❤ ❧➔ ♠ët ❝ỉ♥❣ ❝ư ♥❤÷ ✈➟②✳ ❑❤✉♥❣ ❝❤♦ ♣❤➨♣ t❛ ❜✐➸✉ ❞✐➵♥ ♠é✐ ♣❤➛♥ tû tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♥❤÷ ♠ët tê ❤đ♣ t✉②➳♥ t➼♥❤ ❣✐ú❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❦❤✉♥❣ ♥❤÷♥❣ ❦❤ỉ♥❣ ✤á✐ ❤ä✐ t➼♥❤ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❣✐ú❛ ❝→❝ ♣❤➛♥ tû ❦❤✉♥❣✱ ❝→❝ ❤➺ sè ❦❤æ♥❣ ♥❤➜t t❤✐➳t ❞✉② ♥❤➜t✳ ❈â t❤➸ ♥â✐ ❦❤✉♥❣ ♥❤÷ ♠ët ❝ì sð ♥❤÷♥❣ t❛ ❝â t❤➸ t❤➯♠ tỷ ỡ ữủ ợ t ✶✾✺✸ ❜ð✐ ❉✉❢❢✐♥ ✈➔ ❙❝❤❛r❢❢❡r✳ ✣➳♥ ♥➠♠ ✶✾✽✻✱ ❦❤✐ ❜➔✐ s rss r r ỵ tt ợ t ữủ q t rở r õ ự tr ỷ ỵ t ỵ tt t ỵ tt ữủ tỷ ỳ ▼ët ❦❤✉♥❣ ❝â t❤➸ ①❡♠ ♥❤÷ ♠ët ❝ì sð trü❝ ❝❤✉➞♥ s✉② rë♥❣✳ ◆➳✉ {fi } , i ∈ I ❧➔ ♠ët ❦❤✉♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ V t❤➻ ❜➜t ❦➻ ✈➨❝ tì f ∈ V ♥➔♦ ❝ơ♥❣ ❝â t❤➸ ✈✐➳t ♥❤÷ ♠ët tê ❤đ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ♣❤➛♥ tû fi ✳ ❈→❝ ❤➺ sè ❦❤æ♥❣ ♥❤➜t t❤✐➳t ❞✉② ♥❤➜t ✈➔ ❦❤❛✐ tr✐➸♥ ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ trü❝ ❣✐❛♦✳ ◆❤í t➼♥❤ t❤ø❛ ♠➔ ❦❤✉♥❣ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ q✉❛♥ trå♥❣ tr♦♥❣ ỷ ỵ t õ ❝❤♦ ❝❤ó♥❣ t❛ t➼♥❤ ❜➲♥ ✈ú♥❣✱ ❝❤➜t ❧÷đ♥❣ ❝õ❛ t➼♥ ❤✐➺✉ ❜à ↔♥❤ ❤÷ð♥❣ ➼t ❤ì♥ ❦❤✐ ❝â ♥❤✐➵✉ t✐➳♥❣ ç♥ ✈➔ t➼♥ ❤✐➺✉ ❝â t❤➸ ❦❤ỉ✐ ♣❤ư❝ ❧↕✐ tø õ tữỡ ố t ỵ t❤✉②➳t sâ♥❣ ♥❤ä ❧➔ ❦➳t q✉↔ ❝õ❛ sü ♥é ❧ü❝ ❝õ❛ ♥❤✐➲✉ ♥❣➔♥❤ ✈➔ ❣â♣ ♣❤➛♥ ✤❡♠ ❝→❝ ♥❤➔ t♦→♥ t ỵ sữ ỗ ợ ỳ ự sõ ọ õ t tr ỷ ỵ t➼♥ ❤✐➺✉✱ ❦ÿ t❤✉➟t ♥➙♥❣ ❝❛♦ ❝❤➜t ❧÷đ♥❣ ❤➻♥❤ ↔♥❤✱ ♥➨♥ ❞➜✉ ✈➙♥ t❛②✱ ♥❤➟♥ ❞↕♥❣ ✤è✐ t÷đ♥❣✱ ❦ÿ t❤✉➟t t ỗ t sõ ọ ởt ❦❤✉♥❣ ❝â ❝➜✉ tró❝ ✤➦❝ ❜✐➺t✳ ▲ỵ♣ ❦❤✉♥❣ ♥➔② r➜t ỳ tr ỷ ỵ t ❝→❝ t➼♥❤ ❤✐➺✉ ❝â ✤➦❝ tr÷♥❣ ❤➻♥❤ ❤å❝ ♣❤ù❝ t↕♣✳ ✣➸ ①➙② ❞ü♥❣ ❦❤✉♥❣ sâ♥❣ ♥❤ä ❝â ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❦❤→❝ ♥❤❛✉✳ ▲✉➟♥ ✈➠♥ ①✐♥ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ①➙② ỹ sõ ọ ỹ tr ỵ rở ỗ ữỡ ữỡ ỗ ♣❤➛♥✱ tr➻♥❤ ❜➔② ❧÷đ❝ ✈➲ ❝ì sð ✈➔ ❦❤→✐ ♥✐➺♠ ❦❤✉♥❣✱ ♠ët sè ✤➦❝ t➼♥❤ ❝õ❛ ❦❤✉♥❣ s❛✉ ✤â tr➻♥❤ ❜➔② ✈➼ ❞ư ❦❤✉♥❣ ❦❤➢❝ ✤✐➸♠ ✷ ▼Ư❈ ▲Ư❈ ②➳✉ ❝õ❛ ❝ì sð ❞ü❛ tr➯♥ t➼♥❤ t❤ø❛ ❝õ❛ ❦❤✉♥❣✳ ◆❤í t➼♥❤ t❤ø❛ ♠➔ ❦❤✉♥❣ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ q✉❛♥ trồ tr ỷ ỵ t t ❧÷đ♥❣ ❝õ❛ t➼♥ ❤✐➺✉ ➼t ❜à ↔♥❤ ❤÷ð♥❣ ❤ì♥ ❦❤✐ õ t ỗ t õ t ổ ❧↕✐ tø ❝→❝ ♠➝✉ ❝â ✤ë ❝❤➼♥❤ ①→❝ t÷ì♥❣ ✤è✐ t ữỡ ỗ ởt tr ỵ tt sõ ọ trữ sõ ọ P❤➛♥ ❤❛✐ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ①➙② ❞ü♥❣ ▼❘❆ ✈➔ ✈➼ ❞ư ①➙② ❞ü♥❣ ❦❤✉♥❣ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ▼❘❆✳ ❈❤÷ì♥❣ ỗ ♣❤➛♥✳ P❤➛♥ ♠ët ❧✉➟♥ ✈➠♥ ✤÷❛ r❛ ♠ët sè ❤↕♥ ❝❤➳ ❦❤✐ ①➙② ❞ü♥❣ ❦❤✉♥❣ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ▼❘❆✱ ♣❤➛♥ ❤❛✐ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝❤ t❤ù❝ ①➙② ❞ü♥❣ ❦❤✉♥❣ sâ♥❣ ♥❤ä ❞ü❛ tr➯♥ ♣❤÷ì♥❣ ♣❤→♣ t❤→❝ tr✐➸♥ ✤ì♥ ♥❤➜t✳ P❤➛♥ ❜❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝❤ t❤ù❝ ①➙② ❞ü♥❣ ❦❤✉♥❣ sâ♥❣ ♥❤ä ❞ü❛ tr➯♥ ♣❤÷ì♥❣ ♣❤→♣ t❤→❝ tr✐➸♥ ♥❣❤✐➯♥❣✳ ▼➦❝ ❞ị ✤➣ r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❝❤➢❝ ❝❤➢♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ✤÷đ❝ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚ỉ✐ r➜t ữủ ỳ t õ ỵ tứ qỵ t❤➛② ❝ỉ ✈➔ ❜↕♥ ✤å❝ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ ♥❣➔② ✵✶ t❤→♥❣ ✵✽ ♥➠♠ ✷✵✷✵ ❍å❝ ✈✐➯♥ ❇ò✐ ❚❤à P❤÷đ♥❣ ❇ị✐ ❚❤à P❤÷đ♥❣ ✸ ❚♦→♥ ù♥❣ ❞ư♥❣ ▲í✐ ❝↔♠ ì♥ ✣➲ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✱ ♥❣♦➔✐ ♥❤ú♥❣ sü ❝❤✉➞♥ ❜à ❝õ❛ ❝→ ♥❤➙♥ tỉ✐✱ ❝➛♥ ♥❤í ❝â sü ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ✈➔ ❝❤✉ ✤→♦ ❝õ❛ ❣✐→♦ ✈✐➯♥ ữợ ổ tọ ỏ t ỡ s s tợ P ữớ trỹ t ữợ ú ù tổ t ♥➔②✳ ❚❤➛② ❧✉æ♥ q✉❛♥ t➙♠ ✈➔ ✤ë♥❣ ✈✐➯♥ tæ✐✱ ♥❤➭ õ ỵ tổ sỷ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❑❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥✱ P❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ụ ữ t t qỵ t ổ ❣✐↔♥❣ ❞↕② ❦❤â❛ ❝❛♦ ❤å❝ ✷✵✶✼ ✲ ✷✵✶✾✳ ❳✐♥ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ ✤➣ ❞↕② ❜↔♦ ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❳✐♥ t ỡ ỗ ♥❣❤✐➺♣ ✤➣ ❧✉ỉ♥ ❤é trđ✱ ✤ë♥❣ ✈✐➯♥✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ tæ✐ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❍➔ ◆ë✐✱ ♥❣➔② ✵✶ t❤→♥❣ ✵✽ ♥➠♠ ✷✵✷✵ ❍å❝ ✈✐➯♥ ũ Pữủ ữỡ ợ t ỵ t❤✉②➳t ❦❤✉♥❣ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ❧÷đ❝ ✈➲ ❝ì sð ✈➔ ♠ët ✈➔✐ ✤✐➸♠ ②➳✉ ❝õ❛ ❝ì sð✳ ❉ü❛ tr➯♥ ✤â t❛ ✤÷❛ r❛ ❦❤→✐ ỵ tt tr ổ rt ỗ tớ tr ởt số ♣❤ư❝ ✤✐➸♠ ②➳✉ ❝õ❛ ❝ì sð✳ ✶✳✶ ❙ì ❧÷đ❝ ✈➲ ❝ì sð ✈➔ ♠ët ✈➔✐ ✤✐➸♠ ②➳✉ ❝õ❛ ❝ì sð ❈❤♦ V ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì ❤ú✉ ❤↕♥ ❝❤✐➲✉ ữủ tr ởt t ổ ữợ ợ r ♠ët ❞➣② {ek }m k=1 tr♦♥❣ V ❧➔ ❝ì sð ♥➳✉ ❤❛✐ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥ ✭✐✮ V = s♣❛♥{ek }m k=1 ✳ m ck ek = ✈ỵ✐ ổ ữợ {ck }m k=1 ck = ✭✐✐✮ {ek }m ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ♥➳✉ k=1 ✈ỵ✐ ♠å✐ k = 1, , m ❱ỵ✐ ♠é✐ f ∈ V ✤➲✉ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ q✉❛ ❝→❝ tỷ ỡ s ỗ t t m ổ ữợ {ck }m k=1 s f = ck ek ✳ ✭✶✳✶✮ k=1 ◆➳✉ {ek }m k=1 ❧➔ ♠ët ❝ì sð trü❝ ❝❤✉➞♥ t❤➻ ♥➳✉ k = j ♥➳✉ k = j ek , ej = k,j = ợ ộ ej tú ỵ t ❝â m f, ej ❂ m ck ek , ej ❂ k=1 ck ek , ej ❂cj ✳ k=1 ❉♦ ✤â m f, ek ek ✳ f= k=1 ✺ ✭✶✳✷✮ ị trữ ♠ët ❝ì sð {ek }m k=1 tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì V ❧➔ ✈ỵ✐ ♠é✐ f ∈ V ❝â t❤➸ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ♥❤÷ ♠ët tê ❤đ♣ ❝→❝ ♣❤➛♥ tû ek tr♦♥❣ ❝ì sð m ck (f ) ek ✳ f= k=1 ❱ỵ✐ ❝→❝ ❤➺ sè ck (f ) ❧➔ ❞✉② ♥❤➜t✳ ❉♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ✤➸ ❧➔ ♠ët ❝ì sð ❧➔ r➜t ♠↕♥❤ ♥➯♥ • ❑❤ỉ♥❣ ❞➵ ✤➸ ①➙② ỹ ữủ ỡ s ợ tở t t • ❚❤➟♠ ❝❤➼ ♠ët t❤❛② ✤ê✐ ♥❤ä ❝ô♥❣ ❝â t❤➸ ♣❤→ ❤õ② t❤✉ë❝ t➼♥❤ ❝ì ❜↔♥ ❝õ❛ ❝ì sð✳ ❈ì s ữủ ỹ t tr ợ ❝→❝ ❤➺ sè ck (f ) ❧➔ ❞✉② ♥❤➜t✳ ❈➙✉ ❤ä✐ ✤➦t r❛ ❧➔ sü ❞✉② ♥❤➜t ✤â ❧✐➺✉ ❝â t❤ü❝ sü ❝➛♥❄ ❈➙✉ tr↔ ❧í✐ ❧➔ ❦❤ỉ♥❣✳ ❚❤ỉ♥❣ t❤÷í♥❣ t t sỹ tỗ t ởt số sè ❝ị♥❣ ✈ỵ✐ ♠ët ❝ỉ♥❣ t❤ù❝ ✤➸ t➻♠ r❛ ❝❤ó♥❣ ❧➔ ✤õ✳ ✣➙② ❧➔ ❝❤➻❛ ❦❤â❛ tr♦♥❣ q✉→ tr➻♥❤ ❜✐➳♥ ✤ê✐ tø ❝ì sð s❛♥❣ ❦❤✉♥❣✳ ●✐í t❛ s➩ ✤✐ ❣✐ỵ✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ ❝õ❛ ❦❤✉♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ H✳ ợ t ỵ tt ởt ❞➣② {f } tr♦♥❣ H ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❞➣② ss tỗ t k k=1 ởt số B > s❛♦ ❝❤♦✿ ∞ | f, fk |2 ≤ B f ✱ ∀f ∈ H✳ k=1 B ✤÷đ❝ ❣å✐ ❧➔ ❝➟♥ ❇❡ss❡❧ ❝õ❛ {fk }∞ k=1 ✳ ✣à♥❤ ❧➼ ✶✳✷✳✷✳ ❈❤♦ {f } ∞ k k=1 ❧➔ ♠ët H trữợ B > ✤â {fk }∞ k=1 ❧➔ ♠ët ❞➣② ❇❡ss❡❧ ✈ỵ✐ ❝➟♥ B ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ∞ T : {ck }∞ k=1 → ck f k k=1 ❧➔ ♠ët t♦→♥ tû ①→❝ ✤à♥❤ ❜à ❝❤➦♥ tø l2 (N) ✈➔♦ H ✈➔ T ≤ ❈❤ù♥❣ ♠✐♥❤✿ ❈❤♦ {ck }∞ k=1 √ B✳ ✣➛✉ t✐➯♥ ❣✐↔ sû r➡♥❣ {fk }∞ k=1 ❧➔ ♠ët ❞➣② ❇❡ss❡❧ ✈ỵ✐ ❝➟♥ ❇❡ss❡❧ ❧➔ B ✳ ∞ ∈ l (N)✳ ▼✉è♥ ❝❤➾ r❛ T {fk }∞ k=1 ①→❝ ✤à♥❤ tèt t❤➻ ❝➛♥ ck fk ❤ë✐ tö✳ k=1 ❇ị✐ ❚❤à P❤÷đ♥❣ ✻ ❚♦→♥ ù♥❣ ❞ư♥❣ ✶✳✷✳ ●■❰■ ❚❍■➏❯ ị ợ n, m N, n > m t❛ ❝â n m n ck f k − k=1 ck f k = k=1 ck f k k=m+1 n = sup ck f k , g g =1 k=m+1 n ≤ sup |ck fk , g | g =1 k=m+1 1/2 n ≤ |ck | ≤ | fk , g | sup g =1 k=m+1 k=m+1 1/2 n √ 1/2 n 2 |ck | B k=m+1 ∞ n ❱➟② ❧➔ ♠ët ❞➣② ❈❛✉❝❤② tr♦♥❣ H ♥➯♥ ♥â ❤ë✐ tö✳ ❑❤✐ ✤â T {ck }∞ k=1 ck f k k=1 k=1 = sup | T {ck }∞ k=1 , g | ✈➔ T ❧➔ ❜à ①→❝ ✤à♥❤ tèt✳ ❱➻ T ❧➔ t✉②➳♥ t➼♥❤ ♥➯♥ T {ck }∞ k=1 √ ❝❤➦♥ ✈ỵ✐ T ≤ B ✳ g =1 ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐➲✉ ♥❣÷đ❝ ❧↕✐✱ ❣✐↔ sû r➡♥❣ T ①→❝ ✤à♥❤ tèt ✈➔ T ≤ ♠✐♥❤ {fk }∞ k=1 √ B ✱ t❛ ❝❤ù♥❣ ❧➔ ♠ët ❞➣② ❇❡ss❡❧ ✈ỵ✐ ❝➟♥ B ✳ ❚❤➟t ✈➟②✱ ①➨t t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ ∞ T : l (N) → V ✱ T {ck }∞ k=1 ck f k ✳ = k=1 ❚❛ ❝â t♦→♥ tû T ∗ ✤÷đ❝ ❝❤♦ ❜ð✐ T ∗ : V → l2 (N)✱ T ∗ f = { f, fk }∞ k=1 ❝ô♥❣ ❧➔ ♠ët t♦→♥ tû ❜à ❝❤➦♥ ✈➔ T = T ∗ ✱ ❞♦ ✤â T ∗f ≤ T f ✱ ✈ỵ✐ ♠å✐ f ∈ V ✳ ❱➟② {fk }∞ k=1 ❧➔ ♠ët ❞➣② ❇❡ss❡❧ ✈ỵ✐ ❝➟♥ B ✳ ❍➺ q✉↔ ✶✳✷✳✸✳ ◆➳✉ ∞ {fk }∞ k=1 ❧➔ ♠ët ❞➣② tr♦♥❣ H ✈➔ ∞ {ck }∞ k=1 ∈ l (N) t❤➻ {fk }k=1 ❧➔ ♠ët ❞➣② ❇❡ss❡❧✳ ❍➺ q✉↔ ✶✳✷✳✹✳ ◆➳✉ {f } ∞ k k=1 ∞ ❧➔ ♠ët ❞➣② ❇❡ss❡❧ tr♦♥❣ H✱ t❤➻ ck fk ❤ë✐ tư ✈ỉ ✤✐➲✉ k=1 ❦✐➺♥ ✈ỵ✐ ♠å✐ {ck }∞ k=1 ∈ l (N) ũ Pữủ ck fk tử ợ ♠å✐ ❞➣② k=1 ✼ ❚♦→♥ ù♥❣ ❞ö♥❣ ✶✳✷✳ ●■❰■ ❚❍■➏❯ ▲Þ ❚❍❯❨➌❚ ❑❍❯◆● ❇ê ✤➲ ✶✳✷✳✺✳ ●✐↔ sû {f } ∞ k k=1 ❧➔ ♠ët ❞➣② ❝→❝ ♣❤➛♥ tû tr♦♥❣ H tỗ t số B > s ❝❤♦ ∞ | f, fk |2 ≤ B f ✱ k=1 ✈ỵ✐ ♠å✐ f tr♦♥❣ t➟♣ ❝♦♥ ✤➛② ✤õ V ❝õ❛ H✳ ❑❤✐ ✤â {fk }∞ k=1 ❧➔ ♠ët ❞➣② ❇❡ss❡❧ ✈ỵ✐ ❝➟♥ B ✳ ❈❤ù♥❣ ♠✐♥❤✿ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ❇❡ss❡❧ t❤ä❛ ♠➣♥ ✈ỵ✐ ♠å✐ ♣❤➛♥ tû tr♦♥❣ H✳ ❈❤♦ g ∈ H ✈➔ ❣✐↔ sû ♣❤↔♥ ❝❤ù♥❣ r➡♥❣ ∞ | g, fk |2 > B g k=1 õ tỗ t t ỳ ❤↕♥ F ⊂ N ♠➔ | g, fk |2 > B g ✳ k∈F ❱➻ V ❧➔ t➟♣ ❝♦♥ tr H tỗ t h V s❛♦ ❝❤♦ | h, fk |2 > B h ✳ k∈F ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥✱ ✈➟② {fk }∞ k=1 ❧➔ ❞➣② ❇❡ss❡❧ ✈ỵ✐ ❝➟♥ B ✳ ▼ët ❞➣② ❝→❝ ♣❤➛♥ tû {fk }m k=1 ✱ ♠é✐ ♣❤➛♥ tû f H ữủ t ữ ổ tự ợ ❤➺ sè t÷ì♥❣ ù♥❣ ❦❤ỉ♥❣ ♥❤➜t t❤✐➳t ♣❤↔✐ ❞✉② ♥❤➜t✳ ❉➣② ❝→❝ ♣❤➛♥ tû ♥➔② ❝â t❤➸ ❦❤æ♥❣ ❧➔ ♠ët ❝ì sð✳ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ♠ët ❦❤→✐ ♥✐➺♠ ♠ỵ✐✱ ✤â ❝❤➼♥❤ ❧➔ ❦❤✉♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✻✳ ▼ët ❞➣② ✤➳♠ ✤÷đ❝ ❝→❝ ♣❤➛♥ tû {f } k k∈I tr♦♥❣ V ởt V tỗ t sè A, B > s❛♦ ❝❤♦ A f | f, fk |2 ≤ B f ≤ (1.3) k∈I ❈→❝ sè A, B ✤÷đ❝ ❣å✐ ❧➔ ❝➟♥ ã tr tố ữ tr ♥❤ä ♥❤➜t tr♦♥❣ ❝→❝ ❝➟♥ ❦❤✉♥❣ tr➯♥✳ • ❈➟♥ ❦❤✉♥❣ ữợ tố ữ tr ợ t tr ữợ ã ữủ õ f = ã ởt ữủ ❧➔ ❝❤➦t ♥➳✉ A = B ✳ ❇ị✐ ❚❤à P❤÷đ♥❣ ✽ ❚♦→♥ ù♥❣ ❞ư♥❣ ✶✳✷✳ ●■❰■ ❚❍■➏❯ ▲Þ ❚❍❯❨➌❚ ❑❍❯◆● • ❑❤✉♥❣ P❛rs❡✈❛❧ ❧➔ ❦❤✉♥❣ ❝â A = B = 1✳ ❚❛ ①➨t ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ ♣❤➛♥ tû {fk }m k=1 ✈ỵ✐ m ∈ N✱ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲ ❙❝❤✇❛r③✬ ❜✐➸✉ ❞✐➵♥ ♥❤÷ s❛✉ m m | f, fk | ≤ k=1 fk f ✈ỵ✐ ♠å✐ f ∈ V ✳ k=1 ▼➺♥❤ ✤➲ ✶✳✷✳✼✳ ❈❤♦ {f } m k k=1 ❧➔ ♠ët ❞➣② tr♦♥❣ V ✳ ❑❤✐ ✤â {fk }m k=1 ❧➔ ♠ët ❦❤✉♥❣ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì W = s♣❛♥ {fk }m k=1 ✳ ❈❤ù♥❣ ♠✐♥❤✿ ❱ỵ✐ f m k = 0✱ ✤✐➲✉ tr ữủ tọ ợ fk ✳ ❚❛ ①➨t →♥❤ ①↕ ❧✐➯♥ tö❝ B= k=1 m φ : W → R✱ φ (f ) = | f, fk |2 ✳ k=1 ❍➻♥❤ ❝➛✉ ✤ì♥ ✈à tr♦♥❣ W ❧➔ ❝♦♠♣❛❝t ♥➯♥ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ g ∈ W ✈ỵ✐ g = s❛♦ ❝❤♦ m m | g, fk | = ✐♥❢ A= k=1 | f, fk |2 | f ∈ W, f =1 ✳ k=1 ❘ã r➔♥❣ A > 0✱ ✈ỵ✐ f ∈ W ✈➔ f = 0✱ t❛ ❝â m m f , fk f | f, fk |2 ❂ k=1 k=1 ❍➺ q✉↔ ✶✳✷✳✽✳ ▼ët ❤å ❝→❝ ♣❤➛♥ tû {f } m k k=1 ❦❤✐ V = f ≥A f ✳ tr♦♥❣ V ❧➔ ♠ët ❦❤✉♥❣ ❝õ❛ V ❦❤✐ ✈➔ ❝❤➾ s♣❛♥ {fk }m k=1 ✳ ◆❤÷ ✈➟② • ▼ët ❦❤✉♥❣ ❝â t❤➸ ❝❤ù❛ ♥❤✐➲✉ ♣❤➛♥ tû ❤ì♥ ♠ët ❝ì sð✳ n • ◆➳✉ {fk }m k=1 ❧➔ ♠ët ❦❤✉♥❣ ❝õ❛ V ✈➔ {gk }k=1 ❧➔ ♠ët t➟♣ ❤ú✉ ❤↕♥ ❝→❝ ✈➨❝ tì ❜➜t n ❦➻ tr♦♥❣ V ✱ t❤➻ {fk }m k=1 ∪ {gk }k=1 ❝ô♥❣ ❧➔ ♠ët ❦❤✉♥❣ ❝õ❛ V ✳ ❱➼ ❞ö ✶✳✶✿ ❈❤♦ {e } ∞ k k=1 ❧➔ ♠ët ❝ì sð trü❝ ❝❤✉➞♥ ❝õ❛ H✳ ✭✐✮ ❇➡♥❣ ❝→❝❤ ❧➦♣ ❧↕✐ ❧➛♥ ❝→❝ ♣❤➛♥ tû tr♦♥❣ ❞➣② {ek }∞ k=1 t❛ t❤✉ ữủ ũ Pữủ ự Pì P❍⑩P ❚❍⑩❈ ❚❘■➎◆ ✣❒◆ ◆❍❻❚ ❇ê ✤➲ ✸✳✷✳✸✳ ❈❤♦ ψ ∈ L2 (R) ✈➔ ❣✐↔ sû r➡♥❣ lim ψ0 (γ) = 1✳ ❈❤♦ f ∈ L2 (R) ❧➔ ♠ët γ→0 ❤➔♠ ❜➜t ❦➻ ♠➔ fˆ ∈ Cc (R)✳ ❱ỵ✐ (1 − ) f ❜➜t ❦➻ ✈➔ > 0✱ tỗ t J Z s | f, Dj Tk ψ0 |2 ≤ (1 + ) f ≤ ✱ ✈ỵ✐ ♠å✐ j ≥ J ✳ k∈Z ❈❤ù♥❣ ♠✐♥❤✿ ❱ỵ✐ j ∈ Z ✈➔ f ∈ L (R)✱ ❣✐↔ sû fˆ ∈ C (R)✱ (D fˆ)ψ j c ∈ L1 (R)✳ ❑❤✐ ✤â P((Dj fˆ)ψ0 ) ①→❝ ✤à♥❤ tèt✳ ◆➳✉ γ ∈ T✱ t❤➻ P((Dj fˆ)ψˆ0 ) ∈ L2 (T)✳ ❚❤❡♦ ❦❤❛✐ tr✐➸♥ ❋♦✉r✐❡r✱ f, Dj Tk ψ0 = Ff, FDj Tk ψ0 = Dj fˆ, E−k ψ0 ✳ ✭✸✳✾✮ ⑩♣ ❞ö♥❣ ✤â ❇ê ✤➲ ✸✳✷✳✷✱ t❛ ❝â | f, Dj Tk ψ0 |2 = k∈Z | Dj fˆ, E−k ψ0 |2 k∈Z 1/2 (Dj fˆ)(γ + n)ψ0 (γ + n) dγ = −1/2 n∈Z ❱ỵ✐ > trữợ t õ t b [0, 1/2] ✈➔ |γ| ≤ b s❛♦ ❝❤♦ 1− ≤ |ψ0 (γ)|2 ≤ 1+ ✳ ❇➡♥❣ ❝→❝❤ ❧➜② J ∈ Z ✈➔ j > J s❛♦ ❝❤♦ Dj fˆ ❝â ❣✐→ tr➯♥ [−b, b]✱ t❛ t❤✉ ✤÷đ❝ 1/2 b (Dj fˆ)(γ + n)ψ0 (γ + n) dγ = −1/2 n∈Z (Dj fˆ)(γ)ψ0 (γ) dγ −b ❑❤✐ ✤â (1 − ) Dj fˆ | f, Dj Tk ψ0 |2 ≤ (1 + ) Dj fˆ ✱ ✈ỵ✐ ♠å✐ j ≥ J ✳ ≤ k∈Z ❱➻ Dj ✈➔ ❦❤❛✐ tr✐➸♥ ❋♦✉r✐❡r ❧➔ t♦→♥ tû ✤ì♥ ♥❤➜t ♥➯♥ ❜ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû r➡♥❣ {ψl , Hl }nl=0 ❝â ❝→❝ t➼♥❤ ❝❤➜t ♥❤÷ tr♦♥❣ t❤✐➳t ❧➟♣ ❝❤✉♥❣✳ ❱ỵ✐ ♠é✐ ❤➔♠ f ✈➔ fˆ ∈ Cc (R) t❛ ❝â { f, Dj Tk ψl }j,k∈Z ∈ l2 (Z) ✈ỵ✐ ♠å✐ l = 1, n ✭✸✳✶✵✮ ❚❛ ❝â t❤➸ ①→❝ ✤à♥❤ ♠ët ❤å ❝→❝ ❤➔♠ Fj,l ∈ L2 (T) ❜ð✐ ❝❤✉é✐ ❋♦✉r✐❡r f, Dj Tk ψl E−k , j ∈ Z, l = 0, 1, n✳ Fj,l = ✭✸✳✶✶✮ k∈Z ❚ø ✤â Fj,l ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ψl ✱ ✈➔ ✤÷đ❝ ①→❝ ✤à♥❤ ❧↕✐ t❤❡♦ ψ0 ✈➔ Hl ✳ ũ Pữủ ự Pì PP ❚❍⑩❈ ❚❘■➎◆ ✣❒◆ ◆❍❻❚ ❇ê ✤➲ ✸✳✷✳✹✳ ❈❤♦ {ψ , H } n l l=0 l ❝â t➼♥❤ ❝❤➜t ♥❤÷ tr♦♥❣ t❤✐➳t ❧➟♣ ❝❤✉♥❣✳ ❱ỵ✐ ♠å✐ j ∈ Z ✈➔ l = 0, 1, , n✱ t❛ ❝â Fj−1,l (γ) = 2−1/2 (Hl Fj,0 + T1/2 (Hl Fj,0 ))(γ/2)✳ ❈❤ù♥❣ ♠✐♥❤✿ ❚❛ ❝â f, Dj−1 Tk ψl = D−j f, D−1 Tk ψl = D−j f, T2k D−1 ψl = FD−j f, FT2k D−1 ψl = Dj fˆ, E−2k D−1 ψˆl ❚❤❡♦ ❝æ♥❣ t❤ù❝ ✭✸✳✺✮✱ t❛ ❝â f, Dj−1 Tk ψl = Dj fˆ, E−2k 21/2 Hl ψ0 ∞ 1/2 (D fˆ)Hl ψ0 E2k = 21/2 1/2 P((Dj fˆ)Hl ψ0 )E2k j =2 −∞ 1/2 −1/2 P((Dj fˆ)Hl ψ0 )E2k + T1/2 P((Dj fˆ)Hl ψ0 )E2k = 21/2 1/2 P((Dj fˆ)Hl ψ0 ) + T1/2 P((Dj fˆ)Hl ψ0 ) E2k = 21/2 ❚➼♥❤ t♦→♥ tr➯♥ ❝❤➾ r❛ r➡♥❣ f, Dj−1 Tk ψl ❧➔ ❤➺ sè t❤ù −k tr♦♥❣ ❦❤❛✐ tr✐➸♥ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦➻ 1/2✳ ❱ỵ✐ 21/2 E2k k∈Z = 21/2 e4πik(·) k∈Z ❧➔ ❝ì sð trü❝ ❣✐❛♦ ❝õ❛ L2 (0, 1/2)✱ t❛ ❝â Fj−1,l (γ) = 2−1/2 f, Dj−1 Tk ψl 21/2 E−2k (γ/2) k∈Z −1/2 =2 P((Dj fˆ)Hl ψˆ0 ) + T1/2 P((Dj fˆ)Hl ψˆ0 ) (γ/2) (3.12) ❍➔♠ Hl t✉➛♥ ❤♦➔♥ ❝❤✉ ❦➻ 1✱ ♥➯♥ P((Dj fˆ)Hl ψˆ0 ) = Hl P((Dj fˆ)ψˆ0 )✳ ✭✸✳✶✸✮ ❚r♦♥❣ ❝æ♥❣ t❤ù❝ ✭✸✳✾✮✱ t❛ ❝â f, Dj Tk ψ0 = Dj fˆ, E−k ψˆ0 ✳ ❚❤❡♦ ❜ê ✤➲ ✸✳✷✳✷ Dj fˆ, E−k ψˆ0 E−k = P((Dj fˆ)ψˆ0 )✳ Fj,0 = ✭✸✳✶✹✮ k∈Z ❚❤❛② ❝æ♥❣ t❤ù❝ ✭✸✳✶✸✮ ✈➔ ✭✸✳✶✹✮ ✈➔♦ tr♦♥❣ ❦❤❛✐ tr✐➸♥ ✭✸✳✶✷✮ ❝õ❛ Fj−1,l t❛ t❤✉ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇ị✐ ❚❤à Pữủ ự Pì PP ✣❒◆ ◆❍❻❚ ▼❛ tr➟♥ ❝õ❛ H tr♦♥❣ ❝æ♥❣ t❤ù❝ ✭✸✳✻✮ ❝â ❜✐➸✉ ❞✐➵♥ s❛✉   Fj−1,0 (γ)  Fj−1,1 (γ)      Fj,0 ( γ2 )   = 2−1/2 H( γ ) , ✈ỵ✐ ♠å✐ γ ∈ R   T1/2 Fj,0 ( γ2 )     Fj−1,n (γ) (3.15) ❚r♦♥❣ ♣❤➛♥ ♥➔② ♠❛ tr➟♥ H(γ)∗ H(γ) ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ ✈➔ ✤✐➲✉ ❦✐➺♥ ❧➔ H(γ)∗ H(γ) = I ✈ỵ✐ ♠å✐ γ ∈ T✳ ❇ê ✤➲ ✸✳✷✳✺✳ ❈❤♦ {ψ , H } ❝â t➼♥❤ ❝❤➜t ♥❤÷ tr♦♥❣ t❤✐➳t ❧➟♣ ❝❤✉♥❣✱ ❣✐↔ sû r➡♥❣ H(γ) H(γ) = I ✈ỵ✐ γ ∈ T✳ ❑❤✐ ✤â✱ ✈ỵ✐ j ∈ Z ✈➔ ♠å✐ f ∈ L2 (R) s❛♦ ❝❤♦ fˆ ∈ Cc (R)✱ t❛ n l l=0 l ∗ ❝â n j | f, Dj−1 Tk ψl |2 ✳ | f, D Tk ψ0 | = k∈Z l=0 k∈Z ❇ê ✤➲ s❛✉ s➩ ❝❤➾ r❛ ❤➔♠ t➾ ❧➺ tr♦♥❣ t❤✐➳t ❧➟♣ ❝❤✉♥❣ ❧✉æ♥ t↕♦ r❛ ♠ët ❝❤✉é✐ ❇❡ss❡❧✳ ❇ê ✤➲ ✸✳✷✳✻✳ ❈❤♦ {ψ , H } n l l=0 l ❝â t➼♥❤ ❝❤➜t ♥❤÷ tr♦♥❣ t❤✐➳t ❧➟♣ ❝❤✉♥❣✱ ❣✐↔ sû r➡♥❣ H(γ) H(γ) = I ✈ỵ✐ γ ∈ T✳ ❑❤✐ ✤â ∗ ✭✐✮ {Tk ψ0 }k∈Z ❧➔ ♠ët ❞➣② ❇❡ss❡❧ ✈ỵ✐ P(|ψ0 |2 ) ≤ 1✳ ✭✐✐✮ ◆➳✉ f ∈ L2 (R) t❤➻ lim j→−∞ | f, Dj Tk ψ0 |2 = 0✳ k∈Z ❇➙② ❣✐í t❛ s➩ ①➙② ❞ü♥❣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♥❣✉②➯♥ t➢❝ ❦❤❛✐ tr✐➸♥ ✤ì♥ ♥❤➜t ✭❯❊P✮✳ ✣à♥❤ ❧➼ ✸✳✷✳✼✳ ❈❤♦ {ψ , H } n l l=0 l ❝â t➼♥❤ ❝❤➜t ♥❤÷ tr♦♥❣ t❤✐➳t ❧➟♣ ❝❤✉♥❣✱ ❣✐↔ sû r➡♥❣ H(γ) H(γ) = I ✈ỵ✐ γ ∈ T✳ ❑❤✐ ✤â {Dj Tk ψl }j,k∈Z,l=1, n t↕♦ t❤➔♥❤ ♠ët ❦❤✉♥❣ ❝❤➦t ❝❤♦ ∗ L2 (R) ✈ỵ✐ ❝➟♥ ❦❤✉♥❣ A = ự ợ > trữợ t ❤➔♠ f s❛♦ ❝❤♦ fˆ ∈ Cc (R)✳ ❚❤❡♦ ❜ê ✤➲ ✸✳✷✳✸✱ ❝❤å♥ J > s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ j > J ✱ t❛ ❝â (1 − ) f ❚❤❡♦ ❇ê ✤➲ ✸✳✷✳✺ | f, Dj Tk ψ0 |2 ≤ (1 + ) f ≤ ✭✸✳✶✻✮ k∈Z j n | f, Dj−1 Tk ψl |2 | f, D Tk ψ0 | = k∈Z l=0 k∈Z n | f, D = j−1 | f, Dj−1 Tk ψl |2 Tk ψ0 | + k∈Z ▲➦♣ ❧↕✐ ❧➟♣ ❧✉➟♥ tr➯♥ ❝❤♦ l=1 k∈Z | f, Dj−1 Tk ψ0 |2 ✈ỵ✐ m < j ✱ t❛ ❝â kZ ũ Pữủ ự Pì P❍⑩P ❚❍⑩❈ ❚❘■➎◆ ✣❒◆ ◆❍❻❚ n j m | f, D Tk ψ0 | = j−1 | f, Dp Tk ψl |2 ✳ | f, D Tk ψ0 | + k∈Z l=1 p=m k∈Z k∈Z ❚❤❡♦ ❝æ♥❣ t❤ù❝ ✭✸✳✶✻✮✱ ✈ỵ✐ ♠å✐ j > J ✈➔ m < j t❤➻ n (1 − ) f m ≤ l=1 p=m k∈Z ≤ (1 + ) f ❇ê ✤➲ ✸✳✸✳✺ ✱ t❛ ❝â k∈Z n 2 | f, Dm Tk ψ0 |2 = 0✳ ❑❤✐ ✤â✱ ✈ỵ✐ m → −∞ ✈➔ ♠å✐ lim m→−∞ j > J t❛ ✤÷đ❝ (1 − ) f | f, Dp Tk ψl |2 | f, D Tk ψ0 | + k∈Z ❚❤❡♦ j−1 j−1 ≤ | f, Dp Tk ψl |2 ≤ (1 + ) f ✳ | f, Dp Tk ψl |2 ≤ (1 + ) f ✳ l=1 p=−∞ k∈Z ❈❤♦ j → ∞✱ n (1 − ) f ∞ ≤ l=1 p=−∞ k∈Z ❱ỵ✐ > tũ ỵ t t r n | f, Dp Tk ψl |2 = f ✳ l=1 p∈Z kZ ợ f L2 (R) t ữủ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✸✳✷✳✽✳ ❈❤♦ {ψ , H } n l l=0 l            ❝â t➼♥❤ ❝❤➜t ♥❤÷ tr♦♥❣ t❤✐➳t ❧➟♣ ❝❤✉♥❣✱ ❣✐↔ sû r➡♥❣ n |Hl (γ)|2 =1 l=0 n (3.17) Hl (γ)T1/2 Hl (γ) =0 l=0 ✈ỵ✐ γ ∈ T✳ ❑❤✐ ✤â ❤➺ ✤❛ sâ♥❣ ♥❤ä {Dj Tk ψl }j,k∈Z,l=1, n t↕♦ t❤➔♥❤ ♠ët ❦❤✉♥❣ ❝❤➦t ❝õ❛ L2 (R) ✈ỵ✐ ❝➟♥ ❦❤✉♥❣ A = 1✳ ▼ët ù♥❣ ❞ö♥❣ ❝õ❛ ✣à♥❤ ❧➼ ✸✳✷✳✼ ❧➔ t❛ ❝â t❤➸ ❝❤➾ r❛ ❝→❝❤ ①➙② ❞ü♥❣ ❦❤✉♥❣ ❝❤➦t ✤❛ sâ♥❣ ♥❤ä ❝â ❣✐→ ❝♦♠♣❛❝t ❞ü❛ tr➯♥ ❝→❝ B ✲s♣❧✐♥❡s✳ ❱ỵ✐ ❝→❝ B ✲s♣❧✐♥❡s ❧➔ ❝→❝ ❤➔♠ Bn , n ∈ N ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ∞ Bn+1 (x) = Bn ∗ B1 (x) = Bn (x − t)B1 (t)dt −∞ 1/2 Bn (x − t)dt = −1/2 ❇ị✐ ❚❤à P❤÷đ♥❣ ✸✾ ❚♦→♥ ù♥❣ ❞ư♥❣ ✸✳✷✳ P❍×❒◆● P❍⑩P ❚❍⑩❈ ❚❘■➎◆ ✣❒◆ ◆❍❻❚ ❚r♦♥❣ ✤â B1 (x) = χ[−1/2;1/2] (x)✱ ❝→❝ ♥❤➙♥ tû s✐♥❤ ❧➔ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❤ú✉ ❤↕♥ ❝õ❛ ❝→❝ ♠è✐ ♥è✐ Bm (2x − k), k ∈ Z ❝â ❣✐→ ❝♦♠♣❛❝t✳ ❱➼ ❞ư ✸✳✶✳ ❱ỵ✐ m = 1, 2, ✱ t❛ ①➨t ψ = B2m ❱ỵ✐ n ∈ N✱ t❛ ❝â n eπiγ − e−πiγ 2πiγ Bn (γ) = = sin(πγ) πγ n ❱➟② sin(πγ) πγ ψˆ0 (γ) = 2m ❘ã r➔♥❣ r➡♥❣ lim ψ0 (γ) = ✈➔ γ→0 sin(2πγ) 2πγ ψ0 (2γ) = 2m = sin(πγ) cos(πγ) 2πγ 2m = cos2m (πγ) ❑❤✐ õ tọ ữỡ tr t ợ ❧å❝ H0 (γ) = cos2m (πγ) ❑➼ ❤✐➺✉ ❤➺ sè ♥❤à t❤ù❝ ❧➔ 2m l (2m)! (2m − l)!l! = ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ❤➔♠ H1 , H2m ∈ L∞ (T) ❜ð✐ Hl (γ) = 2m sinl (πγ) cos2m−l (πγ)✱ l = 1, , 2m l ✭✸✳✶✽✮ ❙û ❞ö♥❣ cos(π(γ − 1/2)) = sin(πγ) ✈➔ sin(π(γ − 1/2)) = − cos(πγ) t❛ ✤÷đ❝ ♠❛ tr➟♥ H tr♦♥❣ ❝ỉ♥❣ t❤ù❝ ✭✸✳✻✮   H0 (γ) T1/2 H0 (γ)  H1 (γ) T1/2 H1 (γ)       H(γ) =        H2m (γ) T1/2 H2m (γ)           =         sin2m (πγ) cos2m (πγ) 2m sin(πγ) cos2m−l (πγ) l − 2m sin2 (πγ) cos2m−2 (πγ) 2m sin2m (πγ) 2m   2m  cos(πγ) sin2m−l (πγ) l    2m  cos2 (πγ) sin2m−2 (πγ)            2m 2m cos (πγ) 2m ❳➨t ♠❛ tr➟♥ × 2✱ M = H(γ)∗ H(γ) ✈➔ sû ❞ư♥❣ ❦❤❛✐ tr✐➸♥ ♥❤à t❤ù❝ ❇ị✐ ❚❤à P❤÷đ♥❣ ✹✵ ❚♦→♥ ù♥❣ ❞ư♥❣ ✸✳✷✳ P❍×❒◆● P❍⑩P ❚❍⑩❈ ❚❘■➎◆ ✣❒◆ ◆❍❻❚ 2m 2m (x + y) 2m l 2m−l xy l = l=0 ❙è ❤↕♥❣ ✤➛✉ t✐➯♥ tr♦♥❣ ❤➔♥❣ ✤➛✉ t✐➯♥ ❝õ❛ M ❧➔ 2m M1,1 = l=0 2m sin2l (πγ) cos2(2m−l) (πγ) l = (sin2 (πγ) + cos2 (πγ))2m = ❚÷ì♥❣ tü t❛ ❝â M2,2 = 1✳ ❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❦❤❛✐ tr✐➸♥ ♥❤à t❤ù❝ ✈ỵ✐ x = 1, y = −1 2m 2m 2m + − + 2m M1,2 = sin2m (πγ) cos2m (πγ) − = (sin2m (πγ) cos2m (πγ))(1 − 1)2m = ❚❤❡♦ ✣à♥❤ ❧➼ ✸✳✷✳✶ ✱ ❣✐↔ sû r➡♥❣ ❝→❝ ❤➔♠ ψ , ψ 2m ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ψl (γ) = Hl (γ/2)ψ0 (γ/2) = 2m sin2m+l (πγ/2) cos(2m−l) (πγ/2) l (πγ/2)2m ❈→❝ ❤➔♠ ♥➔② s➩ t↕♦ t❤➔♥❤ ♠ët ❦❤✉♥❣ ❝❤➦t ✤❛ sâ♥❣ ♥❤ä {Dj Tk ψl }j,k∈Z,l=1, ,2m ❝õ❛ L2 (R)✳ ❱➼ ❞ö ✸✳✷✳ ❚❛ t✐➳♣ tö❝ ❱➼ ❞ư ✸✳✶ ✱ ♥❤÷♥❣ t❛ ✤à♥❤ ♥❣❤➽❛ 2m sinl (πγ) cos(2m−l) (πγ)✱ l = 1, , 2m✳ l Hl (γ) = il ✭✸✳✶✾✮ ❚r♦♥❣ ❝æ♥❣ t❤ù❝ tr➯♥✱ Hl ❦❤→❝ tr♦♥❣ ❝æ♥❣ t❤ù❝ ✭✸✳✶✽✮ ð ❤➺ sè ♣❤ù❝ ❝â ✤ë ❧ỵ♥ ❜➡♥❣ 1✳ ❑❤✐ ✤â ❝→❝ ❤➔♠ ψ1 , , ψ2m ✤÷đ❝ ❝❤♦ ❜ð✐ ψl (γ) = Hl (γ/2)ψ0 (γ/2) = il 2m sinl (πγ/2) cos2m−l (πγ/2)ψ0 (γ/2) l (3.20) ❝ô♥❣ t↕♦ r❛ ♠ët ❦❤✉♥❣ ❝❤➦t ✤❛ sâ♥❣ ♥❤ä✳ ❚❛ ✈✐➳t ❧↕✐ Hl (γ/2) ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❊✉❧❡r l Hl (γ/2) = i = 2−2m eπiγ/2 − e−πiγ/2 2i 2m l 2m l l eπiγ/2 − e−πiγ/2 eπiγ/2 + e−πiγ/2 l eπiγ/2 + e−πiγ/2 2m−l 2m−l (3.21) ❱➟② Hl (γ/2) ❧➔ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❤ú✉ ❤↕♥ ❝→❝ ♣❤➛♥ tû ❇ị✐ ❚❤à P❤÷đ♥❣ ✹✶ ❚♦→♥ ù♥❣ ❞ư♥❣ ✸✳✷✳ P❍×❒◆● P❍⑩P ❚❍⑩❈ ❚❘■➎◆ ✣❒◆ ◆❍❻❚ e−πimγ , e−πi(m−1)γ , , eπi(m−1)γ , eπimγ ✱ ✈ỵ✐ t➜t ❝↔ ❝→❝ ❤➺ sè tr♦♥❣ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❧➔ t❤ü❝✳ ❚❛ ✈✐➳t eπikγ = Ek/2 (γ) ✈➔ sû ❞ö♥❣ ψl (γ) = √ 2Hl (γ/2)D−1 ψ0 (γ)✳ ❑❤✐ ✤â ψl ❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❤ú✉ ❤↕♥ ❝→❝ ♣❤➛♥ tû ✈ỵ✐ ❤➺ sè t❤ü❝ Ek/2 D−1 ψ0 = FT−k/2 Dψ0 = FDTk ψ0 ✱ k = −m, , m✳ ❱➔ ψl ❧➔ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❤ú✉ ❤↕♥ ❝→❝ ❤➔♠ ✈ỵ✐ ❤➺ sè t❤ü❝✳ DTk ψ0 ✱ k = −m, , m✳ ◆❤÷ ✈➟② DTm ψ0 ❝â ❣✐→ ♥➡♠ tr♦♥❣ ✤♦↕♥ [0, m]✱ DT−m ψ0 ❝â ❣✐→ ♥➡♠ tr♦♥❣ ✤♦↕♥ [−m, 0]✱ ✈➔ ψl ❝â ❣✐→ ♥➡♠ tr♦♥❣ ✤♦↕♥ [−m, m]✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ m = 1✱ t❛ t❤✉ ✤÷đ❝ ❤➔♠ ψ1 ✈➔ ψ2 ✳ ⑩♣ ❞ư♥❣ ❦❤❛✐ tr✐➸♥ ✭✸✳✷✶✮ ❝❤♦ H1 ψ1 (γ) = H1 (γ/2)ψ0 (γ/2) √ = eπiγ/2 − e−πiγ/2 eπiγ/2 + e−πiγ/2 B2 (γ/2) πiγ e − e−πiγ D−1 B2 (γ) = E1/2 D−1 FB2 (γ) − E−1/2 D−1 FB2 (γ) = = F T−1/2 DB2 − T1/2 DB2 (γ) ❉♦ ✤â T−1/2 DB2 (x) − T1/2 DB2 x)) = √ (B2 (2x + 1) − B2 (2x − 1)) ψ1 (x) = ❚÷ì♥❣ tü ❝❤ó♥❣ r❛ ❝â ❦❤❛✐ tr✐➸♥ ❤➔♠ ψ2 ψ2 (x) = ❇ị✐ ❚❤à P❤÷đ♥❣ (B2 (2x + 1) − 2B2 (2x) + B2 (2x − 1)) ✹✷ ❚♦→♥ ù♥❣ ❞ư♥❣ ✸✳✷✳ P❍×❒◆● P❍⑩P ❚❍⑩❈ ❚❘■➎◆ ✣❒◆ ◆❍❻❚ ❇ị✐ ❚❤à P❤÷đ♥❣ ❍➻♥❤ ✸✳✶✳ ❍➔♠ ψ1 ❍➻♥❤ ✸✳✷✳ ❍➔♠ ψ2 ✹✸ ❚♦→♥ ù♥❣ ❞ö♥❣ ✸✳✸✳ Pì PP Pữỡ t tr ỡ ♥❤➜t ❧➔ ♠ët ❝ỉ♥❣ ❝ư r➜t ❤ú✉ ➼❝❤ ✤➸ ①➙② ỹ t t ỵ tt t ♠ð rë♥❣ ♥➔② ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ✈✐➺❝ ①➙② ❞ü♥❣ ♠❛ tr➟♥ ❜➜t ❦➻ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♥➔♦ ✤â✳ ❚✉② ♥❤✐➯♥✱ ♥❣✉②➯♥ t➢❝ ♥➔② ❝â ♠ët sè ❤↕♥ ❝❤➳✳ ❚r♦♥❣ t➜t ❝↔ ❝→❝ ❝➜✉ tró❝ s♣❧✐♥❡✱ ❝â ➼t ♥❤➜t ♠ët tr♦♥❣ sè ❝→❝ sâ♥❣ ♥❤ä ❜à tr✐➺t t✐➯✉ ✈➔ ❦❤æ♥❣ ❝â ♠ët ❦❤✉♥❣ ♥➔♦ tr♦♥❣ ❝→❝ ❦❤✉♥❣ ♥➔② ❝â ❜➟❝ ①➜♣ ①➾ ❝❛♦ ❤ì♥ 2✳ ✣➸ t❤❛② t❤➳ ♥❣✉②➯♥ t➢❝ ❦❤❛✐ tr✐➸♥ ✤ì♥ ♥❤➜t✱ t❛ ①➙② ❞ü♥❣ ♠ët ♥❣✉②➯♥ t➢❝ ♠ð rë♥❣ tê♥❣ q✉→t ❤ì♥✳ ✣â ❧➔ ♥❣✉②➯♥ t➢❝ t❤→❝ tr✐➸♥ ♥❣❤✐➯♥❣✳ ✸✳✸ P❤÷ì♥❣ ♣❤→♣ t❤→❝ tr✐➸♥ ♥❣❤✐➯♥❣ ◆➳✉ {ψl }nl=1 ✤÷đ❝ ①➙② ❞ü♥❣ t❤❡♦ t❤✐➳t ❧➟♣ ❝❤✉♥❣ ✈➔ ♥❣✉②➯♥ t➢❝ t❤→❝ tr✐➸♥ ✤ì♥ ♥❤➜t✱ t❛ ❝â ψl (γ) = Hl (γ/2)ψˆ0 (γ/2) ✈➔ ψ0 (0) = 1✳ ❚ø ✤➙②✱ sè ❧÷đ♥❣ ❝→❝ ❤➔♠ ψl = t↕✐ γ = ❧➔ ❜➡♥❣ 0✳ ✣✐➲✉ ♥➔② t❤ü❝ sü ❧➔ ♠ët ❤↕♥ ♥➳✉ ①➙② ❞ü♥❣ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ t❤→❝ tr✐➸♥ ✤ì♥ ♥❤➜t✳ ❱➼ ❞ö ✸✳✸✳ ●✐↔ sû ♠è✐ ♥è✐ B 2m ❧å❝ tr♦♥❣ ✈➼ ❞ư ✸✳✶ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ t✛ ❧➺ ✈ỵ✐ ♠➔♥❣ H0 (γ) = cos2m (πγ) ◆➳✉ t❛ ố ỹ ởt t ỵ t tr ✤ì♥ ♥❤➜t✱ ✤✐➲✉ ❦✐➺♥ ❧➔ H(γ)∗ H(γ) = I ✱ ❝ö t❤➸ n |Hl (γ)|2 1= l=0 ❑❤✐ ✤â n |Hl (γ)|2 = − cos2m (πγ)✳ ✭✸✳✷✷✮ l=1 ❙è ❤↕♥❣ t❤ù tr♦♥❣ ❦❤❛✐ tr✐➸♥ ❤➔♠ − cos2m (πγ) ❜➡♥❣ t↕✐ γ = 0✳ ❱➻ ✈➟②✱ ð ❜➯♥ tr→✐ ❝õ❛ ❝ỉ♥❣ t❤ù❝ ✭✸✳✷✷✮✱ ❝❤ó♥❣ t❛ ❝❤➾ ❝â ♥❤➙♥ tû γ ✳ ✣✐➲✉ ♥➔② ♥❣❤➽❛ ❧➔ ❝â ➼t ♥❤➜t ♠ët tr♦♥❣ ❝→❝ ❤➔♠ |Hl |2 ❝â sè ❤↕♥❣ t❤ù ❜➡♥❣ t↕✐ γ = 0✱ ✈➔ ❝â ➼t ♥❤➜t ♠ët tr♦♥❣ ❝→❝ ❤➔♠ ψl = t↕✐ γ = 0✳ ❱➻ ❤↕♥ ❝❤➳ ✤â✱ ❦❤â ❝â t❤➸ ①➙② ❞ü♥❣ ❦❤✉♥❣ ❦❤✐ ❧➔♠ ✈✐➺❝ ✈ỵ✐ ♠ët t➟♣ ❝→❝ ❤➔♠ {ψi , Hi }ni=0 ♥❤÷ tr♦♥❣ ♣❤➛♥ t❤✐➳t ❧➟♣ ❝❤✉♥❣✳ ❚✉② ♥❤✐➯♥✱ ♥➳✉ t❛ t❤❛② ✤ê✐ ❝→❝❤ ❝❤å♥ ❝→❝ ❤➔♠ ♥➔② t❤➻ ❝â t❤➸ ❞➝♥ ✤➳♥ ♠ët ❝→❝❤ ①➙② ❞ü♥❣ ❦❤→❝ ❬✹❪✱ ❬✸❪✳ ❈→❝❤ ①➙② ❞ü♥❣ ♥➔② ❧✐♥❤ ✤à♥❤ ❧➼ ✸✳✸✳✶ ✳ ✣â ❧➔ ♣❤÷ì♥❣ ♣❤→♣ t❤→❝ tr✐➸♥ ♥❣❤✐➯♥❣ ✭❖❊P✮✳ ✣à♥❤ ❧➼ ✸✳✸✳✶✳ ❈❤♦ {ψ , H } ❝â t➼♥❤ ❝❤➜t ♥❤÷ tr♦♥❣ t❤✐➳t ❧➟♣ ❝❤✉♥❣✳ ●✐↔ sỷ r tỗ t ỡ l n l l=0 t↕✐ ♠ët ❤➔♠ ①→❝ ✤à♥❤ θ ∈ L (T) ♠➔ lim θ(γ) = ✈➔ ✈ỵ✐ ♠å✐ γ ∈ T✱ t❛ ❝â γ→0 ❇ị✐ ❚❤à P❤÷đ♥❣ ✹✹ ❚♦→♥ ù♥❣ ❞ư♥❣ ✸✳✸✳ P❍×❒◆● P❍⑩P ❚❍⑩❈ ❚❘■➎◆ ◆●❍■➊◆● n H0 (γ)H0 (γ + ν)θ(2γ) + l=1  θ(γ) ν = Hl (γ)Hl (γ + ν)❂ 0 ν = (3.23) t❤➻ ❞➣② ❤➔♠ {Dj Tk ψl }j,k∈Z,l=1, ,n t↕♦ t❤➔♥❤ ♠ët ❦❤✉♥❣ ❝❤➦t ❝õ❛ L2 (R) ✈ỵ✐ ❝➟♥ ❦❤✉♥❣ A = 1✳ ❈❤ù♥❣ ♠✐♥❤✿ ●✐↔ sû r➡♥❣ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ ✤à♥❤ ❧➼ ✸✳✸✳✶ t❤ä❛ ♠➣♥ ✈➔ ❤➔♠ ψ0 ∈ L2 (R) ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ψ0 (γ) = ✭✸✳✷✹✮ θ(γ)ψ0 (γ) ❱➔ ❝→❝ ❤➔♠ H0 , , Hn t✉➛♥ ❤♦➔♥ ❝❤✉ ❦➻ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ θ(2γ) H0 (γ), Hl (γ) = θ(γ) H0 (γ) = Hl (γ), l = 1, , n θ(γ) ✭✸✳✷✺✮ Þ t÷ð♥❣ ❝õ❛ ❝❤ù♥❣ ♠✐♥❤ ❧➔ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❦❤❛✐ tr✐➸♥ ✤ì♥ ♥❤➜t ❝❤♦ ψ0 , H0 , , Hn ✳ ❑❤✐ ✤â t❛ t❤✉ ✤÷đ❝ Dj Tk ψl j,k∈Z,l=1, ,n ❧➔ ♠ët ❦❤✉♥❣ ❝❤➦t✱ ✈➔ ψl = ψl ✈ỵ✐ l = 1, , n ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ψ0 , H0 , , Hn t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♥❤÷ tr♦♥❣ t❤✐➳t ❧➟♣ ❝❤✉♥❣✳ ✣➛✉ t✐➯♥✱ ψ0 (2γ) = θ(2γ)ψ0 (2γ) = θ(2γ)H0 (γ)ψ0 (γ) = θ(2γ) H0 (γ)ψ0 (γ) θ(γ) = H0 (γ)ψ0 (γ) ❱➻ ✈➟② lim ψ0 (γ) = lim γ→0 θ(γ)ψ0 (γ) = γ→0 ❚ø ❝ỉ♥❣ t❤ù❝ ✭✸✳✷✸✮ ✈➔ ✭✸✳✷✺✮✱ ✈ỵ✐ ν = 0✱ n θ(2γ) |Hl (γ)| = |H0 (γ)|2 + θ(γ) n l=0 l=1 |Hl (γ)|2 = θ(γ) ❑❤✐ ✤â H0 , , Hn ∈ L∞ (T)✳ ❱➻ θ (2(γ + 1/2)) = θ(2γ) ♥➯♥ n Hl (γ)Hl (γ + 1/2) = l=0 + θ(2γ) θ(γ)θ(γ + 1/2) θ(γ)θ(γ + 1/2) H0 (γ)H0 (γ + 1/2) n Hl (γ)Hl (γ + 1/2) l=1 = ❈→❝ ❤➔♠ ψ1 , , ψn ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❇ị✐ ❚❤à P❤÷đ♥❣ ✹✺ ❚♦→♥ ù♥❣ ❞ư♥❣ ✸✳✸✳ P❍×❒◆● P❍⑩P ❚❍⑩❈ ❚❘■➎◆ ◆●❍■➊◆● ψl (2γ) = Hl (γ)ψ0 (γ)✱ ✈ỵ✐ l = 1, , n ❚❤❡♦ ✤à♥❤ ❧➼ ✸✳✸✳✶ ✱ ❝→❝ ❤➔♠ Dj Tk ψl ✭✸✳✷✻✮ j,k∈Z,l=1, ,n t↕♦ t❤➔♥❤ ♠ët ❦❤✉♥❣ ❝❤➦t ❝õ❛ L2 (R) ✈ỵ✐ ❝➟♥ ❦❤✉♥❣ A = 1✳ ❚❛ ❝â ψl (2γ) = Hl (γ)ψ0 (γ) = θ(γ)Hl (γ) θ(γ) ψ0 (γ) = ψl (2γ)✱ ✤✐➲✉ ♥➔② ❝❤➾ r❛ r➡♥❣ ψl = ψl ❇➡♥❣ ❝→❝❤ ❝❤♦ θ = tr♦♥❣ ✤à♥❤ ❧➼ ✸✳✹✳✶ t❛ t❤✉ ✤÷đ❝ ✤à♥❤ ❧➼ ✸✳✸✳✶ ✳ ❈→❝ ❦❤✉♥❣ ❝â t❤➸ ✤÷đ❝ ①➙② ❞ü♥❣ ♥❤÷ ♥❤❛✉ tr♦♥❣ ❤❛✐ ✤à♥❤ ❧➼ ♥➔②✳ ◆❤÷♥❣ ✈✐➺❝ ❧ü❛ ❝❤å♥ tị② þ θ tr♦♥❣ ✤à♥❤ ❧➼ ✸✳✹✳✶ ❝❤♦ t❛ ♠ët ❦➳t q✉↔ tê♥❣ q✉→t ✈➔ ❧✐♥❤ ❤♦↕t ❤ì♥ ✤à♥❤ ❧➼ ✸✳✸✳✶ ✳ ❈❤♦ θ = ✈➔ ❣✐↔ sû r➡♥❣ ψ0 ❧➔ ♠ët ❤➔♠ ❝â ❣✐→ ❝♦♠♣❛❝t t❤ä❛ ♠➣♥ ψ0 (2γ) = H0 (γ)ψ0 (γ)✳ ❚r♦♥❣ ✤â ❝→❝ ❤➔♠ H0 ∈ L∞ (T) ✈➔ ❝→❝ ❤➔♠ θ✱ Hl ✈ỵ✐ l = 1, n ❧➔ ❝→❝ ✤❛ t❤ù❝ ❧÷đ♥❣ ❣✐→❝ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ ✤à♥❤ ❧➼ ✸✳✹✳✶ ✳ ❚❛ ❝â ❜✐➸✉ ❞✐➵♥ H (γ) = ck e2πikγ ✭tê♥❣ ❤ú✉ l k∈Z ❤↕♥✮✱ ✈➔ ψl ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ ckl T−k ψ0 (γ)✳ ψl (2γ) = Hl (γ)ψ0 (γ) = F k∈Z ❑❤è✐ ❧÷đ♥❣ t➼♥❤ t♦→♥ s➩ t➠♥❣ ❧➯♥ t❤❡♦ sè ❧÷đ♥❣ ❝→❝ ♥❤➙♥ tû s✐♥❤ tr♦♥❣ ♠ët ❦❤✉♥❣ ✤❛ sâ♥❣ ♥❤ä✳ ❱➻ ✈➟②✱ ♥❤➻♥ ❝❤✉♥❣ t❛ ♠♦♥❣ ♠✉è♥ ❝â ➼t ♥❤➜t sè ♥❤➙♥ tû s✐♥❤ ❝â t❤➸ ✳ ❚èt ♥❤➜t ❧➔ ①➙② ❞ü♥❣ ♠ët ❝➦♣ ❤➔♠ ψ, ψ s❛♦ ❝❤♦ ❤➺ ✤❛ sâ♥❣ ♥❤ä {Dj Tk ψ}j,k∈Z ✱ Dj Tk ψ j,k∈Z t↕♦ t❤➔♥❤ ♠ët ❦❤✉♥❣ ❦➨♣✳ ✣à♥❤ ❧➼ ✸✳✸✳✷✳ ●✐↔ sû r➡♥❣ {ψ , H } t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t ♥❤÷ t❤✐➳t ❧➟♣ ❝❤✉♥❣ ✈➔ 0 {Tk ψ0 }k∈Z ❧➔ ♠ët ❝❤✉é✐ ❘✐❡s③✳ ◆➳✉ |H0 (− 14 )| = sâ♥❣ ♥❤ä {Dj Tk ψ}j,k∈Z ✱ Dj Tk ψ span {DTk ψ0 }k∈Z ✳ j,k∈Z √1 t❤➻ ổ tỗ t ởt tr õ , õ t V1 = ỵ t❤→❝ tr✐➸♥ ♥❣❤✐➯♥❣ r➜t ❤ú✉ ➼❝❤ tr♦♥❣ ✈✐➺❝ ①➙② ❞ü♥❣ ❝→❝ ❦❤✉♥❣ ✤❛ sâ♥❣ ♥❤ä ❞ü❛ tr➯♥ ❝→❝ B ✲s♣❧✐♥❡✳ ✣à♥❤ ❧➼ ✸✳✸✳✸✳ ❈❤♦ B 2m ❧➔ B ✲s♣❧✐♥❡ ❝➜♣ ✈ỵ✐ ♠➔♥❣ ❧å❝ H0 = cos2m (πγ)✳ ❱ỵ✐ ♠é✐ số ữỡ M 2m tỗ t ởt t❤ù❝ θ ❝â ❞↕♥❣ M −1 cj sin2j (πγ) θ(γ) = + (3.27) j=1 t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤➙② ❇ị✐ ❚❤à P❤÷đ♥❣ ✹✻ ❚♦→♥ ù♥❣ ❞ư♥❣ ✸✳✸✳ Pì PP cj ợ ♠å✐ j = 1, , M − 1✱ ✈➔ θ(γ) > ✈ỵ✐ ♠å✐ γ ∈ R ✭✐✐✮ ◆➳✉ f ∈ L2 (R) t❤➻ lim j→−∞ | f, Dj Tk ψ0 |2 = 0✳ k∈Z ◆➳✉ {Hl , ψl }nl=0 t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ♥❤÷ tr♦♥❣ t❤✐➳t ❧➟♣ ❝❤✉♥❣ ✈➔ {Dj Tk ψl }j,k∈Z,l=1, ,n ❧➔ ♠ët ❦❤✉♥❣ ❝❤➦t ✤÷đ❝ ỹ t ỵ t tr t Vj = s♣❛♥ {DTk ψ0 }k∈Z ●✐↔ sû ❤➔♠ ψ0 ❝â ❜➟❝ ①➜♣ ①➾ s✱ t❛ ①➨t ❤➔♠ ∞ j−1 n |H0 (2m γ)|2 |Hl (γ)| Θ(γ) = m=0 j=0 l=1 ❇➟❝ ①➜♣ ①➾ ❝õ❛ {Dj Tk ψl }j,k∈Z,l=1, ,n ✤↕t ❣✐→ trà ❝ü❝ ✤↕✐ ✭✤↕t ✤➳♥ ❣✐→ trà s✮ ♥➳✉ ❤➔♠ ❤➔♠ Θ t❤ä❛ ♠➣♥ n |Hl (γ)|2 ✳ Θ(γ) = |H0 (γ)|2 Θ(γ) + l=1 ▼ët tr♦♥❣ ❤❛✐ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ ♥❣✉②➯♥ t➢❝ t❤→❝ tr✐➸♥ ♥❣❤✐➯♥❣ t❤ä❛ ♠➣♥ ♥➳✉ ❝❤ó♥❣ t❛ ❝❤å♥ θ = Θ✳ ❱➼ ❞ư ✸✳✹✳ ❈❤ó♥❣ t❛ trð ❧↕✐ ✈➼ ❞ư ✸✳✸ tr♦♥❣ tr÷í♥❣ ❤đ♣ m = 1✱ t❛ ①➨t ψ = T1 B2 ✈➔ ♠➔♥❣ ❧å❝ H0 (γ) = − (1 + e−2πiγ )2 = e−2πiγ cos2 (πγ)✳ ❙û ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ t❤→❝ tr✐➸♥ ♥❣❤✐➯♥❣ ✈➔ ❝→❝ ❤➺ q✉↔✱ ❝→❝ ❤➔♠ H1 ✈➔ H2 ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ √ −2πiγ H1 (γ) = ie sin(πγ) cos(πγ) = √ ie sin(2πγ) = (1 − e−4πiγ ) (1 − e−2πiγ )2 H2 (γ) = −e−2πiγ sin2 (πγ) = −2πiγ √ (i) ❑❤✐ ✤â ❝→❝ ❤➔♠ ψ1 = ψ1 ✈➔ ψ2 t↕♦ ♥➯♥ ♠ët ❦❤✉♥❣ ❝❤➦t ❝❤♦ L2 (R)✳ ❱ỵ✐ (i) ψ1 = √ (B2 (2x + 1) − B2 (2x − 1))✱ ψ2 (x) = (B2 (2x + 1) − 2B2 (2x) + B2 (2x − 1)) ❇ò✐ ❚❤à Pữủ ự Pì PP ◆●❍■➊◆● ❍➻♥❤ ✸✳✸✳ ❍➔♠ ψ1(i) ❍➻♥❤ ✸✳✹✳ ❇ị✐ ❚❤à P❤÷đ♥❣ ✹✽ ❍➔♠ ψ2 ❚♦→♥ ù♥❣ ❞ö♥❣ ❑➳t ❧✉➟♥ ❚r♦♥❣ ❧✉➟♥ ữủ ợ t t tứ ♠➔ ❦❤✉♥❣ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ q✉❛♥ trå♥❣ tr♦♥❣ ①û ỵ t õ ❝❤ó♥❣ t❛ t➼♥❤ ❜➲♥ ✈ú♥❣✿ ❝❤➜t ❧÷đ♥❣ ❝õ❛ t➼♥ ❤✐➺✉ ữ t ỡ õ t ỗ ✈➔ t➼♥ ❤✐➺✉ ❝â t❤➸ ❦❤ỉ✐ ♣❤ư❝ ❧↕✐ tø ❝→❝ ♠➝✉ ❝â ✤ë ❝❤➼♥❤ ①→❝ t÷ì♥❣ ✤è✐ t❤➜♣✳ ❑❤✉♥❣ sâ♥❣ ♥❤ä ❧➔ ♠ët ❦❤✉♥❣ ❝â ❝➜✉ tró❝ ✤➦❝ ❜✐➺t✳ ▲ỵ♣ rt ỳ tr ỷ ỵ t➼♥ ❤✐➺✉ ♥❣➢♥✱ ❝→❝ t➼♥❤ ❤✐➺✉ ❝â ✤➦❝ tr÷♥❣ ❤➻♥❤ ❤å❝ ♣❤ù❝ t↕♣✳ ✣➸ ①➙② ❞ü♥❣ ❦❤✉♥❣ sâ♥❣ ♥❤ä ❝â ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❦❤→❝ ♥❤❛✉✳ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ①➙② ❞ü♥❣ ❦❤✉♥❣ sâ♥❣ ♥❤ä ❞ü❛ tr➯♥ ❝→❝ ỵ rở ứ õ õ t s s ✤÷đ❝ ÷✉ ✤✐➸♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ❤↕♥ ❝❤➳ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤â✳ ✹✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❇❛❧❛③✱ P ✿ ❇❛s✐❝ ❞❡❢✐♥✐t✐♦♥ ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❇❡ss❡❧ ♠✉❧t✐♣❧✐❡rs✳ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✷✺✭✶✮✱ ✺✼✶✲✺✽✺ ✭✷✵✵✼✮✳ ❬✷❪ ❇❡♥❡❞❡tt♦✱ ❏✱ ▲✐✱ ❙✿ ❚❤❡ t❤❡♦r② ♦❢ ♠✉❧t✐r❡s♦❧✉t✐♦♥ ❛♥❛❧②s✐s ❢r❛♠❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❢✐❧t❡r ❜❛♥❦s✳ ❆♣♣❧✳ ❈♦♠♣✉t✳ ❍❛r♠♦♥✳ ❆♥❛❧✳ ✺✱ ✸✽✾✲✹✷✼ ✭✶✾✾✽✮✳ ❬✸❪ ❈❤✉✐✱ ❈✱ ❙❤✐✱ ❳✿ ❖rt❤♦♥♦r♠❛✐❧ ✇❛✈❡❧❡t ❛♥❞ t✐❣❤t ❢r❛♠❡s ✇✐t❤ ❛r❜✐tr❛r② r❡❛❧ ❞✐❧❛✲ t✐♦♥s✳ ❆♣♣❧✳ ❈♦♠♣✉t✳ ❍❛r♠♦♥✳ ❆♥❛❧✳ ✾✭✸✮✱ ✷✹✸✲✷✻✹ ✭✷✵✵✵✮✳ ❬✹❪ ❉❛✉❜❡❝❤✐❡s✱ ■✱ ❍❛♥✱ ❇✱ ❘♦♥✱ ❆✱ ❙❤❡♥✱ ❩✳✿ ❋r❛♠❡❧❡t✿ ▼❘❆✲❜❛s❡❞ ❝♦♥str✉❝t✐♦♥s ♦❢ ✇❛✈❡❧❡t ❢r❛♠❡s✳ ❆♣♣❧✳ ❈♦♠♣✳ ❍❛r♠✳ ❆♥❛❧✳ ✶✹✭✶✮ ✶✲✹✻ ✭✷✵✵✸✮✳ ❬✺❪ ❉❛✉❜❡❝❤✐❡s✱ ■✿ ❚❡♥ ▲❡❝t✉r❡s ♦♥ ❲❛✈❡❧❡ts✳ ❙■❆▼✱ P❤✐❧❛❞❡♥♣❤✐❛ ✭✶✾✾✷✮✳ ❬✻❪ ▼❛❧❧❛t✱ ❙✿ ▼✉❧t✐r❡s♦❧✉t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ✇❛✈❡❧❡t ♦rt❤♦♥♦r♠❛❧ ❜❛s❡s ♦❢ L2 (R)✳ ❚r❛♥s✳ ❆♠✳ ▼❛t❤✳ ❙♦❝✳ ✸✶✺✭✶✮✱ ✻✾✲✽✼ ✭✶✾✽✾✮✳ ❬✼❪ ❖❧❡ ❈❤r✐st❡♥s❡♥✿ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❢r❛♠❡s ❛♥❞ ❘✐❡s③ ❜❛s✐❝✳ ❬✽❪ P❛❧❡②✱ ▼✱ ❙✐❦✐❝✱ ❍✱ ❲❡✐ss✱ ●✱ ❳✐❛♦✱ ❙✿ ●❡♥❡r❛❧✐③❡❞ ❧♦✇ ♣❛ss ❢✐❧t❡rs ❛♥❞ ▼❘❆ ❢r❛♠❡ ✇❛✈❡❧❡ts✳ ❏✳ ●❡♦♠✳ ❆♥❛❧✳ ✶✶✭✷✮✱ ✸✶✶✲✸✹✷ ✭✷✵✵✶✮✳ ❬✾❪ ❘♦♥✱ ❆✱ ❙❤❡♥✱ ❩✿ ❆❢❢✐♥❡ s②st❡♠s ✐♥ L2 (Rd )✿ ❚❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❛♥❛❧②s✐s ♦♣❡r❛t♦r✳ ❏✳ ❋✉♥❝t✳ ❆♥❛❧✳ ✶✹✽✱ ✹✵✽✲✹✹✼ ✭✶✾✾✼✮✳ ✺✵