❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❇➳❝❤ ❦❤♦❛ ❍➭ ◆é✐ ❇ï✐ ❳✉➞♥ ❉✐Ö✉ ▼è✐ ❧✐➟♥ ❤Ö ❣✐÷❛ ❤Ư ➤é♥❣ ❧ù❝ rê✐ r➵❝ ✈➭ ❧✐➟♥ tơ❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sÜ ❑❤♦❛ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ❝➠♥❣ ♥❣❤Ö ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿ ✶✳ P●❙✳ ❚❙❑❍✳ ◆❣✉②Ơ♥ ❱➝♥ ▼✐♥❤ ✷✳ ❚❙✳ ❍➭ ❇×♥❤ ▼✐♥❤ ❍➭ ◆é✐ ✲ ✷✵✶✵ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ❝➠♥❣ tr×♥❤ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ r✐➟♥❣ t➠✐✳ ❈➳❝ ❦Õt q✉➯ ✈✐Õt ❝❤✉♥❣ ✈í✐ ❝➳❝ t➳❝ ❣✐➯ ❦❤➳❝ ➤➲ ➤➢ỵ❝ sù ♥❤✃t trÝ ❝ñ❛ ❝➳❝ ➤å♥❣ t➳❝ ❣✐➯ ❦❤✐ ➤➢❛ ✈➭♦ ❧✉❐♥ ✈➝♥✳ ❈➳❝ ❦Õt q✉➯ ♥➟✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ❧➭ tr✉♥❣ t❤ù❝ ✈➭ ❧➭ ❦Õt q✉➯ ❝đ❛ q✉➳ tr×♥❤ ❧➭♠ ✈✐Ư❝ ♥❣❤✐➟♠ tó❝ ❝đ❛ t➠✐ ❝ï♥❣ ✈í✐ ♥❤ã♠ ♥❣❤✐➟♥ ❝ø✉ t➵✐ ❑❤♦❛ ❚♦➳♥ ✲ ❚✐♥ ø♥❣ ❞ô♥❣✱ ➜➵✐ ❤ä❝ ❇➳❝❤ ❑❤♦❛ ❍➭ ◆é✐✳ ❚➳❝ ❣✐➯ ❇ï✐ ❳✉➞♥ ❉✐Ö✉ ✷ ▼ơ❝ ❧ơ❝ ❚r❛♥❣ ♣❤ơ ❜×❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▲ê✐ ❝❛♠ ➤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❇➯♥❣ ❦ý ❤✐Ö✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ▼ë ➤➬✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ▼ô❝ ❧ô❝ ✳ ✳ ✳ ✳ ▲ê✐ ❝➯♠ ➡♥ ✳ ❈❤➢➡♥❣ ✶ ✿ ❚æ♥❣ q✉❛♥ ✶✳✶ ❍➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ❇♦❝❤♥❡r ✶✳✶✳✷ ❑❤➠♥❣ ❣✐❛♥ ❝➳❝ ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✸ ➜Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ❇♦❤r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✹ ➜Þ♥❤ ♥❣❤Ü❛ t❤❡♦ ❦✐Ĩ✉ ❤é✐ tơ tõ♥❣ ➤✐Ĩ♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✳ ✳ ✳ ✳ ✳ ✳ ❉➲② sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ✸ ✳ ✳ ✳ ✶✳✹ ✳ ✳ ✳ ❑❤➠♥❣ ❣✐❛♥ ❝➳❝ ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ✳ ✳ ✳ ✶✳✸ ✳ ✳ ✳ ❉➲② sè ❤➬✉ t✉➬♥ ❤♦➭♥ ✳ ✳ ✳ ✶✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❱➭✐ ♥Ðt ✈Ị ❧Þ❝❤ sư ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ❈❤➢➡♥❣ ✷ ✿ ◗✉❛♥ s➳t ♥❣❤✐Ư♠ ❤➬✉ t✉➬♥ ❤♦➭♥ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✷✸ ✈✐ ♣❤➞♥ ✷✳✶ ●✐í✐ t❤✐Ư✉ ❜➭✐ t♦➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ❈➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ ▼ét sè ø♥❣ ❞ô♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✳ ❈❤➢➡♥❣ ✸ ✿ ◗✉❛♥ s➳t ♥❣❤✐Ö♠ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ❝đ❛ ♣❤➢➡♥❣ ✸✸ tr×♥❤ ✈✐ ♣❤➞♥ ✸✳✶ ●✐í✐ t❤✐Ö✉ ❜➭✐ t♦➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷ ❈➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ❑✐Õ♥ ♥❣❤Þ ❝❤♦ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉ t✐Õ♣ t❤❡♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ❑Õt ❧✉❐♥ ❝❤✉♥❣ ✳ ✳ ❈➳❝ ❝➠♥❣ tr×♥❤ ❝đ❛ t➳❝ ❣✐➯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➯♥❣ ❦ý ❤✐Ö✉ ✈➭ ❝❤÷ ✈✐Õt t➽t Z R C Rn X AA(X) KAA(X) AP(X) BC(X) BUC(X) l∞ (X) aa(X) kaa(X) H(f ) α Tα T (f |R , ε) T (ϕ|Z , ε) Tˆ(ϕ|{tn } , ε) Rf {tn } [t] {t} ❚❐♣ ❤ỵ♣ ❝➳❝ sè ♥❣✉②➟♥ ❚❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝ ❚❐♣ ❤ỵ♣ ❝➳❝ sè ♣❤ø❝ ❑❤➠♥❣ ❣✐❛♥ t❤ù❝ n ❝❤✐Ị✉ Ox1 x2 xn ❑❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❑❤➠♥❣ ❣✐❛♥ ❝➳❝ ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ♥❤❐♥ ❣✐➳ trÞ tr➟♥ X ❑❤➠♥❣ ❣✐❛♥ ❝➳❝ ❤➭♠ sè ❝♦♠♣❛❝t ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ♥❤❐♥ ❣✐➳ trÞ tr➟♥ ❑❤➠♥❣ ❣✐❛♥ ❝➳❝ ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ ♥❤❐♥ ❣✐➳ trÞ tr➟♥ X ❑❤➠♥❣ ❣✐❛♥ ❝➳❝ ❤➭♠ sè ❧✐➟♥ tơ❝✱ ❜Þ ❝❤➷♥ ♥❤❐♥ ❣✐➳ trÞ tr➟♥ X ❑❤➠♥❣ ❣✐❛♥ ❝➳❝ ❤➭♠ sè ❧✐➟♥ tơ❝ ➤Ị✉✱ ❜Þ ❝❤➷♥ ♥❤❐♥ ❣✐➳ trÞ tr➟♥ ❑❤➠♥❣ ❣✐❛♥ t✃t ❝➯ ❝➳❝ ❞➲② sè ❜Þ ❝❤➷♥ tr➟♥ X X ❑❤➠♥❣ ❣✐❛♥ t✃t ❝➯ ❝➳❝ ❞➲② sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ♥❤❐♥ ❣✐➳ trÞ tr➟♥ X X ❑❤➠♥❣ ❣✐❛♥ t✃t ❝➯ ❝➳❝ ❞➲② sè ❝♦♠♣❛❝t ❛✉t♦♠♦r♣❤✐❝ ♥❤❐♥ ❣✐➳ trÞ tr➟♥ ❇❛♦ ❝đ❛ ❤➭♠ sè f ❉➲② sè ✈í✐ ♣❤➬♥ tư tỉ♥❣ q✉➳t ❧➭ αn ❚♦➳♥ tư ❞Þ❝❤ ❝❤✉②Ĩ♥ f ❚❐♣ ❞Þ❝❤ ❝❤✉②Ĩ♥ ❝đ❛ ❚❐♣ ❞Þ❝❤ ❝❤✉②Ĩ♥ ❝đ❛ ❞➲② sè ❚❐♣ ❞Þ❝❤ ❝❤✉②Ĩ♥ ❝đ❛ ❞➲② sè ❚❐♣ ❣✐➳ trÞ ❝đ❛ ❤➭♠ sè ϕn ϕ tr➟♥ {tn } f ❉➲② ❝➳❝ q✉❛♥ s➳t ➤➢ỵ❝ s➽♣ ①Õ♣ t❤❡♦ t❤ø tù t➝♥❣ ❞➬♥ P❤➬♥ ♥❣✉②➟♥ ❝ñ❛ sè t❤ù❝ P❤➬♥ ♣❤➞♥ ❝ñ❛ sè t❤ù❝ ✺ t t X ▼ë ➤➬✉ ▼è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❝➳❝ ❤Ư ➤é♥❣ ❧ù❝ rê✐ r➵❝ ✈➭ ❧✐➟♥ tơ❝✱ ✈í✐ ý t➢ë♥❣ ❝❤Ý♥❤ ❝đ❛ ➳♥❤ ①➵ P♦✐♥❝❛rÐ✱ ➤ã♥❣ ♠ét ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ❧ý t❤✉②Õt ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥✳ ❈➳❝ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ❝➳❝ ❤Ư ➤é♥❣ ❧ù❝ ♥➭② ➤➲ ❝ã ♠ét ❧Þ❝❤ sư ♣❤➳t tr✐Ĩ♥ ❧➞✉ ➤ê✐✱ ❦❤ë✐ ♥❣✉å♥ tõ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉ t✐➟♥ ♣❤♦♥❣ ❝ñ❛ ▲②❛♣✉♥♦✈ ✈➭ P♦✐♥❝❛rÐ✳ ❈❤♦ ➤Õ♥ ♥❛②✱ ➤➲ ❝ã r✃t ♥❤✐Ò✉ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ ❦❤➳❝ ♥❤❛✉ ✈Ị ❝➳❝ ❤Ư ➤é♥❣ ❧ù❝ rê✐ r➵❝ ✈➭ ❧✐➟♥ tơ❝✱ ♥❤➢ sù tå♥ t➵✐✱ tÝ♥❤ ❞✉② ♥❤✃t ♥❣❤✐Ư♠✱ tÝ♥❤ ❧✐➟♥ tơ❝ ➤Ị✉✱ tÝ♥❤ ❜Þ ❝❤➷♥✱ tÝ♥❤ ỉ♥ ➤Þ♥❤✱ tÝ♥❤ ỉ♥ ➤Þ♥❤ t✐Ư♠ ❝❐♥✱ tÝ♥❤ t✉➬♥ ❤♦➭♥✱ tÝ♥❤ ❤➬✉ t✉➬♥ ❤♦➭♥✱ tÝ♥❤ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ❝đ❛ ♥❣❤✐Ư♠✳ ❚✉② ♥❤✐➟♥✱ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❤❛✐ ❤Ư ➤é♥❣ ❧ù❝ ♥➭② ✈➱♥ ❝❤➢❛ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ♠ét ❝➳❝❤ ➤➬② ➤đ ♥❤✃t✳ ▲✉❐♥ ✈➝♥ ♥➭② ❝đ❛ t➳❝ ❣✐➯ ♥❤➺♠ ♠ơ❝ ➤Ý❝❤ ❣✐➯✐ q✉②Õt ♠ét ♣❤➬♥ ❝đ❛ ❜➭✐ t♦➳♥ tr➟♥✳ ◆ã✐ r✐➟♥❣✱ ❝❤ó♥❣ t➠✐ q✉❛♥ t ế t tì ố ệ ữ tí ❤➬✉ t✉➬♥ ❤♦➭♥✱ tÝ♥❤ ❛❧♠♦st ❛✉t♦♠♦r♣❤②✱ ✈➭ tÝ♥❤ ❜Þ ❝❤➷♥ ❝đ❛ ❝➳❝ ♥❣❤✐Ư♠ tr♦♥❣ ❝➳❝ ❤Ư ➤é♥❣ ❧ù❝ ♥➭②✳ ✻ ❑Õt ❝✃✉ ❧✉❐♥ ✈➝♥ ▲✉❐♥ ✈➝♥ ♥➭② ❣å♠ ✸ ❝❤➢➡♥❣✿ trì ế tứ ị ❣✐➯ ♥➟✉ r❛ ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛ ❦❤➳❝ ♥❤❛✉ ❝đ❛ ➤Þ♥❤ ♥❣❤Ü❛ ♠ét ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥✱ ❞➲② sè ❤➬✉ t✉➬♥ ❤♦➭♥✱ ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✱ ❞➲② sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② t➳❝ ❣✐➯ ❝ò♥❣ ❝❤Ø r❛ ♠ét t✐➟✉ ❝❤✉➮♥ ✭➤✐Ị✉ ❦✐Ư♥ ➤đ✮ ➤Ĩ ❦✐Ĩ♠ tr❛ ❦❤✐ ♥➭♦ ♠ét ❞➲② ❝➳❝ q✉❛♥ s➳t {x(tn )}n∈Z ❧➭ ❤➬✉ t✉➬♥ ❤♦➭♥✳ ❈❤➢➡♥❣ ✷ ✈➭ ❝❤➢➡♥❣ ✸ ❧➭ ❝➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ t➳❝ ❣✐➯✳ ❈❤➢➡♥❣ ✷ t➳❝ ❣✐➯ ♥❣❤✐➟♥ ❝ø✉ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❤➬✉ t✉➬♥ ❤♦➭♥ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ♥❣❤✐Ư♠ ❝❤Ø ♥Õ✉ ❞➲② q✉❛♥ s➳t x(t) {x(tn )}n∈Z ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❧➭ ❤➬✉ t✉➬♥ ❤♦➭♥ ♥Õ✉ ✈➭ ❧➭ ❤➬✉ t✉➬♥ ❤♦➭♥✳ ❈❤➢➡♥❣ ✸ t➳❝ ❣✐➯ ♥❣❤✐➟♥ ❝ø✉ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ♥❣❤✐Ư♠ ♣❤✐❝ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❞➲② q✉❛♥ s➳t x(t) ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❧➭ ❛❧♠♦st ❛✉t♦♠♦r✲ {x(n)}n∈Z ✼ ❧➭ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ▲ê✐ ❝➯♠ ➡♥ ➜➬✉ t✐➟♥ t➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ tỏ ò ết s s ủ ì ế ❝➳❝ t❤➬② ❣✐➳♦ P●❙✳❚❙❑❍✳ ◆❣✉②Ô♥ ❱➝♥ ▼✐♥❤ ✈➭ ❚❙✳ ❍➭ ❇×♥❤ ▼✐♥❤ ✈× sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤✱ tr✉②Ị♥ t❤ơ ♥❤÷♥❣ ❦✐Õ♥ t❤ø❝ ✈➭ ❦✐♥❤ ♥❣❤✐Ư♠ q✉ý ❜➳✉ tr♦♥❣ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ ❝ò♥❣ ♥❤➢ sù q✉❛♥ t➞♠ t❤➢ê♥❣ ①✉②➟♥✱ ❣✐ó♣ ➤ì ➞♥ t×♥❤✱ ➤é♥❣ ✈✐➟♥ t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ t❤ù❝ ❤✐Ư♥ ❧✉❐♥ ✈➝♥✳ ❚➳❝ ❣✐➯ ❝ị♥❣ ①✐♥ tr➞♥ trä♥❣ ❝➯♠ ➡♥ ❚❙✳ ◆❣✉②Ơ♥ ❚❤✐Ư✉ ❍✉② ✈➭ t✃t ❝➯ ❝➳❝ t❤➭♥❤ ✈✐➟♥ ❝đ❛ ❳➟✲♠✐✲♥❛ ✬✬ ❉➳♥❣ ➤✐Ư✉ t✐Ư♠ ❝❐♥ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ✈➭ ø♥❣ ❞ơ♥❣✬✬✱ ❦❤♦❛ ❚♦➳♥ ✲ ❚✐♥ ø♥❣ ❞ơ♥❣ ➤➲ ❣✐ó♣ ➤ì✱ ➤é♥❣ ✈✐➟♥ ✈➭ ➤ã♥❣ ❣ã♣ ♥❤✐Ò✉ ý ❦✐Õ♥ q✉ý ❜➳✉ ❝❤♦ ♥é✐ ❞✉♥❣ ❝ñ❛ ❧✉❐♥ ✈➝♥✳ ❚➳❝ ❣✐➯ tr➞♥ trä♥❣ ❝➯♠ ➡♥ ❇❛♥ ●✐➳♠ ❤✐Ư✉✱ ❇❛♥ ❈❤đ ♥❤✐Ư♠ ❑❤♦❛ ❚♦➳♥ ✲ ❚✐♥ ø♥❣ ❞ơ♥❣✱ P❤ß♥❣ ❚ỉ ❝❤ø❝ ❝➳♥ ❜é✱ ❱✐Ư♥ ➜➭♦ t➵♦ s❛✉ ➜➵✐ ❤ä❝ ➤➲ ❣✐ó♣ ➤ì✱ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❝❤♦ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❚➳❝ ❣✐➯ ❝ò♥❣ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❝➳❝ t❤➬②✱ ❝➠ ❣✐➳♦ ✈➭ ❝➳❝ ❛♥❤ ❝❤Þ ❡♠ ➤å♥❣ ♥❣❤✐Ư♣ ❑❤♦❛ ❚♦➳♥ ✲ ❚✐♥ ø♥❣ ❞ô♥❣ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❇➳❝❤ ❦❤♦❛ ❍➭ ◆é✐ ✈Ị sù q✉❛♥ t➞♠ ❣✐ó♣ ➤ì✱ ➤é♥❣ ✈✐➟♥ t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ t❤ù❝ ❤✐Ư♥ ❧✉❐♥ ✈➝♥✳ ❍➭ ◆é✐✱ t❤➳♥❣ ✶✵ ♥➝♠ ✷✵✶✵ ❚➳❝ ❣✐➯ ❇ï✐ ❳✉➞♥ ❉✐Ö✉ ✽ ❈❤➢➡♥❣ ✶ ❚æ♥❣ q✉❛♥ ✶✳✶ ❍➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ ❚r♦♥❣ ♠ơ❝ ♥➭② ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ♠ét ✈➭✐ ➤Þ♥❤ ♥❣❤Ü❛ t➢➡♥❣ ➤➢➡♥❣ ❦❤➳❝ ♥❤❛✉ ❝ñ❛ ♠ét ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ♥ã✳ ❈➳❝ ➤Þ♥❤ ♥❣❤Ü❛ ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ ❞➢í✐ ❇❛♥❛❝❤ X ➤➞② ✳ ế ợ f t trì f ❤➭♠ ❧➭ ❤➭♠ sè ♥❤❐♥ ❣✐➳ trÞ t❤ù❝✱ tr♦♥❣ ♥❤❐♥ Rk trị tr ứ tì ị ĩ ũ ợ ể ột t➢➡♥❣ tù✳ ✶✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ❇♦❝❤♥❡r ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶✳ sè t❤ù❝ {αn }n∈Z ❍➭♠ sè f : R → X ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã t❤Ĩ trÝ❝❤ ➤➢ỵ❝ ♠ét ❞➲② ❝♦♥ ❤➬✉ t✉➬♥ ❤♦➭♥ ♥Õ✉ tõ ❜✃t ❦× ❞➲② {αn }n∈Z s❛♦ ❝❤♦ lim f (t + αn ) n→∞ ❤é✐ tơ ➤Ị✉ tr➟♥ ➤➢ê♥❣ t❤➻♥❣ t❤ù❝✳ ◆Õ✉ f ❧➭ ♠ét ❤➭♠ sè t✉➬♥ ❤♦➭♥ ✈í✐ ❝❤✉ ❦× ❚❤❐t ✈❐②✱ tõ ❜✃t ❦× ❞➲② sè ❝❤♦ {αn (modT )} → α0 ✳ {αn }n∈Z ❑❤✐ ➤ã T t❤× ♥ã ❧➭ ♠ét ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥✳ ❝❤ó♥❣ t❛ ❝ã t❤Ĩ ❝❤ä♥ ♠ét ❞➲② ❝♦♥ lim f (t + αn ) = f (t + α0 ) n→∞ ♠ét ❤➭♠ sè t tì ó ị r ♥➝♠ ✶✾✻✷✳ ✾ {αn }n∈Z ◆❣♦➭✐ r❛ ♥Õ✉ s❛♦ f ❧➭ ❙❛✉ ➤➞② ❧➭ ♠ét ✈➭✐ ❦Ý ❤✐Ö✉ ❞♦ ❇♦❝❤♥❡r ❚♦➳♥ tư ❞Þ❝❤ ❝❤✉②Ĩ♥✳ β⊂α ♥❣❤Ü❛ ❧➭ {−αn } ✳ ❚❛ βn = βn(k) ✳ β ♥ã✐ {αn } sÏ ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❧➭ ❧➭ ♠ét ❞➲② sè ❝♦♥ ❝đ❛ ❞➲② α β ✈➭ f ◆Õ✉ ➤Þ♥❤ ♥❣❤Ü❛ ❜ë✐ ❉➲② sè ❧➭ ❝➳❝ ➤å♥❣ ❞➲② α ✳ ❑Ý ❤✐Ö✉ ❝♦♥ ❝ñ❛ α α ✳ ◆Õ✉ β = {βn } α + β = {αn + βn }, −α = β ✈➭ ♥Õ✉ αn = αn(k) ❧➭ ♠ét ❤➭♠ sè ❤➬✉ t tì t tử ị ể T f = g ♥Õ✉ g(t) = lim f (t + αn ) n tì ết T ợ ộ tụ ề ❞ơ♥❣ ❝➳❝ ❦Ý ❤✐Ư✉ ♥➭②✱ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ ➤➢ỵ❝ ✈✐Õt ❧➵✐ ♥❤➢ s❛✉✿ ❤➭♠ sè s❛♦ ❝❤♦ f ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➬✉ t✉➬♥ ❤♦➭♥ ♥Õ✉ ✈í✐ ♠ä✐ ❞➲② sè Tα f ✱ tå♥ t➵✐ ❞➲② ❝♦♥ α⊂α tå♥ t➵✐✳ ❇❛♦ ❝ñ❛ ❤➭♠ sè✳ ❇❛♦ ❝ñ❛ ❤➭♠ sè H(f ) = {g : ∃α ➜Þ♥❤ ❧ý f ị ý ế f ợ ➤Þ♥❤ ♥❣❤Ü❛ ♥❤➢ s❛✉✿ s❛♦ ❝❤♦ Tα f = g ❤é✐ tơ ➤Ị✉ ❧➭ ♠ét ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ f ❧➭ ♠ét ❤➭♠ sè ❤➬✉ t✉➬♥ t❤× ✈í✐ ♠ä✐ } H(f ) ❧➭ t❐♣ ❝♦♠♣❛❝t✳ g ∈ H(f )✱ H(g) = H(f )✳ ❑❤➠♥❣ ❣✐❛♥ ❝➳❝ ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ ➜➷t f = sup |f (t)| ✈➭ AP(X) = {f : f ❧➭ ♠ét ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ } ✱ ♥ã t ➤➢ỵ❝ tr❛♥❣ ❜Þ ❝❤✉➮♥ ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥ AP(Rk ) f : R → R ❤♦➷❝ ▼ét ❝➳❝❤ t➢➡♥❣ tù✱ ❦❤➠♥❣ ❣✐❛♥ ❝➳❝ f : R → Rk ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❧➭ AP(R) ✈➭ t➢➡♥❣ ø♥❣✳ ➜Þ♥❤ ❧ý ✶✳✶✳✸✳ ợ ế ị ĩ tr AP(X) X = C✮✱ ❧➭ ♠ét ➤➵✐ sè tr➟♥ X✳ ◆❣❤Ü❛ ❧➭ ♥ã ➤ã♥❣ ✈í✐ ♣❤Ð♣ ❧✃② ❧✐➟♥ ➤ã♥❣ ✈í✐ ❝➳❝ ♣❤Ð♣ t♦➳♥ ❝é♥❣✱ ♥❤➞♥ ✈➭ ♥❤➞♥ ✈í✐ ✈➠ ❤➢í♥❣✳ ❍➡♥ ✶✵ ❈❤➢➡♥❣ ✸ ◗✉❛♥ s➳t ♥❣❤✐Ư♠ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ✸✳✶ ●✐í✐ t❤✐Ư✉ ❜➭✐ t♦➳♥ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ①Ðt ♣❤➢➡♥❣ tr×♥❤ dx = f (x, t), dt ë ➤ã f (x, t) ❝♦♠♣❛❝t ❝ñ❛ ❧➭ ♠ét ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ t❤❡♦ Rk ✭✸✳✶✮ t ➤Ò✉ t❤❡♦ x ✳ ❇➭✐ t♦➳♥ ❝❤ó♥❣ t➠✐ q✉❛♥ t➞♠ ➤Õ♥ ❧➭ ❦❤✐ ệ tr ỗ t x(t) ủ trì ✭✸✳✶✮ ❧➭ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ♥Õ✉ ❜✐Õt r➺♥❣ ❞➲② ❝➳❝ q✉❛♥ s➳t ❝ñ❛ ♥ã tr➟♥ Z ❧➭ ♠ét ❞➲② ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ▼ét sè ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ ❣➬♥ ➤➞② ✈Ò ❜➭✐ t♦➳♥ ♥➭② ❝ã t❤Ó ❦Ó ➤Õ♥ ♥❤➢ s❛✉✿ ◆➝♠ ✶✾✻✾✱ tr♦♥❣ ❬✻❪✱ t➳❝ ❣✐➯ ①Ðt ♣❤➢➡♥❣ tr×♥❤ dx(t) = F (t, x), t ∈ R, dt ë ➤ã F ❧➭ ♠ét ❤➭♠ sè ❧✐➟♥ tô❝ ✈➭ t❤á❛ ♠➲♥ ✭✸✳✷✮ F (t, x) = F (t + 1, x) ✳ ♣❤➢➡♥❣ tr×♥❤ ✭✸✳✷✮ ❧➭ ❞✉② ♥❤✃t ♥❣❤✐Ư♠ ✈í✐ ❜➭✐ t♦➳♥ ❣✐➳ trị ủ trì ó ϕ(n) ϕ(t) ●✐➯ sư t❤➟♠ ϕ ❧➭ ♥❣❤✐Ư♠ ❧➭ ❝♦♠♣❛❝t ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❧➭ ♠ét ❞➲② ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ✸✸ ◆➝♠ ✷✵✵✻✱ tr♦♥❣ ❬✶✸❪✱ ❝➳❝ t➳❝ ❣✐➯ ①Ðt ♣❤➢➡♥❣ tr×♥❤ dx(t) = A(t)x(t) + f (t), t ∈ R, dt ë ➤ã f ❧➭ ♠ét ❤➭♠ sè ❛❧♠♦st tr trị tr q trì tế ó sử u(t) ♥Õ✉ ✭✸✳✸✮ (U (t, s))t≥s ✱ ❧➭ t♦➳♥ tö s✐♥❤ t✉➬♥ ❤♦➭♥ ❝❤✉ ❦× ✶ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧➭ ♠ét ♥❣❤✐Ư♠ ❜Þ ❝❤➷♥ ❝đ❛ ✭✸✳✸✮✳ ❑❤✐ ➤ã u(n)n∈Z X A(t) u(t) X ✳ ●✐➯ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ♥Õ✉ ✈➭ ❝❤Ø ❧➭ ♠ét ❞➲② ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ❚✐Õ♣ t❤❡♦✱ ✈➭♦ ♥➝♠ ✷✵✵✼✱ tr♦♥❣ ❬✶✹❪ ❝➳❝ t➳❝ ❣✐➯ ①Ðt ♣❤➢➡♥❣ tr×♥❤ dx(t) = Ax ([t]) + f (t), t ∈ R, dt ë ➤ã A ✭✸✳✹✮ ❧➭ t♦➳♥ tư t✉②Õ♥ tÝ♥❤ ❜Þ ❝❤➷♥ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✱ ✈➭ [t] ❧➭ ♣❤➬♥ ♥❣✉②➟♥ ❝đ❛ t ✳ ●✐➯ sư u ♣❤➢➡♥❣ tr×♥❤ ✭✸✳✹✮✳ ❈➳❝ t➳❝ ❣✐➯ ❝ị♥❣ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❞➲② ❝➳❝ q✉❛♥ s➳t {u(n)}n∈Z X f ✱ ❧➭ ♠ét ❤➭♠ sè ❧➭ ♠ét ♥❣❤✐Ư♠ ❜Þ ❝❤➷♥ ❝đ❛ u(t) ❧➭ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ❧➭ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ◆❤÷♥❣ ♥❣➢ê✐ q✉❛♥ t➞♠ ❝ã t❤Ĩ ①❡♠ ❝➳❝ ❦Õt q✉➯ ✈Ị ❜➭✐ t♦➳♥ ♥➭② tr♦♥❣ ❬✺✱ ✾✱ ✶✼✱ ✶✽✱ ✶✾❪✳ ▼ô❝ ➤Ý❝❤ ❝đ❛ ❝❤ó♥❣ t➠✐ tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ❧➭ sÏ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ❦Õt q✉➯ tr➟♥ ✈➱♥ ➤ó♥❣ ♥Õ✉ f (x, t) t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ▲✐♣s❝❤✐t③ t❤❡♦ x ✱ ➤Ị✉ t❤❡♦ t ✳ ▼ét ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ t❤× ❦❤➠♥❣ ♥❤✃t t❤✐Õt ♣❤➯✐ ❧✐➟♥ tơ❝ ➤Ị✉✱ ❝❤Ý♥❤ ✈× ✈❐② ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ tr➢í❝ ➤➞② ➤è✐ ✈í✐ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❤➬✉ t✉➬♥ ❤♦➭♥ ❞ù❛ tr➟♥ tÝ♥❤ ❧✐➟♥ tơ❝ ➤Ị✉ ❝đ❛ ♥❣❤✐Ư♠ ✈➭ ❝đ❛ ✈Õ ♣❤➯✐ ✸✹ f ❧➭ ❦❤➠♥❣ ❝ß♥ ❤✐Ư✉ q✉➯✳ ➜Ĩ ❜✐Õt t❤➟♠ ✈Ị ♣❤➢➡♥❣ ♣❤➳♣ ♥➭②✱ ❝ã t❤Ó ①❡♠ t❤➟♠ tr♦♥❣ ❬✶✱ ✼✱ ✶✶✱ ✶✻✱ ✷✵✱ ✶✺❪ ✳ ❍➡♥ ♥÷❛✱ ➤è✐ ✈í✐ ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥✱ ❝❤ó♥❣ t❛ ❝ã t✐➟✉ ❝❤✉➮♥ ❇♦❤r ➤Ĩ ❦✐Ĩ♠ tr❛ ♠ét ❤➭♠ sè ❝ã ❤➬✉ t✉➬♥ ❤♦➭♥ ❤❛② ❦❤➠♥❣ ❞ù❛ tr➟♥ ✈✐Ư❝ ❦✐Ĩ♠ tr❛ tÝ♥❤ trï ♠❐t t➢➡♥❣ ➤è✐ ❝đ❛ t❐♣ ✲❞Þ❝❤ ❝❤✉②Ĩ♥ ❝đ❛ ♥ã✳ ❚r♦♥❣ ❦❤✐ ➤ã ➤è✐ ✈í✐ ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✱ t❤❡♦ ❤✐Ĩ✉ ❜✐Õt ❝đ❛ t➳❝ ❣✐➯✱ ❤✐Ö♥ ♥❛② ❝❤➢❛ ❝ã ♠ét t✐➟✉ ❝❤✉➮♥ ♥➭♦ t➢➡♥❣ tù ♥❤➢ ✈❐②✳ ➜➞② ❝ò♥❣ ❝❤Ý♥❤ ❧➭ ❝➳❝ ❦❤ã ❦❤➝♥ ♠➭ t➳❝ ❣✐➯ ❣➷♣ ♣❤➯✐ tr♦♥❣ q✉➳ tr×♥❤ ❝❤ø♥❣ ♠✐♥❤✳ ✸✳✷ ❈➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✷✳✶✳ ❍➭♠ sè f : RìX X ợ ọ st tr t t R ề t x tr ỗ t ❝♦♠♣❛❝t ❝đ❛ X ♥Õ✉ ✈í✐ ♠ä✐ ❞➲② sè t❤ù❝ (sk )✱ tå♥ t➵✐ ♠ét ❞➲② sè ❝♦♥ (sk ) s❛♦ ❝❤♦ lim f (t + sk , x) = g(t, x), ỗ tR ọ x X, lim g(t − sk , x) = f (t, x), ỗ tR ọ x X k ✈➭ k→∞ ❇ỉ ➤Ị ✸✳✷✳✶✳ t❤❡♦ ◆Õ✉ f : R×X → X ❧➭ ♠ét ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ t❤❡♦ x tr ỗ t t ủ X ế f x ➤Ị✉ t❤❡♦ t✱ ❦❤✐ ➤ã ❤➭♠ sè ➤✐Ị✉ ❦✐Ư♥ ▲✐♣s❝❤✐t③ t❤❡♦ ❈❤ø♥❣ ♠✐♥❤✳ g t ➤Ò✉ t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ư♥ ▲✐♣s❝❤✐t③ t❤❡♦ ①➳❝ ➤Þ♥❤ tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✷✳✶ tr➟♥ ❝ị♥❣ t❤á❛ ♠➲♥ x ➤Ị✉ t❤❡♦ t✳ ❳❡♠ ❬✶✼✱ ➜Þ♥❤ ❧ý ✷✳✷✳✺ ❪✳ ✸✺ ➜Þ♥❤ ❧ý ✸✳✷✳✶✳ ➤Ị✉ t❤❡♦ t❤❡♦ ❈❤♦ f : R × X → X ❧➭ ♠ét ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ t❤❡♦ t ∈ R x tr➟♥ ỗ t t ủ Rk sử f t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ▲✐♣s❝❤✐t③ x ➤Ị✉ t❤❡♦ t ∈ R✳ ●✐➯ sư u(t) ❧➭ ♠ét ♥❣❤✐Ư♠ ❜Þ ❝❤➷♥ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ➤ã✱ ♥❣❤✐Ư♠ ✭✸✳✶✮✳ ❑❤✐ u(t) ❧➭ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❞➲② ❝➳❝ q✉❛♥ s➳t {u(n)}n∈Z ❧➭ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ❈❤ø♥❣ ♠✐♥❤✳ t❤× ➜✐Ị✉ ❦✐Ư♥ ❝➬♥✿ ❍✐Ĩ♥ ♥❤✐➟♥✱ ♥Õ✉ {u(n)}n∈Z u(t) ❧➭ ♠ét ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ❧➭ ♠ét ❞➲② sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ➜✐Ị✉ ❦✐Ư♥ ➤đ✿ ●✐➯ sư ♠✐♥❤ u(t) {u(n)}n∈Z ❧➭ ♠ét ❞➲② sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ❚❛ sÏ ❝❤ø♥❣ ❧➭ ♠ét ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ❈❤ø♥❣ ♠✐♥❤ ❝đ❛ ❝❤ó♥❣ t➠✐ ➤➢ỵ❝ ❝❤✐❛ ❧➭♠ ✸ ❜➢í❝ ♥❤➢ s❛✉✳ ❇➢í❝ ✶✳ ➜➬✉ t✐➟♥✱ ❣✐➯ sư ❞➲② sè ❝♦♥ {nk } ✈➭ {nk } {v(n)} ❧➭ ♠ét ❞➲② sè ♥❣✉②➟♥ ❝❤♦ tr➢í❝✳ ❑❤✐ ➤ã tå♥ t➵✐ ♠ét s❛♦ ❝❤♦ lim v(n − nk ) = u(n), lim u(n + nk ) = v(n), k→∞ k→∞ lim g(t − nk , x) = f (t, x), lim f (t + nk , x) = g(t, x), k ỗ tR ❝è ➤Þ♥❤✱ ❦Ý ❤✐Ư✉ {t} = t − [t] ✮✳ ❑❤✐ ➤ã ✈í✐ ❱× g k→∞ [t] i∈Z ❧➭ ♣❤➬♥ ♥❣✉②➟♥ ❝✉❛t ✱ ❣ä✐ (v i ) (t) = g(t, v i ), ∀n ∈ Z v i (t) t ✈➭ {t} ✭✸✳✺✮ ∀t ∈ R ✭✸✳✻✮ ❧➭ ♣❤➬♥ ♣❤➞♥ ❝đ❛ t ✱ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❈❛✉❝❤② s❛✉✿ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ❜❛♥ ➤➬✉ v i |t=i = u(i) t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ▲✐♣❝❤✐t③ ♥➟♥ ❝➳❝ ❜➭✐ t♦➳♥ ❈❛✉❝❤② tr➟♥ ➤Ị✉ ❝ã ♥❣❤✐Ư♠ ❞✉② ♥❤✃t✱ ✈➭ ❝➳❝ ♥❣❤✐Ư♠ v i (t) ợ ị tr ✸✻ (i − δi , i + δi ) ✳ ❍➡♥ ♥÷❛✱ ❝❤ó♥❣ t❛ ❝ã t❤Ĩ ❝❤ä♥ t❤❡♦ t ✳ ❤➭♠ sè δi = δ ✈í✐ ♠ä✐ ❇➺♥❣ ❝➳❝ ✬♥è✐✬ ❝➳❝ ❤➭♠ sè v(t), t ∈ R i∈Z ❞♦ g t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ t❤❡♦ v i (t), t ∈ (i − δ, i + δ) x ➤Ị✉ ✈í✐ ♥❤❛✉✱ t❛ ➤Þ♥❤ ♥❣❤Ü❛ ♥❤➢ s❛✉✿ ✭❛✮ v(t) ✭❜✮ v(t) = v i (t), ∀t ∈ (i − δ, i + δ) ❧➭ ❤➭♠ sè ❧✐➟♥ tô❝ tr➟♥ R ❚❤❡♦ ❝➠♥❣ t❤ø❝ ❜✐Õ♥ t❤✐➟♥ ❤➺♥❣ sè✱ ✱ ✈➭ ✳ v i (t) t❤á❛ ♠➲♥ ♣❤➢➡♥❣ tr×♥❤ t i g(s, v i (s))ds, ∀t ∈ (i, i + δ) v (t) = u([t]) + [t] ❉♦ ➤ã✱ ❤➭♠ sè v(t) sÏ t❤á❛ ♠➲♥ ♣❤➢➡♥❣ tr×♥❤ s❛✉ ➤➞② t g(s, v(s))ds, ∀t ∈ v(t) = u([t]) + (i, i + δ) [t] ❈❤ó♥❣ t❛ sÏ ❝❤Ø r❛ i∈Z lim u(t + nk ) = v(t) , ∀t ∈ k→∞ (i, i + δ) ✳ i∈Z ✸✼ ❚❤❐t ✈❐②✱ ✈í✐ k ➤đ ❧í♥✱ t❛ ❝ã u(t + nk ) − v(t) = t+nk t f (s, u(s))ds − u([t]) − = u([t] + nk ) + g(s, v(s))ds [t]+nk [t] t+nk ≤ u([t] + nk ) − u([t]) + t f (s, u(s))ds − g(s, v(s))ds [t]+nk [t] t = u([t] + nk ) − u([t]) + f (s + nk , u(s + nk )) − g(s, v(s)) ds [t] t u([t] + nk ) − u([t]) = f (s + nk , u(s + nk )) − g(s, u(s + nk )) ds + [t] ≤ε✭ t❤❡♦ ❣✐➯ t❤✐Õt ❝đ❛ ❜➢í❝ ✶✮ ≤ε✭t❤❡♦ ❣✐➯ t❤✐Õt ❝đ❛ ❜➢í❝ ✶✮ t g(s, u(s + nk )) − g(s, v(s)) ds + [t] ≤L u(s+nk )−v(s) ✭g ▲✐♣s❝❤✐t③✮ t ≤ 2ε + L u(s + nk ) − v(s) ds, ∀t ∈ [t] (i, i + δ) i∈Z ❚❤❡♦ ❜ỉ ➤Ị ●r♦❧❧✇❛❧❧✱ t❛ ❝ã u(t + nk ) − v(t) ≤ 2εeL , ∀t ∈ (i, i + δ) i∈Z ❉♦ ➤ã✱ lim u(t + nk ) = v(t), ∀t ∈ k→∞ (i, i + δ) ✳ i∈Z ❚➢➡♥❣ tù✱ ❝ã t❤Ó ❝❤Ø r❛ lim v(t − nk ) = u(t), ∀t ∈ k→∞ ❇➢í❝ ✷✳ ❇➞② ❣✐ê ①Ðt tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t✱ ë ➤ã ♥❤✃t t❤✐Õt ❧➭ ❞➲② sè ♥❣✉②➟♥✳ {sk } k ✳ ❱× {tk } ✳ ❧➭ ♠ét ❞➲② sè ❜✃t ❦×✱ ❦❤➠♥❣ ❱Ị ❝➡ ❜➯♥ ❝❤ø♥❣ ♠✐♥❤ ❝ị♥❣ sÏ t➢➡♥❣ tù ♥❤➢ ❜➢í❝ ✶✱ ❝é♥❣ t❤➟♠ ✈í✐ tÝ♥❤ t✐Ị♥ ❝♦♠♣❛❝t ❝đ❛ ủ số ỗ (i, i + ) i∈Z ❧➭ ♠ét ❞➲② sè ♥➺♠ tr♦♥❣ ❦❤♦➯♥❣ ✸✽ f ✳ ➜➷t [0, 1) nk = [sk ] ✈➭ tk = {sk } ♥➟♥ ❝ã t❤Ĩ ❝❤ä♥ ➤➢ỵ❝ ❞➲② sè ❝♦♥ {nk } tõ {nk } s❛♦ ❝❤♦ lim v(n − nk ) = u(n), lim u(n + nk ) = v(n), k→∞ k→∞ ∀n ∈ Z, lim g(t − nk , x) = f (t, x), lim f (t + nk , x) = g(t, x), k→∞ ✭✸✳✼✮ ∀t ∈ R, k→∞ ✭✸✳✽✮ lim tk = t0 ∈ [0, 1] ✭✸✳✾✮ k→∞ ❳Ðt ❤❛✐ tr➢ê♥❣ ❤ỵ♣✿ ❚❍✶✮ {t + t0 } > ◆Õ✉ ✱ t❛ sÏ ❝❤Ø r❛ lim u(t + sk ) = v(t + t0 ) , ∀t + t0 ∈ k→∞ ➜➬✉ t✐➟♥ t❛ sÏ ❝❤Ø r❛✱ (i, i + δ) i∈Z lim u(t + sk ) = lim u(t + t0 + nk ) ✳ ❚❤❐t ✈❐②✱ ✈í✐ k→∞ k→∞ k ➤đ ❧í♥✱ t❛ ❝ã u(t + sk ) − u(t + t0 + nk ) = u(t + tk + nk ) − u(t + t0 + nk ) ≤ u([t + tk ] + nk ) − u([t + t0 ] + nk ) t+tk +nk t+t0 +nk f (s, u(s))ds − + [t+tk ]+nk f (s, u(s))ds [t+t0 ]+nk t+tk +nk = u([t + tk ] + nk ) − u([t + t0 ] + nk ) + f (s, u(s))ds t+t0 +nk ≤ε✭✈í✐ k ➤đ ❧í♥✮ ≤ε✭✈í✐ k ≤ 2ε, ∀t + t0 ∈ ➤đ ❧í♥✮ (i, i + δ) i∈Z ▲➷♣ ❧➵✐ ❝➳❝ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ❜➢í❝ ✶ ♥ã✐ tr➟♥✱ t❛ ❝ã lim u(t + t0 + nk ) = v(t + k→∞ t0 ), ∀t + t0 ∈ (i, i + δ) ✳ i∈Z ❚❍✷✮ ◆Õ✉ {t + t0 } = ✱ ❦❤✐ ➤ã✱ t + t0 ❧➭ ♠ét sè ♥❣✉②➟♥✳ ◆Õ✉ ❝❤ó♥❣ t❛ ❝ã t❤Ĩ ❧➷♣ ❧➵✐ ❝➳❝ ❜➢í❝ ❝❤ø♥❣ ♠✐♥❤ ♥❤➢ ë tr➟♥✳ ❈ß♥ ♥Õ✉ ✸✾ t + tk ↓ t + t0 t + tk ↑ t + t0 ✱ ✱ t❛ ❝ã [t + tk ] = t + t0 − ✈➭ {t + tk } → ✈➭ tÝ♥❤ t✐Ò♥ ❝♦♠♣❛❝t ❝ñ❛ ❞➯✐ ❝ñ❛ ❤➭♠ sè ✳ ❉♦ ➤ã✱ tõ tÝ♥❤ ❜Þ ❝❤➷♥ ❝đ❛ ❞➲② sè f {u(n)} ❞➱♥ ➤Õ♥ u(t + sk ) − u(t + t0 + nk ) = u(t + tk + nk ) − u(t + t0 + nk ) = t+tk +nk = u([t + tk ] + nk ) − u([t + t0 ] + nk ) + f (s, u(s))ds t+t0 −1+nk ≤ε✭✈í✐ k ➤đ ❧í♥ ✮ ≤ε✭✈í✐ k ≤ 2ε, ∀t + t0 ∈ ➤đ ❧í♥✮ (i, i + δ) i∈Z ❈✉è✐ ❝ï♥❣✱ ➤Ĩ ý t❤✃② r➺♥❣ ❜✃t ❦× ♠ét ❞➲② sè ❤é✐ tô ♥➭♦✱ ❤♦➷❝ ♥ã ❧➭ ♠ét ❞➲② sè ➤➡♥ ➤✐Ư✉✱ ❤♦➷❝ ❝ã t❤Ĩ ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ❤❛✐ ❞➲② sè ❝♦♥ ➤➡♥ ➤✐Ö✉✱ ♠ét ❞➲② ❝♦♥ ➤➡♥ ➤✐Ö✉ t➝♥❣ ✈➭ ♠ét ❞➲② ❝♦♥ ➤➡♥ ➤✐Ö✉ ❣✐➯♠✳ ◆❤❐♥ ①Ðt ♥➭② ➤➲ ❦Õt t❤ó❝ ❝❤ø♥❣ ♠✐♥❤ ❜➢í❝ sè ✷ ❝đ❛ ❝❤ó♥❣ t❛✳ ❇➢í❝ ✸✳ ❚r♦♥❣ ❝➳❝ ❇➢í❝ ✶ ✈➭ ✷ ♥ã✐ tr➟♥✱ ❝❤ó♥❣ t❛ ➤➲ ❝❤Ø r❛ r➺♥❣✱ ♥Õ✉ ❧➭ ♠ét ❞➲② sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ t❤× ✈í✐ ♠ä✐ ❞➲② sè t❤ù❝ ❝♦♥ (sk ) lim u(t + sk ) = v1 (t), ∀t ∈ ✱ tå♥ t➵✐ ♠ét ❞➲② sè lim v1 (t − sk ) = u(t), ∀t ∈ ①Ðt ❞➲② ♠í✐ {u(n + δ/2)}n∈Z (i, i + δ), ✈➭ i∈Z k→∞ tô❝ (sk ) s❛♦ ❝❤♦ k→∞ ❚✐Õ♣ {u(n)}n∈Z (i, i + δ) i∈Z ✳ ❉➲② sè ♥➭② ❝ò♥❣ ❧➭ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ t❤❡♦ ♥❤➢ ❦Õt q✉➯ ✈õ❛ ❝❤Ø r❛ ë tr➟♥✳ ❇➺♥❣ ❝➳❝❤ ❧➷♣ ❧➵✐ ❝➳❝ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ ❝➳❝ ❜➢í❝ ✶ ✈➭ ✷ ë tr➟♥✱ ❝❤ó♥❣ t❛ ❝ã t❤Ĩ ❝❤Ø r❛✱ ✈í✐ ♠ä✐ ❞➲② sè t❤ù❝ ✹✵ (sk ) ✱ tå♥ t➵✐ ♠ét ❞➲② sè ❝♦♥ (sk ) s❛♦ ❝❤♦ lim u(t + sk ) = v2 (t), ∀t ∈ k→∞ (i + δ/2, i + 3δ/2), ✈➭ i∈Z lim v2 (t − sk ) = u(t), ∀t ∈ k→∞ (i + δ/2, i + 3δ/2) i∈Z ế tụ q trì ế ợ ❝đ❛ ❝➳❝ t❐♣ ❤ỵ♣ ♥ã✐ tr➟♥ ❧✃♣ ➤➬② trơ❝ sè R ✳ N ◆❤➢ ✈❐② ❝❤ó♥❣ t❛ ➤➲ ❝❤Ø r❛ ➤➢ỵ❝ ❝ã u(t + sk ) → vj (t) ✈➭ vj (t − sk ) → u(t) ❤➭♠ sè ✈í✐ ♠ä✐ v1 (t), v2 (t), , vN (t) t∈ (i + i∈Z ❚✐Õ♣ t❤❡♦ t❛ ①➞② ❞ù♥❣ ột số tụ tr ỗ t (i + i∈Z v(t) (2j−3)δ ,i + v(t) ①➳❝ ➤Þ♥❤ tr➟♥ (2j−1)δ )✱ ✈í✐ ♠ä✐ (2j−3)δ ,i R s❛♦ ❝❤♦ + (sk ) (2j−1)δ )✳ v(t) = vj (t) j = 1, , N ①➞② ❞ù♥❣ t❤❡♦ ❝➳❝❤ ♥➭② t❤á❛ ♠➲♥ ➤✐Ò✉ s❛✉ ➤➞②✿ ỗ số tự t ột số s❛♦ ❝❤♦ ✳ ❍➭♠ sè (sk ) s❛♦ ❝❤♦ lim u(t + sk ) = v(t), ∀t ∈ R, ✈➭ k→∞ lim v(t − sk ) = u(t), ∀t ∈ R k→∞ ❉♦ ➤ã✱ u ❧➭ ♠ét ❤➭♠ sè ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ tr➟♥ ✹✶ R ✳ ➜✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✱ tå♥ ❑Õt ❧✉❐♥ ❈➳❝ ❦Õt q✉➯ ❝➡ ❜➯♥ ❝ñ❛ ❧✉❐♥ ✈➝♥ ❧➭✿ ✶✳ ❳➞② ❞ù♥❣ ➤➢ỵ❝ ♠ét t✐➟✉ ❝❤✉➮♥ ✭➤✐Ị✉ ❦✐Ư♥ ➤đ✮ ➤Ĩ ❦✐Ĩ♠ tr❛ ❦❤✐ ♥➭♦ ♠ét ❞➲② ❝➳❝ q✉❛♥ s➳t ✷✳ {x(tn )}n∈Z ❧➭ ❤➬✉ t✉➬♥ ❤♦➭♥ ✭▼Ö♥❤ ➤Ị ✶✳✷✳✶✮✳ ▼ë ré♥❣ ❦Õt q✉➯ ❝đ❛ ❬✼✱ ➜Þ♥❤ ❧ý ✾✳✼❪ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ❞➲② ❝➳❝ q✉❛♥ s➳t ♠➲♥ tn − n → ❦❤✐ n→∞ tn t❤á❛ ✭➤Þ♥❤ ❧ý ✷✳✷✳✶✮✳ ❍Ư ❝đ❛ trù❝ t✐Õ♣ ❝đ❛ ❦Õt q✉➯ ♥➭② ❧➭ t➳❝ ❣✐➯ ➤➲ ♠ë ré♥❣ ♠ét sè ❦Õt q✉➯ ➤➲ ❜✐Õt tr♦♥❣ ❬✶✶❪ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ❞➲② ❝➳❝ q✉❛♥ s➳t tn t❤á❛ ♠➲♥ tn − n → ❦❤✐ n→∞ ✭①❡♠ ❝➳❝ ➤Þ♥❤ ❧ý ✷✳✷✳✷ ✈➭ ✷✳✷✳✸ tr♦♥❣ ❧✉❐♥ ✈➝♥✮✳ ✸✳ ▼ë ré♥❣ ❦Õt q✉➯ ❝ñ❛ ❝➳❝ t➳❝ ❣✐➯ tr♦♥❣ ❬✶✹✱ ❇ỉ ➤Ị ✸✳✸❪ ✈➭ ❬✶✸✱ ❇ỉ ➤Ị ✸✳✶❪ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ❤➭♠ sè f (x, t) ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ t❤❡♦ ✹✷ t ✱ ➤Ị✉ t❤❡♦ x ✳ ❉❛♥❤ ♠ơ❝ ❝➠♥❣ tr×♥❤ ❝đ❛ t➳❝ ❣✐➯ ❝ã ❧✐➟♥ q✉❛♥ tí✐ ❧✉❐♥ ✈➝♥ ✶✳ ❇✉✐ ❳✉❛♥ t❤❡ ❉✐❡✉✱ ❍❛ r❡❧❛t✐♦♥s❤✐♣ ❇✐♥❤ ❜❡t✇❡❡♥ ▼✐♥❤✱ ❉♦ ❞✐s❝r❡t❡ ❉✉❝ ❛♥❞ ❚❤✉❛♥ ❛♥❞ ❝♦♥t✐♥✉♦✉s ◆❣✉②❡♥ ❱❛♥ ♦❜s❡r✈❛t✐♦♥s ♦❢ ▼✐♥❤✱ ✬✬❖♥ ❞②♥❛♠✐❝❛❧ s②st❡♠s✬✬✱ s✉❜♠✐tt❡❞ t♦ ❆❜str❛❝t ❛♥❞ ❆♣♣❧✐❡❞ ❆♥❛❧②s✐s ✭❆❆❆✴✷✹✽✻✵✷✮ ♦♥ ✵✸✱ ❖❝t♦❜❡r✱ ✷✵✶✵✳ ✷✳ ❍❛ ❇✐♥❤ ▼✐♥❤✱ ❇✉✐ ❳✉❛♥ ❉✐❡✉✱ ❛♥❞ ❉♦ ❉✉❝ ❚❤✉❛♥✱ ✬✬❖♥ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ♦❢ ❛❧♠♦st ❛✉t♦♠♦r♣❤② ❜❡t✇❡❡♥ ❝♦♥t✐♥✉♦✉s ❛♥❞ ✶✵✴✷✵✶✵✳ ✹✸ ❞✐s❝r❡t❡ ❞②♥❛♠✐❝❛❧ s②st❡♠s✬✬✱ ♣r❡♣r✐♥t ❑✐Õ♥ ♥❣❤Þ ❝❤♦ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉ t✐Õ♣ t❤❡♦ • ❚✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉ s➞✉ t❤➟♠ ✈Ị ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❝➳❝ q✉❛♥ s➳t rê✐ r➵❝ ✈➭ ❧✐➟♥ tơ❝ ❝đ❛ ❤Ư ➤é♥❣ ❧ù❝✳ s➳t {tn } ❍✐Ư♥ ♥❛②✱ ❜➭✐ t♦➳♥ tỉ♥❣ q✉➳t ①Ðt ①❡♠ ✈í✐ t❐♣ ❝➳❝ q✉❛♥ ♥❤➢ tế tì ó tể ị ệ x(t) ủ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❤➬✉ t✉➬♥ ❤♦➭♥ ✭✷✳✶✮ ❧➭ ❤➬✉ t✉➬♥ ❤♦➭♥ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❞➲② ❝➳❝ q✉❛♥ s➳t {x(tn )}n∈Z ❧➭ ♠ét ❞➲② sè ❤➬✉ t✉➬♥ ❤♦➭♥✱ t❤❡♦ ❤✐Ĩ✉ ❜✐Õt ❝đ❛ t➳❝ ❣✐➯✱ ✈➱♥ ➤❛♥❣ ❧➭ ♠ét ❜➭✐ t♦➳♥ ♠ë ✈➭ ❝ã ♥❤✐Ò✉ ý ♥❣❤Ü❛ tr♦♥❣ t❤ù❝ tÕ✳ • ❳➞② ❞ù♥❣ ♠ét t✐➟✉ ❝❤✉➮♥ ✭➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ✮ ➤Ĩ ❦✐Ĩ♠ tr❛ ①❡♠ ❦❤✐ ♥➭♦ t❤× ❞➲② ❝➳❝ q✉❛♥ s➳t • {x(tn )}n∈Z ❧➭ ♠ét ❞➲② sè ❤➬✉ t✉➬♥ ❤♦➭♥✳ ➜è✐ ✈í✐ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ✭✸✳✶✮✱ t✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉ ①❡♠ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ♥➭♦ t❤× ♥❣❤✐Ư♠ x(t) ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✭✸✳✶✮ ❧➭ ❛❧♠♦st ❛✉t♦✲ ♠♦r♣❤✐❝ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❞➲② ❝➳❝ q✉❛♥ s➳t • {x(tn )}n∈Z ❧➭ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝✳ ◆❣❤✐➟♥ ❝ø✉ ①❡♠ ❧✐Ö✉ ❝ã t❤Ó ➤➢❛ r❛ ♠ét t✐➟✉ ❝❤✉➮♥ ➤Ó ❦✐Ó♠ tr❛ ❦❤✐ ♥➭♦ ♠ét ❤➭♠ sè✱ ❞➲② sè ❧➭ ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ❣✐è♥❣ ♥❤➢ t✐➟✉ ❝❤✉➮♥ ❇♦❤r ❝❤♦ ❤➭♠ sè ❤➬✉ t✉➬♥ ❤♦➭♥✳ ✹✹ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❈✳❏✳❑✳ ❇❛tt②✱ ❲✳ ❍✉tt❡r✱ ❋✳ ❘⑥ ❛❜✐❣❡r ✭✶✾✾✾✮✱ ✬✬❆❧♠♦st ♣❡r✐♦❞✐❝✐t② ♦❢ ♠✐❧❞ s♦❧✉t✐♦♥s ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ♣❡r✐♦❞✐❝ ❈❛✉❝❤② ♣r♦❜❧❡♠s✬✬✱ ❏♦✉r♥❛❧ ♦❢ ❉✐❢✲ ❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❱♦❧✳✶✺✻✱ ♣♣✳ ✸✵✾✲✸✷✼✳ ❬✷❪ ❙❡r❣✐♦ ❇✐tt❛♥t✐✱ P❛tr✐③✐♦ ❈♦❧❛♥❡r✐ ✭✷✵✵✵✮✱ ✬✬■♥✈❛r✐❛♥t r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❞✐s❝r❡t❡✲t✐♠❡ ♣❡r✐♦❞✐❝ s②st❡♠s✬✬✱ ❆✉t♦♠❛t✐❝❛✱ ❱♦❧✳✸✻✱ ♣♣✳ ✶✼✼✼✲✶✼✾✸✳ ❬✸❪ ▼✳ ❇♦❤♥❡r✱ ❆✳ P❡t❡rs♦♥ ✭✷✵✵✶✮✱ ❉②♥❛♠✐❝ ❡q✉❛t✐♦♥s ♦♥ t✐♠❡ s❝❛❧❡s✳ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✱ ❇✐r❦❤❛✉s❡r ❇♦st♦♥✱ ■♥❝✳✱ ❇♦st♦♥✱ ▼❆✳ ❬✹❪ ▼✳ ❇♦❤♥❡r ✭✶✾✻✷✮✱ ✬✬❆ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ ❛❧♠♦st ♣❡r✐♦❞✐❝✐t②✬✬✱ Pr♦❝✳ ◆❛t✳ ❆❝❛❞✳ ❙❝✐✳ ❯❙❆✱ ♣♣✳ ✷✵✸✾✲✷✵✹✸✳ ❬✺❪ ❚✳ ❉✐❛❣❛♥❛✱ t♦♠♦r♣❤✐❝ ●✳ ◆✬●✉❡r ➆ ❡❦❛t❛✱ ➆ s♦❧✉t✐♦♥s ♦❢ ◆❣✉②❡♥ ❡✈♦❧✉t✐♦♥ ❱❛♥ ▼✐♥❤ ❡q✉❛t✐♦♥s✬✬✱ ✭✷✵✵✹✮✱ Pr♦❝✳ ✬✬❆❧♠♦st ❆♠❡r✳ ▼❛t❤✳ ❛✉✲ ❙♦❝✳ ❱♦❧✳✶✸✷ ✱ ♣♣✳ ✸✷✽✾✲✸✷✾✽✳ ❬✻❪ ❆✳▼✳ ❋✐♥❦ ✭✶✾✻✾✮✱ ✬✬❊①t❡♥s✐♦♥s ♦❢ ❆❧♠♦st ❆✉t♦♠♦r♣❤✐❝ ❙❡q✉❡♥❝❡s✬✬✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✱ ❱♦❧✳✷✼✱ ♣♣✳ ✺✶✾✲✺✷✸✳ ❬✼❪ ❆✳▼✳ ❋✐♥❦ ✭✶✾✼✹✮✱ ❆❧♠♦st P❡r✐♦❞✐❝ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✳ ❬✽❪ ❚❡ts✉♦ ❋✉r✉♠♦❝❤✐✱ ❚♦s❤✐❦✐ ◆❛✐t♦ ✭✷✵✵✾✮✱ ✬✬P❡r✐♦❞✐❝ s♦❧✉t✐♦♥s ❡♥❝❡ ❡q✉❛t✐♦♥s✬✬✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ❱♦❧✳✼✶✱ ♣♣✳ ✷✷✶✼✲✷✷✷✷✳ ✹✺ ♦❢ ❞✐❢❢❡r✲ ❬✾❪ ❏✳ ▲✐✉✱ ◆✬●✉❡r ◆❣✉②❡♥ ❱❛♥ ▼✐♥❤ ✭✷✵✵✹✮✱ ✬✬❆ ▼❛ss❡r❛ t②♣❡ t❤❡♦✲ ➆ ❡❦❛t❛✱ ➆ r❡♠ ❢♦r ❛❧♠♦st ❛✉t♦♠♦r♣❤✐❝ s♦❧✉t✐♦♥s ♦❢ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✬✬✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ❱♦❧✳✷✵✾✭✷✮✱ ♣♣✳ ✺✽✼✲✺✷✽✳ ❬✶✵❪ ❏✳ ▲✐✉✱ ◆✬●✉❡r ◆❣✉②❡♥ ❱❛♥ ▼✐♥❤ ✭✷✵✵✽✮✱ ❚♦♣✐❝s ♦♥ st❛❜✐❧✐t② ❛♥❞ ➆ ❡❦❛t❛✱ ➆ ♣❡r✐♦❞✐❝✐t② ✐♥ ❛❜str❛❝t ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❲♦r❧❞ ❙❝✐❡♥t✐❢✐❝ P✉❜❧✐s❤✐♥❣ ❈♦✳ Pt❡✳ ▲t❞✳ ❬✶✶❪ ●✳ ❍✳ ▼❡✐st❡rs ✭✶✾✺✾✮✱ ✬✬❖♥ ❆❧♠♦st P❡r✐♦❞✐❝ ❙♦❧✉t✐♦♥s ♦❢ ❛ ❈❧❛ss ♦❢ ❉✐❢✲ ❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✬✬✱ Pr♦❝✳ ♦❢ t❤❡ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ❱♦❧✳✶✵✭✶✮✱ ♣♣✳ ✶✶✸✲ ✶✶✾✳ ❬✶✷❪ ◆❣✉②❡♥ ❱❛♥ ▼✐♥❤✱ ●❛st♦♥ ◆✬●✉❡r ❙t❡❢❛♥ ❙✐❡❣♠✉♥❞ ✭✷✵✵✾✮✱ ✬✬❈✐r✲ ➆ ❡❦❛t❛✱ ➆ ❝✉❧❛r s♣❡❝tr✉♠ ❛♥❞ ❜♦✉♥❞❡❞ s♦❧✉t✐♦♥s ♦❢ ♣❡r✐♦❞✐❝ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s✬✬✱ ❏♦✉r♥❛❧ ♦❢ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❱♦❧✳✷✹✻✱ ♣♣✳ ✸✵✽✾✲✸✶✵✽✳ ❬✶✸❪ ◆❣✉②❡♥ ❱❛♥ ▼✐♥❤✱ ❚✳ ◆❛✐t♦✱ ●✳ ▼✳ ◆✬●✉❡r ➆ ❡❦❛t❛ ➆ ✭✷✵✵✻✮✱ ✬✬❆ ❙♣❡❝tr❛❧ ❈♦✉♥t❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ❢♦r ❆❧♠♦st ❆✉t♦♠♦r♣❤② ♦❢ ❙♦❧✉t✐♦♥s ♦❢ ❉✐❢❢❡r✲ ❡♥t✐❛❧ ❊q✉❛t✐♦♥s✬✬✱ Pr♦❝✳ ♦❢ t❤❡ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ❱♦❧✳✶✸✹✱ ♣♣✳ ✸✷✺✼✲✸✷✻✻✳ ❬✶✹❪ ◆❣✉②❡♥ ♦❢ ❱❛♥ ❜♦✉♥❞❡❞ ▼✐♥❤✱ ❚r❛♥ s♦❧✉t✐♦♥s ♦❢ ❚❛t ❉❛t ✭✷✵✵✼✮✱ ❞✐❢❢❡r❡♥t✐❛❧ ✬✬❖♥ ❡q✉❛t✐♦♥s t❤❡ ✇✐t❤ ❛❧♠♦st ♣✐❡❝❡✇✐s❡ ❛r❣✉♠❡♥t✬✬✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✱ ❱♦❧✳✸✷✻✱ ♣♣✳ ✶✻✺✲✶✼✽✳ ✹✻ ❛✉t♦♠♦r♣❤② ❝♦♥st❛♥t ❬✶✺❪ ❙✳ ▼✉r❛❦❛♠✐✱ ◆❛✐t♦✱ ◆❣✉②❡♥ ❱❛♥ ▼✐♥❤ ✭✷✵✵✹✮✱ ✬✬▼❛ss❡r❛ t❤❡♦r❡♠ ❢♦r ❛❧♠♦st ♣❡r✐♦❞✐❝ s♦❧✉t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✬✬✱ ❏✳ ▼❛t❤✳ ❙♦❝✳ ❏❛♣❛♥✱ ❱♦❧✳✺✻✭✶✮✱ ♣♣✳ ✷✹✼✲✷✻✽✳ ❬✶✻❪ ❚♦s❤✐❦✐ ◆❛✐t♦✱ ◆❣✉②❡♥ ❱❛♥ ▼✐♥❤ ✭✶✾✾✾✮✱ ✬✬❙❡♠✐❣r♦✉♣s ❛♥❞ ❙♣❡❝tr❛❧ ❈r✐✲ t❡r✐❛ ❢♦r ❆❧♠♦st P❡r✐♦❞✐❝ ❙♦❧✉t✐♦♥s ♦❢ P❡r✐♦❞✐❝ ❊✈♦❧✉t✐♦♥ ❊q✉❛t✐♦♥s✬✬ ❏♦✉r♥❛❧ ♦❢ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❱♦❧✳✶✺✷✱ ♣♣✳ ✸✺✽✲✸✼✻✳ ❬✶✼❪ ●✳ ▼✳ ◆✬●✉❡r ➆ ❡❦❛t❛ ➆ ✭✷✵✵✶✮✱ ❆❧♠♦st ❆✉t♦♠♦r♣❤✐❝ ❛♥❞ ❆❧♠♦st P❡r✐♦❞✐❝ ❋✉♥❝t✐♦♥s ✐♥ ❆❜str❛❝t ❙♣❛❝❡s✱ ❑❧✉✇❡r✱ ❆♠st❡r❞❛♠✳ ❬✶✽❪ ●✳ ▼✳ ◆✬●✉❡r ✭✷✵✵✺✮✱ ❚♦♣✐❝s ✐♥ ❆❧♠♦st ❆✉t♦♠♦r♣❤②✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ➆ ❡❦❛t❛ ➆ ❨♦r❦✳ ❬✶✾❪ ●✳ ▼✳ ◆✬●✉❡r ✭✶✾✾✾✮✱ ✬✬❆❧♠♦st ❛✉t♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ ❛♣♣❧✐❝❛✲ ➆ ❡❦❛t❛ ➆ t✐♦♥s t♦ ❛❜str❛❝t ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s✬✬✱ ❈♦♥t❡♠♣♦r❛r② ▼❛t❤✱ ❱♦❧✳✷✺✷✱ ♣♣✳ ✼✶✲✼✻✳ ❬✷✵❪ ❘♦♥❣ ❨✉❛♥ ✭✷✵✵✼✮✱ ✬✬❖♥ ❛❧♠♦st ♣❡r✐♦❞✐❝ s♦❧✉t✐♦♥s ♦❢ ❧♦❣✐st✐❝ ❞❡❧❛② ❞✐❢❢❡r✲ ❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ❛❧♠♦st ♣❡r✐♦❞✐❝ t✐♠❡ ❞❡♣❡♥❞❡♥❝❡✬✬✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✱ ❱♦❧✳✸✸✵✱ ♣♣✳ ✼✽✵✲✼✾✽✳ ✹✼