❙♦♠❡ ♣r♦❜❧❡♠s✿ ❍♦✇ t♦ ✜♥❞ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s❄ ❍♦✇ t♦ ❜♦✉♥❞ t❤❡ ❡rr♦r ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥❄ ❊①✐st❡♥❝❡ ❛♥❞ ❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ❙♦❧✉t✐♦♥ ❚❤❡♦r❡♠ ✭P✐❝❛r❞✲▲✐♥❞❡❧ö❢ ✕ ❧✐♥❡❛r ❝❛s❡✮ ❚❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ ( ▲ , ❇ t ✵ ) : C r ( I ) → C ✵ ( I ) × R r , ✐s ❛ ✭❜✐❝♦♥t✐♥✉♦✉s✮ ✐s♦♠♦r♣❤✐s♠✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t✿ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r♠ ❛ r ✲❞✐♠❡♥s✐♦♥❛❧ ❛✣♥❡ s♣❛❝❡✳ ❋♦r ✜①❡❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛t t ✵ ✱ t❤❡r❡ ✐s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ s♦❧✉t✐♦♥✳ ✻✴✷✺
❊①✐st❡♥❝❡ ❛♥❞ ❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ❙♦❧✉t✐♦♥ ❚❤❡♦r❡♠ ✭P✐❝❛r❞✲▲✐♥❞❡❧ö❢ ✕ ❧✐♥❡❛r ❝❛s❡✮ ❚❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ ( ▲ , ❇ t ✵ ) : C r ( I ) → C ✵ ( I ) × R r , ✐s ❛ ✭❜✐❝♦♥t✐♥✉♦✉s✮ ✐s♦♠♦r♣❤✐s♠✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t✿ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r♠ ❛ r ✲❞✐♠❡♥s✐♦♥❛❧ ❛✣♥❡ s♣❛❝❡✳ ❋♦r ✜①❡❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛t t ✵ ✱ t❤❡r❡ ✐s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ s♦❧✉t✐♦♥✳ ❙♦♠❡ ♣r♦❜❧❡♠s✿ ❍♦✇ t♦ ✜♥❞ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s❄ ❍♦✇ t♦ ❜♦✉♥❞ t❤❡ ❡rr♦r ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥❄ ✻✴✷✺
❘✐❣♦r♦✉s ❛r✐t❤♠❡t✐❝s✿ ■♥t❡r✈❛❧ ❛♥❛❧②s✐s ✭▼♦♦r❡✮ ❘✐❣♦r♦✉s P♦❧②♥♦♠✐❛❧ ❆♣♣r♦①✐♠❛t✐♦♥s ✭❘P❆s✮✿ ✉❧tr❛✲❛r✐t❤♠❡t✐❝s ✭❊♣st❡✐♥✱ ▼✐r❛♥❦❡r✱ ❘✐✈❧✐♥✮✱ ❚❛②❧♦r ♠♦❞❡❧s ✭▼❛❦✐♥♦ ❛♥❞ ❇❡r③✮✱ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❇r✐s❡❜❛rr❡✱ ❏♦❧❞❡s ✱ ✮✳ ❆ ♣♦st❡r✐♦r✐ ✈❛❧✐❞❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✿ ◗✉❛s✐✲◆❡✇t♦♥ ✜①❡❞✲♣♦✐♥t ♠❡t❤♦❞s ✭❨❛♠❛♠♦t♦✱ ▲❡ss❛r❞✮ ❉✲✜♥✐t❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ✭❇❡♥♦✐t✱ ❏♦❧❞❡s ✱ ✱ ▼❡③③❛r♦❜❜❛✮ Pr❡✈✐♦✉s ❲♦r❦s ❙♣❡❝tr❛❧ ♠❡t❤♦❞s✿ ❙❡♠✐♥❛❧ ✇♦r❦s ❜② ❖rs③❛❣✱ ❚r❡❢❡t❤❡♥ ❛♥❞ ♦t❤❡rs ❋❛st ❛❧❣♦r✐t❤♠ ❢♦r ▲❖❉❊s✿ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞ ✼✴✷✺
❆ ♣♦st❡r✐♦r✐ ✈❛❧✐❞❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✿ ◗✉❛s✐✲◆❡✇t♦♥ ✜①❡❞✲♣♦✐♥t ♠❡t❤♦❞s ✭❨❛♠❛♠♦t♦✱ ▲❡ss❛r❞✮ ❉✲✜♥✐t❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ✭❇❡♥♦✐t✱ ❏♦❧❞❡s ✱ ✱ ▼❡③③❛r♦❜❜❛✮ Pr❡✈✐♦✉s ❲♦r❦s ❙♣❡❝tr❛❧ ♠❡t❤♦❞s✿ ❙❡♠✐♥❛❧ ✇♦r❦s ❜② ❖rs③❛❣✱ ❚r❡❢❡t❤❡♥ ❛♥❞ ♦t❤❡rs ❋❛st ❛❧❣♦r✐t❤♠ ❢♦r ▲❖❉❊s✿ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞ ❘✐❣♦r♦✉s ❛r✐t❤♠❡t✐❝s✿ ■♥t❡r✈❛❧ ❛♥❛❧②s✐s ✭▼♦♦r❡✮ ❘✐❣♦r♦✉s P♦❧②♥♦♠✐❛❧ ❆♣♣r♦①✐♠❛t✐♦♥s ✭❘P❆s✮✿ ✉❧tr❛✲❛r✐t❤♠❡t✐❝s ✭❊♣st❡✐♥✱ ▼✐r❛♥❦❡r✱ ❘✐✈❧✐♥✮✱ ❚❛②❧♦r ♠♦❞❡❧s ✭▼❛❦✐♥♦ ❛♥❞ ❇❡r③✮✱ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❇r✐s❡❜❛rr❡✱ ❏♦❧❞❡s ✱ ✮✳ ✼✴✷✺
Pr❡✈✐♦✉s ❲♦r❦s ❙♣❡❝tr❛❧ ♠❡t❤♦❞s✿ ❙❡♠✐♥❛❧ ✇♦r❦s ❜② ❖rs③❛❣✱ ❚r❡❢❡t❤❡♥ ❛♥❞ ♦t❤❡rs ❋❛st ❛❧❣♦r✐t❤♠ ❢♦r ▲❖❉❊s✿ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞ ❘✐❣♦r♦✉s ❛r✐t❤♠❡t✐❝s✿ ■♥t❡r✈❛❧ ❛♥❛❧②s✐s ✭▼♦♦r❡✮ ❘✐❣♦r♦✉s P♦❧②♥♦♠✐❛❧ ❆♣♣r♦①✐♠❛t✐♦♥s ✭❘P❆s✮✿ ✉❧tr❛✲❛r✐t❤♠❡t✐❝s ✭❊♣st❡✐♥✱ ▼✐r❛♥❦❡r✱ ❘✐✈❧✐♥✮✱ ❚❛②❧♦r ♠♦❞❡❧s ✭▼❛❦✐♥♦ ❛♥❞ ❇❡r③✮✱ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❇r✐s❡❜❛rr❡✱ ❏♦❧❞❡s ✱ ✮✳ ❆ ♣♦st❡r✐♦r✐ ✈❛❧✐❞❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✿ ◗✉❛s✐✲◆❡✇t♦♥ ✜①❡❞✲♣♦✐♥t ♠❡t❤♦❞s ✭❨❛♠❛♠♦t♦✱ ▲❡ss❛r❞✮ ❉✲✜♥✐t❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ✭❇❡♥♦✐t✱ ❏♦❧❞❡s ✱ ✱ ▼❡③③❛r♦❜❜❛✮ ✼✴✷✺
❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ❑ ✇❤❡r❡✿ ❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r ▲❡t ϕ = f ( r ) ∈ C ✵ ( I ) ✇✐t❤ f ( t ✵ ) = v ✵ . . . f ( r − ✶ ) ( t ✵ ) = v r − ✶ ✳ ❚❤❡♥ ❢♦r i ∈ [ ✵ , r − ✶ ] ✿ � t � s ✶ � s r − ✶ − i r − ✶ � ( t − t ✵ ) j − i f ( i ) ( t ) = v j + . . . ϕ ❞ s ✶ . . . ❞ s r − i . ( j − i )! t ✵ t ✵ t ✵ j = i ✽✴✷✺
❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ❑ ✇❤❡r❡✿ ❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r ▲❡t ϕ = f ( r ) ∈ C ✵ ( I ) ✇✐t❤ f ( t ✵ ) = v ✵ . . . f ( r − ✶ ) ( t ✵ ) = v r − ✶ ✳ ❚❤❡♥ ❢♦r i ∈ [ ✵ , r − ✶ ] ✿ � t � s ✶ � s r − ✶ − i � r − ✶ ( t − t ✵ ) j − i f ( i ) ( t ) = v j + . . . ϕ ❞ s ✶ . . . ❞ s r − i . ( j − i )! t ✵ t ✵ t ✵ j = i � �� � � t ( t − s ) r − ✶ − i = ( r − ✶ − i )! ϕ ❞ s t ✵ ✽✴✷✺
❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ❑ ✇❤❡r❡✿ ❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r ▲❡t ϕ = f ( r ) ∈ C ✵ ( I ) ✇✐t❤ f ( t ✵ ) = v ✵ . . . f ( r − ✶ ) ( t ✵ ) = v r − ✶ ✳ ❚❤❡♥ ❢♦r i ∈ [ ✵ , r − ✶ ] ✿ � t r − ✶ � ( t − t ✵ ) j − i ( t − s ) r − ✶ − i f ( i ) ( t ) = v j + ( r − ✶ − i )! ϕ ❞ s . ( j − i )! t ✵ j = i ✽✴✷✺
❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r ▲❡t ϕ = f ( r ) ∈ C ✵ ( I ) ✇✐t❤ f ( t ✵ ) = v ✵ . . . f ( r − ✶ ) ( t ✵ ) = v r − ✶ ✳ ❚❤❡♥ ❢♦r i ∈ [ ✵ , r − ✶ ] ✿ � t r − ✶ � ( t − t ✵ ) j − i ( t − s ) r − ✶ − i f ( i ) ( t ) = v j + ( r − ✶ − i )! ϕ ❞ s . ( j − i )! t ✵ j = i ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ ✽✴✷✺
❑ ✐s ❛ ❝♦♠♣❛❝t ♦♣❡r❛t♦r✳ ✭s♦♠❡ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✬s✮ ❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r ▲❡t ϕ = f ( r ) ∈ C ✵ ( I ) ✇✐t❤ f ( t ✵ ) = v ✵ . . . f ( r − ✶ ) ( t ✵ ) = v r − ✶ ✳ ❚❤❡♥ ❢♦r i ∈ [ ✵ , r − ✶ ] ✿ � t r − ✶ � ( t − t ✵ ) j − i ( t − s ) r − ✶ − i f ( i ) ( t ) = v j + ( r − ✶ − i )! ϕ ❞ s . ( j − i )! t ✵ j = i ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ � t r − ✶ � ( t − s ) r − ✶ − i ❑ · ϕ ( t ) = a i ( t ) ( r − ✶ − i )! ϕ ( s ) ❞ s . t ✵ i = ✵ ✽✴✷✺
✭s♦♠❡ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✬s✮ ❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r ▲❡t ϕ = f ( r ) ∈ C ✵ ( I ) ✇✐t❤ f ( t ✵ ) = v ✵ . . . f ( r − ✶ ) ( t ✵ ) = v r − ✶ ✳ ❚❤❡♥ ❢♦r i ∈ [ ✵ , r − ✶ ] ✿ � t r − ✶ � ( t − t ✵ ) j − i ( t − s ) r − ✶ − i f ( i ) ( t ) = v j + ( r − ✶ − i )! ϕ ❞ s . ( j − i )! t ✵ j = i ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ � t r − ✶ � ( t − s ) r − ✶ − i ❑ · ϕ ( t ) = a i ( t ) ( r − ✶ − i )! ϕ ( s ) ❞ s . t ✵ i = ✵ ⇒ ❑ ✐s ❛ ❝♦♠♣❛❝t ♦♣❡r❛t♦r✳ ✽✴✷✺
❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r ▲❡t ϕ = f ( r ) ∈ C ✵ ( I ) ✇✐t❤ f ( t ✵ ) = v ✵ . . . f ( r − ✶ ) ( t ✵ ) = v r − ✶ ✳ ❚❤❡♥ ❢♦r i ∈ [ ✵ , r − ✶ ] ✿ � t r − ✶ � ( t − t ✵ ) j − i ( t − s ) r − ✶ − i f ( i ) ( t ) = v j + ( r − ✶ − i )! ϕ ❞ s . ( j − i )! t ✵ j = i ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ � t r − ✶ � ( t − s ) r − ✶ − i ❑ · ϕ ( t ) = a i ( t ) ( r − ✶ − i )! ϕ ( s ) ❞ s . t ✵ i = ✵ ⇒ ❑ ✐s ❛ ❝♦♠♣❛❝t ♦♣❡r❛t♦r✳ ψ ( t ) = g ( t ) + ✭s♦♠❡ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ v j ✬s✮ . ✽✴✷✺
✸ ✸ ❑ ✹ ❞ ✹ ❞ ✳ ✶ ❝♦s ✶ ❝♦s ✵ ✵ ✸ ✹ ✳ ✵ ✵ ✵ ✶ ❝♦s ❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r ▲❡t ϕ = f ( r ) ∈ C ✵ ( I ) ✇✐t❤ f ( t ✵ ) = v ✵ . . . f ( r − ✶ ) ( t ✵ ) = v r − ✶ ✳ ❚❤❡♥ ❢♦r i ∈ [ ✵ , r − ✶ ] ✿ � t r − ✶ � ( t − t ✵ ) j − i ( t − s ) r − ✶ − i f ( i ) ( t ) = v j + ( r − ✶ − i )! ϕ ❞ s . ( j − i )! t ✵ j = i ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ ❊①❛♠♣❧❡ ✽✴✷✺
✸ ✹ ✳ ✵ ✵ ✵ ✶ ❝♦s ❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r ▲❡t ϕ = f ( r ) ∈ C ✵ ( I ) ✇✐t❤ f ( t ✵ ) = v ✵ . . . f ( r − ✶ ) ( t ✵ ) = v r − ✶ ✳ ❚❤❡♥ ❢♦r i ∈ [ ✵ , r − ✶ ] ✿ � t r − ✶ � ( t − t ✵ ) j − i ( t − s ) r − ✶ − i f ( i ) ( t ) = v j + ( r − ✶ − i )! ϕ ❞ s . ( j − i )! t ✵ j = i ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ ❊①❛♠♣❧❡ � � � t � � � t ✸ ✸ ❑ · ϕ = t ✹ − ϕ ( s ) ❞ s − ✹ − s ϕ ( s ) ❞ s ✳ ✶ + e ❝♦s t ✶ + e ❝♦s t t ✵ t ✵ ✽✴✷✺
❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r ▲❡t ϕ = f ( r ) ∈ C ✵ ( I ) ✇✐t❤ f ( t ✵ ) = v ✵ . . . f ( r − ✶ ) ( t ✵ ) = v r − ✶ ✳ ❚❤❡♥ ❢♦r i ∈ [ ✵ , r − ✶ ] ✿ � t r − ✶ � ( t − t ✵ ) j − i ( t − s ) r − ✶ − i f ( i ) ( t ) = v j + ( r − ✶ − i )! ϕ ❞ s . ( j − i )! t ✵ j = i ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ ❊①❛♠♣❧❡ � � � t � � � t ✸ ✸ ❑ · ϕ = t ✹ − ϕ ( s ) ❞ s − ✹ − s ϕ ( s ) ❞ s ✳ ✶ + e ❝♦s t ✶ + e ❝♦s t t ✵ t ✵ � � ✸ ψ ( t ) = c − ( z ( t ✵ ) + ( t − t ✵ ) z ′ ( t ✵ )) ✹ − ✳ ✶ + e ❝♦s t ✽✴✷✺
❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✐♥ ▲✐♥❡❛r ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✶ ❆♣♣r♦①✐♠❛t✐♥❣ ❋✉♥❝t✐♦♥s ✇✐t❤ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ✷ ❚❤❡ ❱❛❧✐❞❛t✐♦♥ ❆❧❣♦r✐t❤♠ ✸ ✾✴✷✺
✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . ✶✵✴✷✺
✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . T ✵ ( X ) = ✶ ✶✵✴✷✺
✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . T ✵ ( X ) = ✶ T ✶ ( X ) = X ✶✵✴✷✺
✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . T ✵ ( X ) = ✶ T ✶ ( X ) = X T ✷ ( X ) = ✷ X ✷ − ✶ ✶✵✴✷✺
✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . T ✵ ( X ) = ✶ T ✶ ( X ) = X T ✷ ( X ) = ✷ X ✷ − ✶ T ✸ ( X ) = ✹ X ✸ − ✸ X ✶✵✴✷✺
✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . T ✵ ( X ) = ✶ T ✶ ( X ) = X T ✷ ( X ) = ✷ X ✷ − ✶ T ✸ ( X ) = ✹ X ✸ − ✸ X T ✹ ( X ) = ✽ X ✹ − ✽ X ✷ + ✶ ✶✵✴✷✺
✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . T ✵ ( X ) = ✶ T ✶ ( X ) = X T ✷ ( X ) = ✷ X ✷ − ✶ T ✸ ( X ) = ✹ X ✸ − ✸ X T ✹ ( X ) = ✽ X ✹ − ✽ X ✷ + ✶ T ✺ ( X ) = ✶✻ X ✺ − ✷✵ X ✸ + ✺ X ✶✵✴✷✺
❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ✶ ✶ ✶ ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ T n ( ❝♦s ϑ ) = ❝♦s n ϑ. ✶✵✴✷✺
❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ T n ( ❝♦s ϑ ) = ❝♦s n ϑ. ⇒ ∀ t ∈ [ − ✶ , ✶ ] , | T n ( t ) | ≤ ✶ . ✶✵✴✷✺
▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ T n ( ❝♦s ϑ ) = ❝♦s n ϑ. ⇒ ∀ t ∈ [ − ✶ , ✶ ] , | T n ( t ) | ≤ ✶ . ❲❡ ❞❡✜♥❡ T − n = T n ❢♦r n ≥ ✵✳ ✶✵✴✷✺
■♥t❡❣r❛t✐♦♥✿ ✶ ✶ ✶ ✷ ✶ ✶ ❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ T n ( ❝♦s ϑ ) = ❝♦s n ϑ. ⇒ ∀ t ∈ [ − ✶ , ✶ ] , | T n ( t ) | ≤ ✶ . ❲❡ ❞❡✜♥❡ T − n = T n ❢♦r n ≥ ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ T n T m = ✶ ✷ ( T n + m + T n − m ) . ✶✵✴✷✺
❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s ❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T ✵ ( X ) = ✶ , T ✶ ( X ) = X , − ✶ ✶ T n + ✷ ( X ) = ✷ XT n + ✶ ( X ) − T n ( X ) . ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ T n ( ❝♦s ϑ ) = ❝♦s n ϑ. ⇒ ∀ t ∈ [ − ✶ , ✶ ] , | T n ( t ) | ≤ ✶ . ❲❡ ❞❡✜♥❡ T − n = T n ❢♦r n ≥ ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ T n T m = ✶ ✷ ( T n + m + T n − m ) . ■♥t❡❣r❛t✐♦♥✿ � T n + ✶ � � T n = ✶ n + ✶ − T n − ✶ . ✷ n − ✶ ✶✵✴✷✺
❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ ✵ ✐❢ ✐❢ ✵ ✐❢ ✵ ✷ ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ ✶ ❝♦s ❝♦s ❞ ✵ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ ✶ ✶ ▼❛✐♥ q✉❡st✐♦♥✿ ✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮ ❝♦s ❝♦s ❞ ✵ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ❙❝❛❧❛r ♣r♦❞✉❝t✿ � ✶ f ( t ) g ( t ) � f , g � = √ ✶ − t ✷ ❞ t − ✶ ✶✶✴✷✺
✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮ ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ ✵ ✐❢ ✐❢ ✵ ✐❢ ✵ ✷ ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ ✶ ❝♦s ❝♦s ❞ ✵ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ ✶ ✶ ▼❛✐♥ q✉❡st✐♦♥✿ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ❙❝❛❧❛r ♣r♦❞✉❝t✿ � ✶ � π f ( t ) g ( t ) � f , g � = √ ✶ − t ✷ ❞ t = f ( ❝♦s ϑ ) g ( ❝♦s ϑ ) ❞ ϑ. − ✶ ✵ ✶✶✴✷✺
✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮ ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ ✶ ❝♦s ❝♦s ❞ ✵ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ ✶ ✶ ▼❛✐♥ q✉❡st✐♦♥✿ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ❙❝❛❧❛r ♣r♦❞✉❝t✿ � ✶ � π f ( t ) g ( t ) � f , g � = √ ✶ − t ✷ ❞ t = f ( ❝♦s ϑ ) g ( ❝♦s ϑ ) ❞ ϑ. − ✶ ✵ ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ ✵ ✐❢ n � = ± m , � T n , T m � = π ✐❢ n = m = ✵ , π ✐❢ n = ± m � = ✵ . ✷ ✶✶✴✷✺
✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ ✶ ✶ ▼❛✐♥ q✉❡st✐♦♥✿ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ❙❝❛❧❛r ♣r♦❞✉❝t✿ � ✶ � π f ( t ) g ( t ) � f , g � = √ ✶ − t ✷ ❞ t = f ( ❝♦s ϑ ) g ( ❝♦s ϑ ) ❞ ϑ. − ✶ ✵ ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ ✵ ✐❢ n � = ± m , � T n , T m � = π ✐❢ n = m = ✵ , π ✐❢ n = ± m � = ✵ . ✷ ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ � π a n = ✶ f ( ❝♦s ϑ ) ❝♦s n ϑ ❞ ϑ, n ∈ Z . π ✵ ✶✶✴✷✺
✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮ ▼❛✐♥ q✉❡st✐♦♥✿ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ❙❝❛❧❛r ♣r♦❞✉❝t✿ � ✶ � π f ( t ) g ( t ) � f , g � = √ ✶ − t ✷ ❞ t = f ( ❝♦s ϑ ) g ( ❝♦s ϑ ) ❞ ϑ. − ✶ ✵ ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ ✵ ✐❢ n � = ± m , � T n , T m � = π ✐❢ n = m = ✵ , π ✐❢ n = ± m � = ✵ . ✷ ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ � π a n = ✶ f ( ❝♦s ϑ ) ❝♦s n ϑ ❞ ϑ, n ∈ Z . π ✵ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ � � f ( t ) = a n T n ( t ) , t ∈ [ − ✶ , ✶ ] . n ∈ Z ✶✶✴✷✺
✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ❙❝❛❧❛r ♣r♦❞✉❝t✿ � ✶ � π f ( t ) g ( t ) � f , g � = √ ✶ − t ✷ ❞ t = f ( ❝♦s ϑ ) g ( ❝♦s ϑ ) ❞ ϑ. − ✶ ✵ ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ ✵ ✐❢ n � = ± m , � T n , T m � = π ✐❢ n = m = ✵ , π ✐❢ n = ± m � = ✵ . ✷ ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ � π a n = ✶ f ( ❝♦s ϑ ) ❝♦s n ϑ ❞ ϑ, n ∈ Z . π ✵ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ � � f ( t ) = a n T n ( t ) , t ∈ [ − ✶ , ✶ ] . n ∈ Z ▼❛✐♥ q✉❡st✐♦♥✿ f = � ? f ✶✶✴✷✺
❈❤❡❜②s❤❡✈ ❙❡r✐❡s ❙❝❛❧❛r ♣r♦❞✉❝t✿ � ✶ � π f ( t ) g ( t ) � f , g � = √ ✶ − t ✷ ❞ t = f ( ❝♦s ϑ ) g ( ❝♦s ϑ ) ❞ ϑ. − ✶ ✵ ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ ✵ ✐❢ n � = ± m , � T n , T m � = π ✐❢ n = m = ✵ , π ✐❢ n = ± m � = ✵ . ✷ ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ � π a n = ✶ f ( ❝♦s ϑ ) ❝♦s n ϑ ❞ ϑ, n ∈ Z . π ✵ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ � � f ( t ) = a n T n ( t ) , t ∈ [ − ✶ , ✶ ] . n ∈ Z ▼❛✐♥ q✉❡st✐♦♥✿ f = � ? ✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮ f ✶✶✴✷✺
× ✶ × ✶ × ✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ✶ × ✶ ✵ ❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶ ✮✱ t❤❡♥ ✳ ❚❤❡♦r❡♠ ✷ ✶ ✷ ♦✈❡r ■❢ ✐s ✐♥ ✶ ✶ ✶ ✱ t❤❡♥ ❝♦♥✈❡r❣❡s t♦ ✐♥ ✷ ✶ ✷ ✳ ✶ ❚❤❡♦r❡♠ ■❢ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t ✶ ✶ ✱ t❤❡♥ ❛s ✳ ✶✲♥♦r♠ ✐♥ × ✶ ✿ × ✶ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s f [ N ] = � ▲❡t � | n |≤ N a n T n ✳ ✶✷✴✷✺
× ✶ × ✶ × ✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ✶ × ✶ ✵ ❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶ ✮✱ t❤❡♥ ✳ ❚❤❡♦r❡♠ ■❢ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t ✶ ✶ ✱ t❤❡♥ ❛s ✳ ✶✲♥♦r♠ ✐♥ × ✶ ✿ × ✶ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s f [ N ] = � ▲❡t � | n |≤ N a n T n ✳ ❚❤❡♦r❡♠ √ f [ N ] ❝♦♥✈❡r❣❡s t♦ f ✐♥ ✶ − t ✷ ) ♦✈❡r [ − ✶ , ✶ ] ✱ t❤❡♥ � ■❢ f ✐s ✐♥ L ✷ ( ✶ / √ L ✷ ( ✶ / ✶ − t ✷ ) ✳ ✶✷✴✷✺
× ✶ × ✶ × ✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ✶ × ✶ ✵ ❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶ ✮✱ t❤❡♥ ✳ ✶✲♥♦r♠ ✐♥ × ✶ ✿ × ✶ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s f [ N ] = � ▲❡t � | n |≤ N a n T n ✳ ❚❤❡♦r❡♠ √ f [ N ] ❝♦♥✈❡r❣❡s t♦ f ✐♥ ✶ − t ✷ ) ♦✈❡r [ − ✶ , ✶ ] ✱ t❤❡♥ � ■❢ f ✐s ✐♥ L ✷ ( ✶ / √ L ✷ ( ✶ / ✶ − t ✷ ) ✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [ − ✶ , ✶ ] ✱ t❤❡♥ � f [ N ] ( x ) → f ( x ) ❛s N → ∞ ✳ ✶✷✴✷✺
× ✶ × ✶ × ✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ✶ × ✶ ✵ ❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶ ✮✱ t❤❡♥ ✳ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s f [ N ] = � ▲❡t � | n |≤ N a n T n ✳ ❚❤❡♦r❡♠ √ f [ N ] ❝♦♥✈❡r❣❡s t♦ f ✐♥ ✶ − t ✷ ) ♦✈❡r [ − ✶ , ✶ ] ✱ t❤❡♥ � ■❢ f ✐s ✐♥ L ✷ ( ✶ / √ L ✷ ( ✶ / ✶ − t ✷ ) ✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [ − ✶ , ✶ ] ✱ t❤❡♥ � f [ N ] ( x ) → f ( x ) ❛s N → ∞ ✳ ✶✲♥♦r♠ ✐♥ × ✶ ✿ � � f � × ✶ = | a n | n ∈ Z ✶✷✴✷✺
× ✶ × ✶ × ✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ✶ × ✶ ✵ ❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶ ✮✱ t❤❡♥ ✳ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s f [ N ] = � ▲❡t � | n |≤ N a n T n ✳ ❚❤❡♦r❡♠ √ f [ N ] ❝♦♥✈❡r❣❡s t♦ f ✐♥ ✶ − t ✷ ) ♦✈❡r [ − ✶ , ✶ ] ✱ t❤❡♥ � ■❢ f ✐s ✐♥ L ✷ ( ✶ / √ L ✷ ( ✶ / ✶ − t ✷ ) ✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [ − ✶ , ✶ ] ✱ t❤❡♥ � f [ N ] ( x ) → f ( x ) ❛s N → ∞ ✳ ✶✲♥♦r♠ ✐♥ × ✶ ✿ � � f � × ✶ = | a n | ≥ � f � ∞ n ∈ Z ✶✷✴✷✺
× ✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ✶ × ✶ ✵ ❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶ ✮✱ t❤❡♥ ✳ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s f [ N ] = � ▲❡t � | n |≤ N a n T n ✳ ❚❤❡♦r❡♠ √ f [ N ] ❝♦♥✈❡r❣❡s t♦ f ✐♥ ✶ − t ✷ ) ♦✈❡r [ − ✶ , ✶ ] ✱ t❤❡♥ � ■❢ f ✐s ✐♥ L ✷ ( ✶ / √ L ✷ ( ✶ / ✶ − t ✷ ) ✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [ − ✶ , ✶ ] ✱ t❤❡♥ � f [ N ] ( x ) → f ( x ) ❛s N → ∞ ✳ ✶✲♥♦r♠ ✐♥ × ✶ ✿ � × ✶ = { f | � f � × ✶ < ∞} � f � × ✶ = | a n | ≥ � f � ∞ n ∈ Z ✶✷✴✷✺
✶ × ✶ ✵ ❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶ ✮✱ t❤❡♥ ✳ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s f [ N ] = � ▲❡t � | n |≤ N a n T n ✳ ❚❤❡♦r❡♠ √ f [ N ] ❝♦♥✈❡r❣❡s t♦ f ✐♥ ✶ − t ✷ ) ♦✈❡r [ − ✶ , ✶ ] ✱ t❤❡♥ � ■❢ f ✐s ✐♥ L ✷ ( ✶ / √ L ✷ ( ✶ / ✶ − t ✷ ) ✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [ − ✶ , ✶ ] ✱ t❤❡♥ � f [ N ] ( x ) → f ( x ) ❛s N → ∞ ✳ ✶✲♥♦r♠ ✐♥ × ✶ ✿ � × ✶ = { f | � f � × ✶ < ∞} � f � × ✶ = | a n | ≥ � f � ∞ n ∈ Z × ✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ✶✷✴✷✺
❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶ ✮✱ t❤❡♥ ✳ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s f [ N ] = � ▲❡t � | n |≤ N a n T n ✳ ❚❤❡♦r❡♠ √ f [ N ] ❝♦♥✈❡r❣❡s t♦ f ✐♥ ✶ − t ✷ ) ♦✈❡r [ − ✶ , ✶ ] ✱ t❤❡♥ � ■❢ f ✐s ✐♥ L ✷ ( ✶ / √ L ✷ ( ✶ / ✶ − t ✷ ) ✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [ − ✶ , ✶ ] ✱ t❤❡♥ � f [ N ] ( x ) → f ( x ) ❛s N → ∞ ✳ ✶✲♥♦r♠ ✐♥ × ✶ ✿ � × ✶ = { f | � f � × ✶ < ∞} � f � × ✶ = | a n | ≥ � f � ∞ n ∈ Z × ✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ C ✶ ⊂ × ✶ ⊂ C ✵ ✶✷✴✷✺
❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s f [ N ] = � ▲❡t � | n |≤ N a n T n ✳ ❚❤❡♦r❡♠ √ f [ N ] ❝♦♥✈❡r❣❡s t♦ f ✐♥ ✶ − t ✷ ) ♦✈❡r [ − ✶ , ✶ ] ✱ t❤❡♥ � ■❢ f ✐s ✐♥ L ✷ ( ✶ / √ L ✷ ( ✶ / ✶ − t ✷ ) ✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [ − ✶ , ✶ ] ✱ t❤❡♥ � f [ N ] ( x ) → f ( x ) ❛s N → ∞ ✳ ✶✲♥♦r♠ ✐♥ × ✶ ✿ � × ✶ = { f | � f � × ✶ < ∞} � f � × ✶ = | a n | ≥ � f � ∞ n ∈ Z × ✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ C ✶ ⊂ × ✶ ⊂ C ✵ ❚❤❡♦r❡♠ ■❢ f ✐s C r ✭ r ≥ ✶ ✮✱ t❤❡♥ a n = O ( n − r ) ✳ ✶✷✴✷✺
❆♣♣r♦①✐♠❛t✐♥❣ ♦✉r ❊①❛♠♣❧❡ ✸ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t �→ ✹ − ✶ + e ❝♦s t ♦✈❡r [ − ✶ , ✶ ] ✭ e = ✵ . ✺✮✿ ✷ ✷ − ✶ ✶ . ✺ ✶ − ✶ ✶ . ✺ ✶ α ( t ) ✶✸✴✷✺
❆♣♣r♦①✐♠❛t✐♥❣ ♦✉r ❊①❛♠♣❧❡ ✸ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t �→ ✹ − ✶ + e ❝♦s t ♦✈❡r [ − ✶ , ✶ ] ✭ e = ✵ . ✺✮✿ ✷ ✷ − ✶ ✶ . ✺ ✶ − ✶ ✶ . ✺ ✶ α ( t ) ✶ . ✽✷ | α ( t ) − ✶ . ✽✷ | ≤ ✵ . ✷ ✶✸✴✷✺
❆♣♣r♦①✐♠❛t✐♥❣ ♦✉r ❊①❛♠♣❧❡ ✸ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t �→ ✹ − ✶ + e ❝♦s t ♦✈❡r [ − ✶ , ✶ ] ✭ e = ✵ . ✺✮✿ ✷ ✷ − ✶ ✶ . ✺ ✶ − ✶ ✶ . ✺ ✶ α ( t ) ✶ . ✽✷ − ✵ . ✶✽ T ✷ ( t ) | α ( t ) − ( ✶ . ✽✷ − ✵ . ✶✽ T ✷ ( t )) | ≤ ✵ . ✵✵✼ ✶✸✴✷✺
❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✐♥ ▲✐♥❡❛r ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✶ ❆♣♣r♦①✐♠❛t✐♥❣ ❋✉♥❝t✐♦♥s ✇✐t❤ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ✷ ❚❤❡ ❱❛❧✐❞❛t✐♦♥ ❆❧❣♦r✐t❤♠ ✸ ✶✹✴✷✺
✷ × ✶ ✶ × ✶ ✶ ✵ ❞❡❣ ✵ ✵ ✵ ❞❡❣ ✶ ❞❡❣ ✶ ✶ ✶ ❞❡❣ ♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ✶ ❞ ✵ ✵ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s t ✵ j = ✵ ✶✺✴✷✺
✷ × ✶ ✶ × ✶ ✶ ✵ ❞❡❣ ✵ ✵ ✵ ❞❡❣ ✶ ❞❡❣ ✶ ✶ ✶ ❞❡❣ ♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t r − ✶ � � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . t ✵ t ✵ j = ✵ j = ✵ ✶✺✴✷✺
✷ × ✶ ✶ × ✶ ✶ ✵ ❞❡❣ ✵ ✵ ❞❡❣ ✶ ❞❡❣ ✶ ✶ ✶ ❞❡❣ ♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t r − ✶ � � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . t ✵ t ✵ j = ✵ j = ✵ T i ✵ i ✶✺✴✷✺
✷ × ✶ ✶ × ✶ ✶ ✵ ❞❡❣ ✵ ❞❡❣ ✶ ❞❡❣ ✶ ✶ ✶ ❞❡❣ ♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t r − ✶ � � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . t ✵ t ✵ j = ✵ j = ✵ T i T j T i ✵ ✵ i − j i i + j ✶✺✴✷✺
✷ × ✶ × ✶ ✶ ❞❡❣ ❞❡❣ ✶ ❞❡❣ ✶ ✶ ❞❡❣ ♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t r − ✶ � � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . t ✵ t ✵ j = ✵ j = ✵ � t T i T j T i t ✵ T j T i ✵ ✵ ✵ i − j − ✶ i − j i i + j i + j + ✶ ✶✺✴✷✺
✷ × ✶ × ✶ ✶ ❞❡❣ ❞❡❣ ✶ ❞❡❣ ✶ ❞❡❣ ♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t r − ✶ � � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . t ✵ t ✵ j = ✵ j = ✵ � t T i T j T i t ✵ T j T i ✵ ✵ ✵ i − j − ✶ i − j ✶ / i i i + j i + j + ✶ ✶✺✴✷✺
✷ × ✶ × ✶ ✶ ❞❡❣ ❞❡❣ ✶ ❞❡❣ ✶ ❞❡❣ ♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t r − ✶ � � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . t ✵ t ✵ j = ✵ j = ✵ � t T i T j T i t ✵ T j T i ✶ / i ✵ ✵ ✵ i − j − ✶ i − j ✶ / i i i + j i + j + ✶ ✶✺✴✷✺
✷ × ✶ × ✶ ✶ ❞❡❣ ❞❡❣ ✶ ❞❡❣ ✶ ❞❡❣ ♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t � r − ✶ � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . − ✶ − ✶ j = ✵ j = ✵ � t T i T j T i − ✶ T j T i ✶ / i ✷ ✵ ✵ ✵ i − j − ✶ i − j ✶ / i i i + j i + j + ✶ ✶✺✴✷✺
✷ × ✶ × ✶ ♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t � r − ✶ � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . − ✶ − ✶ j = ✵ j = ✵ � t � t T i T j T i β j − ✶ T j T i − ✶ T j T i − ❞❡❣ β j ✶ / i ✷ ✵ ✵ ✵ ❞❡❣ β j i − j − ✶ − ❞❡❣ β j i − j − ✶ i − j ✶ / i i i + j i + j + ✶ i + j + ✶ + ❞❡❣ β j ✶✺✴✷✺
✷ × ✶ ♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t � r − ✶ � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . − ✶ − ✶ j = ✵ j = ✵ � t � t T i T j T i β j − ✶ T j T i − ✶ T j T i − ❞❡❣ β j ✶ / i ✷ ✵ ✵ ✵ ❞❡❣ β j i − j − ✶ − ❞❡❣ β j i − j − ✶ i − j ✶ / i � β j � × ✶ / i i i + j i + j + ✶ i + j + ✶ + ❞❡❣ β j ✶✺✴✷✺
♠❛① ❞❡❣ ♠❛① ✶ ❞❡❣ ✵ ✵ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t � r − ✶ � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . − ✶ − ✶ j = ✵ j = ✵ � t � t T i T j T i β j − ✶ T j T i − ✶ T j T i − ❞❡❣ β j � β j � × ✶ / i ✷ ✶ / i ✷ ✵ ✵ ✵ ❞❡❣ β j i − j − ✶ − ❞❡❣ β j i − j − ✶ i − j ✶ / i � β j � × ✶ / i i i + j i + j + ✶ i + j + ✶ + ❞❡❣ β j ✶✺✴✷✺
❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t � r − ✶ � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . − ✶ − ✶ j = ✵ j = ✵ � t � t T i T j T i β j − ✶ T j T i − ✶ T j T i − ❞❡❣ β j � β j � × ✶ / i ✷ ✶ / i ✷ ✵ ✵ ✵ ❞❡❣ β j i − j − ✶ − ❞❡❣ β j i − j − ✶ i − j ✶ / i � β j � × ✶ / i i i + j i + j + ✶ i + j + ✶ + ❞❡❣ β j h = ♠❛① ✵ ≤ j < r ❞❡❣ β j , d = ♠❛① ✵ ≤ j < r ( j + ✶ + ❞❡❣ β j ) . ✶✺✴✷✺
❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ � t � t � r − ✶ � r − ✶ ( t − s ) r − ✶ − j ❑ · ϕ ( t ) = a j ( t ) ( r − ✶ − j )! ϕ ( s ) ❞ s = β j ( t ) T j ( s ) ϕ ( s ) ❞ s . − ✶ − ✶ j = ✵ j = ✵ � t � t T i T j T i β j − ✶ T j T i − ✶ T j T i − ❞❡❣ β j � β j � × ✶ / i ✷ ✶ / i ✷ ✵ ✵ ✵ ❞❡❣ β j i − j − ✶ − ❞❡❣ β j i − j − ✶ i − j ✶ / i � β j � × ✶ / i i i + j i + j + ✶ i + j + ✶ + ❞❡❣ β j h = ♠❛① ✵ ≤ j < r ❞❡❣ β j , d = ♠❛① ✵ ≤ j < r ( j + ✶ + ❞❡❣ β j ) . ⇒ ❑ ✐s ❛♥ ( h , d ) ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳ ✶✺✴✷✺
❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ → ↓ j = ✵ i = ✵ ❚❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ♦♣❡r❛t♦r ❑ ✳ ✶✻✴✷✺
❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ → ↓ j = ✵ i = ✵ ❚❤❡ ✜♥❛❧✲❞✐♠❡♥s✐♦♥❛❧ tr✉♥❝❛t✐♦♥ ❑ [ N ] ✳ ✶✻✴✷✺
❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ ✲ ❊①❛♠♣❧❡ � � � t � � � t ✸ ✸ ❑ · ϕ = t ✹ − ϕ ( s ) ❞ s + − ✹ + s ϕ ( s ) ❞ s ✶ + e ❝♦s t ✶ + e ❝♦s t t ✵ t ✵ ✶✼✴✷✺
❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ ✲ ❊①❛♠♣❧❡ � t � t ❑ · ϕ ≈ t ( ✶ . ✽✷ − ✵ . ✶✽ T ✷ ( t )) ϕ ( s ) ❞ s + ( − ✶ . ✽✷ + ✵ . ✶✽ T ✷ ( t )) s ϕ ( s ) ❞ s t ✵ t ✵ ✶✼✴✷✺
❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ ✲ ❊①❛♠♣❧❡ � t � t ❑ · ϕ ≈ ( ✶ . ✼✸ T ✶ ( t ) − ✵ . ✵✾ T ✸ ( t )) ϕ ( s ) ❞ s + ( − ✶ . ✽✷ + ✵ . ✶✽ T ✷ ( t )) s ϕ ( s ) ❞ s � �� � � �� � t ✵ t ✵ β ✵ ( t ) β ✶ ( t ) ✶✼✴✷✺
❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ ✲ ❊①❛♠♣❧❡ � t � t ❑ · ϕ ≈ ( ✶ . ✼✸ T ✶ ( t ) − ✵ . ✵✾ T ✸ ( t )) ϕ ( s ) ❞ s + ( − ✶ . ✽✷ + ✵ . ✶✽ T ✷ ( t )) s ϕ ( s ) ❞ s � �� � � �� � t ✵ t ✵ β ✵ ( t ) β ✶ ( t ) ✶✼✴✷✺
◆ ❊q✉✐✈❛❧❡♥t t♦ ■ ❑ ✇❤❡r❡ ✳ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ ❑ ✳ ✵ ✽✷ ✶ ✼✸ ✵ ✶✽ ✵ ✵✾ ✸ ✳ ✵ ✶ ✷ ❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿ ✵ ✻ ✶ ✶✾ ✵ ✻✷ ✵ ✶✼ ✵ ✵✺ ✵ ✵✶ ✵ ✶ ✷ ✸ ✹ ✺ ✸ ✸ ✺ ✻ ✻ ✷ ✶ ✶✵ ✸ ✷ ✶✵ ✺ ✽ ✶✵ ✼ ✻ ✶✵ ✶ ✷ ✶✵ ✻ ✼ ✽ ✾ ✶✵ ✼ ✽ ✾ ✶✵ ✶✶ ✶ ✹ ✶✵ ✶ ✾ ✶✵ ✷ ✵ ✶✵ ✷ ✻ ✶✵ ✷ ✺ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✸ ✵ ✶✵ ✷ ✻ ✶✵ ✸ ✵ ✶✵ ✷ ✺ ✶✵ ✷ ✻ ✶✵ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡ � � ❲❡ ✇❛♥t t♦ s♦❧✈❡ z ′′ ( t ) + ✸ ✹ − z ( t ) = c ✇✐t❤ z ( − ✶ ) = ✵✱ ✶ + ✵ . ✺ ❝♦s t z ′ ( − ✶ ) = ✶ ❛♥❞ c = ✶✳ ✶✽✴✷✺
◆ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ ❑ ✳ ✵ ✽✷ ✶ ✼✸ ✵ ✶✽ ✵ ✵✾ ✸ ✳ ✵ ✶ ✷ ❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿ ✵ ✻ ✶ ✶✾ ✵ ✻✷ ✵ ✶✼ ✵ ✵✺ ✵ ✵✶ ✵ ✶ ✷ ✸ ✹ ✺ ✸ ✸ ✺ ✻ ✻ ✷ ✶ ✶✵ ✸ ✷ ✶✵ ✺ ✽ ✶✵ ✼ ✻ ✶✵ ✶ ✷ ✶✵ ✻ ✼ ✽ ✾ ✶✵ ✼ ✽ ✾ ✶✵ ✶✶ ✶ ✹ ✶✵ ✶ ✾ ✶✵ ✷ ✵ ✶✵ ✷ ✻ ✶✵ ✷ ✺ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✸ ✵ ✶✵ ✷ ✻ ✶✵ ✸ ✵ ✶✵ ✷ ✺ ✶✵ ✷ ✻ ✶✵ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡ � � ❲❡ ✇❛♥t t♦ s♦❧✈❡ z ′′ ( t ) + ✸ ✹ − z ( t ) = c ✇✐t❤ z ( − ✶ ) = ✵✱ ✶ + ✵ . ✺ ❝♦s t z ′ ( − ✶ ) = ✶ ❛♥❞ c = ✶✳ � � · ϕ = ψ ✇❤❡r❡ ϕ = z ′′ ✳ ❊q✉✐✈❛❧❡♥t t♦ ■ + ❑ ✶✽✴✷✺
❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ ❑ ✳ ✵ ✽✷ ✶ ✼✸ ✵ ✶✽ ✵ ✵✾ ✸ ✳ ✵ ✶ ✷ ❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿ ✵ ✻ ✶ ✶✾ ✵ ✻✷ ✵ ✶✼ ✵ ✵✺ ✵ ✵✶ ✵ ✶ ✷ ✸ ✹ ✺ ✸ ✸ ✺ ✻ ✻ ✷ ✶ ✶✵ ✸ ✷ ✶✵ ✺ ✽ ✶✵ ✼ ✻ ✶✵ ✶ ✷ ✶✵ ✻ ✼ ✽ ✾ ✶✵ ✼ ✽ ✾ ✶✵ ✶✶ ✶ ✹ ✶✵ ✶ ✾ ✶✵ ✷ ✵ ✶✵ ✷ ✻ ✶✵ ✷ ✺ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✸ ✵ ✶✵ ✷ ✻ ✶✵ ✸ ✵ ✶✵ ✷ ✺ ✶✵ ✷ ✻ ✶✵ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡ � � ❲❡ ✇❛♥t t♦ s♦❧✈❡ z ′′ ( t ) + ✸ ✹ − z ( t ) = c ✇✐t❤ z ( − ✶ ) = ✵✱ ✶ + ✵ . ✺ ❝♦s t z ′ ( − ✶ ) = ✶ ❛♥❞ c = ✶✳ � ■ + ❑ [ ◆ ] � · ϕ = ψ ✇❤❡r❡ ϕ = z ′′ ✳ ≈ ❊q✉✐✈❛❧❡♥t t♦ ✶✽✴✷✺
✵ ✽✷ ✶ ✼✸ ✵ ✶✽ ✵ ✵✾ ✸ ✳ ✵ ✶ ✷ ❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿ ✵ ✻ ✶ ✶✾ ✵ ✻✷ ✵ ✶✼ ✵ ✵✺ ✵ ✵✶ ✵ ✶ ✷ ✸ ✹ ✺ ✸ ✸ ✺ ✻ ✻ ✷ ✶ ✶✵ ✸ ✷ ✶✵ ✺ ✽ ✶✵ ✼ ✻ ✶✵ ✶ ✷ ✶✵ ✻ ✼ ✽ ✾ ✶✵ ✼ ✽ ✾ ✶✵ ✶✶ ✶ ✹ ✶✵ ✶ ✾ ✶✵ ✷ ✵ ✶✵ ✷ ✻ ✶✵ ✷ ✺ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✸ ✵ ✶✵ ✷ ✻ ✶✵ ✸ ✵ ✶✵ ✷ ✺ ✶✵ ✷ ✻ ✶✵ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡ � � ❲❡ ✇❛♥t t♦ s♦❧✈❡ z ′′ ( t ) + ✸ ✹ − z ( t ) = c ✇✐t❤ z ( − ✶ ) = ✵✱ ✶ + ✵ . ✺ ❝♦s t z ′ ( − ✶ ) = ✶ ❛♥❞ c = ✶✳ � ■ + ❑ [ ◆ ] � · ϕ = ψ ✇❤❡r❡ ϕ = z ′′ ✳ ≈ ❊q✉✐✈❛❧❡♥t t♦ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ + ❑ [ N ] ✳ ✶✽✴✷✺
❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿ ✵ ✻ ✶ ✶✾ ✵ ✻✷ ✵ ✶✼ ✵ ✵✺ ✵ ✵✶ ✵ ✶ ✷ ✸ ✹ ✺ ✸ ✸ ✺ ✻ ✻ ✷ ✶ ✶✵ ✸ ✷ ✶✵ ✺ ✽ ✶✵ ✼ ✻ ✶✵ ✶ ✷ ✶✵ ✻ ✼ ✽ ✾ ✶✵ ✼ ✽ ✾ ✶✵ ✶✶ ✶ ✹ ✶✵ ✶ ✾ ✶✵ ✷ ✵ ✶✵ ✷ ✻ ✶✵ ✷ ✺ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✸ ✵ ✶✵ ✷ ✻ ✶✵ ✸ ✵ ✶✵ ✷ ✺ ✶✵ ✷ ✻ ✶✵ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡ � � ❲❡ ✇❛♥t t♦ s♦❧✈❡ z ′′ ( t ) + ✸ ✹ − z ( t ) = c ✇✐t❤ z ( − ✶ ) = ✵✱ ✶ + ✵ . ✺ ❝♦s t z ′ ( − ✶ ) = ✶ ❛♥❞ c = ✶✳ � ■ + ❑ [ ◆ ] � · ϕ = ψ ✇❤❡r❡ ϕ = z ′′ ✳ ≈ ❊q✉✐✈❛❧❡♥t t♦ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ + ❑ [ N ] ✳ ψ ≈ − ✵ . ✽✷ T ✵ − ✶ . ✼✸ T ✶ + ✵ . ✶✽ T ✷ + ✵ . ✵✾ T ✸ ✳ ✶✽✴✷✺
✵ ✻ ✶ ✶✾ ✵ ✻✷ ✵ ✶✼ ✵ ✵✺ ✵ ✵✶ ✵ ✶ ✷ ✸ ✹ ✺ ✸ ✸ ✺ ✻ ✻ ✷ ✶ ✶✵ ✸ ✷ ✶✵ ✺ ✽ ✶✵ ✼ ✻ ✶✵ ✶ ✷ ✶✵ ✻ ✼ ✽ ✾ ✶✵ ✼ ✽ ✾ ✶✵ ✶✶ ✶ ✹ ✶✵ ✶ ✾ ✶✵ ✷ ✵ ✶✵ ✷ ✻ ✶✵ ✷ ✺ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✸ ✵ ✶✵ ✷ ✻ ✶✵ ✸ ✵ ✶✵ ✷ ✺ ✶✵ ✷ ✻ ✶✵ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡ � � ❲❡ ✇❛♥t t♦ s♦❧✈❡ z ′′ ( t ) + ✸ ✹ − z ( t ) = c ✇✐t❤ z ( − ✶ ) = ✵✱ ✶ + ✵ . ✺ ❝♦s t z ′ ( − ✶ ) = ✶ ❛♥❞ c = ✶✳ � ■ + ❑ [ ◆ ] � · ϕ = ψ ✇❤❡r❡ ϕ = z ′′ ✳ ≈ ❊q✉✐✈❛❧❡♥t t♦ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ + ❑ [ N ] ✳ ψ ≈ − ✵ . ✽✷ T ✵ − ✶ . ✼✸ T ✶ + ✵ . ✶✽ T ✷ + ✵ . ✵✾ T ✸ ✳ ❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿ ✶✽✴✷✺
❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡ � � ❲❡ ✇❛♥t t♦ s♦❧✈❡ z ′′ ( t ) + ✸ ✹ − z ( t ) = c ✇✐t❤ z ( − ✶ ) = ✵✱ ✶ + ✵ . ✺ ❝♦s t z ′ ( − ✶ ) = ✶ ❛♥❞ c = ✶✳ � ■ + ❑ [ ◆ ] � · ϕ = ψ ✇❤❡r❡ ϕ = z ′′ ✳ ≈ ❊q✉✐✈❛❧❡♥t t♦ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ + ❑ [ N ] ✳ ψ ≈ − ✵ . ✽✷ T ✵ − ✶ . ✼✸ T ✶ + ✵ . ✶✽ T ✷ + ✵ . ✵✾ T ✸ ✳ ❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿ ϕ = − ✵ . ✻ T ✵ − ✶ . ✶✾ T ✶ + ✵ . ✻✷ T ✷ + ✵ . ✶✼ T ✸ − ✵ . ✵✺ T ✹ − ✵ . ✵✶ T ✺ � + ✷ . ✶ · ✶✵ − ✸ T ✻ + ✸ . ✷ · ✶✵ − ✸ T ✼ − ✺ . ✽ · ✶✵ − ✺ T ✽ − ✼ . ✻ · ✶✵ − ✻ T ✾ + ✶ . ✷ · ✶✵ − ✻ T ✶✵ + ✶ . ✹ · ✶✵ − ✼ T ✶✶ − ✶ . ✾ · ✶✵ − ✽ T ✶✷ − ✷ . ✵ · ✶✵ − ✾ T ✶✸ + ✷ . ✻ · ✶✵ − ✶✵ T ✶✹ + ✷ . ✺ · ✶✵ − ✶✶ T ✶✺ − ✸ . ✵ · ✶✵ − ✶✷ T ✶✻ − ✷ . ✻ · ✶✵ − ✶✸ T ✶✼ + ✸ . ✵ · ✶✵ − ✶✹ T ✶✽ + ✷ . ✺ · ✶✵ − ✶✺ T ✶✾ − ✷ . ✻ · ✶✵ − ✶✻ T ✷✵ ✶✽✴✷✺
❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥✿ ❑ ❚ ✶ ✐♥❥❡❝t✐✈❡ ❚ ❆ ❑ ❆ ■ ❑ ■❢ ❉ ❚ × ✶ ■ ❆ ■ ❑ ✶✱ ❚ ✐s ❝♦♥tr❛❝t✐✈❡ ❛♥❞ ✇❡ × ✶ ❣❡t ❛ t✐❣❤t ❡♥❝❧♦s✉r❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✿ ❚ ❚ × ✶ × ✶ × ✶ ✶ ✶ ●❡♥❡r❛❧ ■❞❡❛s ❢♦r ❱❛❧✐❞❛t✐♦♥ ♦❢ ▲✐♥❡❛r Pr♦❜❧❡♠s ❘❡❝❛❧❧✿ ❋♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ϕ ( ■ + ❑ ) · ϕ = ψ, ✇❡ ✇❛♥t t♦ ✈❛❧✐❞❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ � ϕ ✿ ϕ − ϕ ∗ � × ✶ . � � ✶✾✴✷✺
❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥✿ ❑ ❚ ✶ ✐♥❥❡❝t✐✈❡ ❚ ❆ ❑ ❆ ■ ❑ ■❢ ❉ ❚ × ✶ ■ ❆ ■ ❑ ✶✱ ❚ ✐s ❝♦♥tr❛❝t✐✈❡ ❛♥❞ ✇❡ × ✶ ❣❡t ❛ t✐❣❤t ❡♥❝❧♦s✉r❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✿ ❚ ❚ × ✶ × ✶ × ✶ ✶ ✶ ●❡♥❡r❛❧ ■❞❡❛s ❢♦r ❱❛❧✐❞❛t✐♦♥ ♦❢ ▲✐♥❡❛r Pr♦❜❧❡♠s ❘❡❝❛❧❧✿ ❋♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ϕ ( ■ + ❑ ) · ϕ = ψ, ✇❡ ✇❛♥t t♦ ✈❛❧✐❞❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ � ϕ ✿ ϕ − ϕ ∗ � × ✶ . � � ◆❛✐✈❡ ♠❡t❤♦❞✿ ϕ − ϕ ∗ � × ✶ ≤ � ( ■ + ❑ ) − ✶ � × ✶ � � � � ϕ + ❑ · � ϕ − ψ � × ✶ . ❇✉t ♣r♦❜❧❡♠s✿ ❍♦✇ t♦ ❝♦♠♣✉t❡ � ( ■ + ❑ ) − ✶ � × ✶ r✐❣♦r♦✉s❧②❄ ❈♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡ ✐ss✉❡s✳ ❇✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s ❞✉❡ t♦ ✐♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝s✳ ❚✐❣❤t♥❡ss ♦❢ t❤❡ ❜♦✉♥❞❄ ✶✾✴✷✺
❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥✿ ❑ ❚ ✶ ✐♥❥❡❝t✐✈❡ ❚ ❆ ❑ ❆ ■ ❑ ■❢ ❉ ❚ × ✶ ■ ❆ ■ ❑ ✶✱ ❚ ✐s ❝♦♥tr❛❝t✐✈❡ ❛♥❞ ✇❡ × ✶ ❣❡t ❛ t✐❣❤t ❡♥❝❧♦s✉r❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✿ ❚ ❚ × ✶ × ✶ × ✶ ✶ ✶ ●❡♥❡r❛❧ ■❞❡❛s ❢♦r ❱❛❧✐❞❛t✐♦♥ ♦❢ ▲✐♥❡❛r Pr♦❜❧❡♠s ❘❡❝❛❧❧✿ ❋♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ϕ ( ■ + ❑ ) · ϕ = ψ, ✇❡ ✇❛♥t t♦ ✈❛❧✐❞❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ � ϕ ✿ ϕ − ϕ ∗ � × ✶ . � � ◆❛✐✈❡ ♠❡t❤♦❞✿ ϕ − ϕ ∗ � × ✶ ≤ � ( ■ + ❑ ) − ✶ � × ✶ � � � � ϕ + ❑ · � ϕ − ψ � × ✶ . ❇✉t ♣r♦❜❧❡♠s✿ ❍♦✇ t♦ ❝♦♠♣✉t❡ � ( ■ + ❑ ) − ✶ � × ✶ r✐❣♦r♦✉s❧②❄ ❈♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡ ✐ss✉❡s✳ ❇✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s ❞✉❡ t♦ ✐♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝s✳ ❚✐❣❤t♥❡ss ♦❢ t❤❡ ❜♦✉♥❞❄ ✶✾✴✷✺
■❢ ❉ ❚ × ✶ ■ ❆ ■ ❑ ✶✱ ❚ ✐s ❝♦♥tr❛❝t✐✈❡ ❛♥❞ ✇❡ × ✶ ❣❡t ❛ t✐❣❤t ❡♥❝❧♦s✉r❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✿ ❚ ❚ × ✶ × ✶ × ✶ ✶ ✶ ●❡♥❡r❛❧ ■❞❡❛s ❢♦r ❱❛❧✐❞❛t✐♦♥ ♦❢ ▲✐♥❡❛r Pr♦❜❧❡♠s ❘❡❝❛❧❧✿ ❋♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ϕ ( ■ + ❑ ) · ϕ = ψ, ✇❡ ✇❛♥t t♦ ✈❛❧✐❞❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ � ϕ ✿ ϕ − ϕ ∗ � × ✶ . � � ❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ ⇔ ❚ · ϕ = ϕ, ❆ ≈ ( ■ + ❑ ) − ✶ ✐♥❥❡❝t✐✈❡ . ❚ · ϕ = ϕ − ❆ · ( ϕ + ❑ · ϕ − ψ ) , ✶✾✴✷✺
●❡♥❡r❛❧ ■❞❡❛s ❢♦r ❱❛❧✐❞❛t✐♦♥ ♦❢ ▲✐♥❡❛r Pr♦❜❧❡♠s ❘❡❝❛❧❧✿ ❋♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ϕ ( ■ + ❑ ) · ϕ = ψ, ✇❡ ✇❛♥t t♦ ✈❛❧✐❞❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ � ϕ ✿ ϕ − ϕ ∗ � × ✶ . � � ❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ ⇔ ❚ · ϕ = ϕ, ❆ ≈ ( ■ + ❑ ) − ✶ ✐♥❥❡❝t✐✈❡ . ❚ · ϕ = ϕ − ❆ · ( ϕ + ❑ · ϕ − ψ ) , ■❢ � ❉ ❚ � × ✶ = � ■ − ❆ ( ■ + ❑ ) � × ✶ = k < ✶✱ ❚ ✐s ❝♦♥tr❛❝t✐✈❡ ❛♥❞ ✇❡ ❣❡t ❛ t✐❣❤t ❡♥❝❧♦s✉r❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✿ � ❚ · � ϕ − � ϕ � × ✶ ϕ − ϕ ∗ � × ✶ ≤ � ❚ · � ϕ − � ϕ � × ✶ ≤ � � . ✶ + k ✶ − k ✶✾✴✷✺
❈♦♠♣✉t✐♥❣ t❤❡ ✭❞❡♥s❡✮ ✐♥✈❡rs❡✱ ✉s✐♥❣ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞✬s ✷ ❛❧❣♦r✐t❤♠✿ ✳ ❈♦♠♣✉t✐♥❣ ❛♥ ❛❧♠♦st✲❜❛♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡✿ ✳ ❚✇♦ ♣♦ss✐❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ❈♦♠♣✉t✐♥❣ ❛♥ ❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ▼❛tr✐① ❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡ ♠❛tr✐①✿ ❆ ≈ ( ■ + ❑ ) − ✶ . ✷✵✴✷✺
❈♦♠♣✉t✐♥❣ t❤❡ ✭❞❡♥s❡✮ ✐♥✈❡rs❡✱ ✉s✐♥❣ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞✬s ✷ ❛❧❣♦r✐t❤♠✿ ✳ ❈♦♠♣✉t✐♥❣ ❛♥ ❛❧♠♦st✲❜❛♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡✿ ✳ ❚✇♦ ♣♦ss✐❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ❈♦♠♣✉t✐♥❣ ❛♥ ❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ▼❛tr✐① ❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡ ♠❛tr✐①✿ ❆ ≈ ( ■ + ❑ [ N ] ) − ✶ . ✷✵✴✷✺
❈♦♠♣✉t✐♥❣ t❤❡ ✭❞❡♥s❡✮ ✐♥✈❡rs❡✱ ✉s✐♥❣ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞✬s ✷ ❛❧❣♦r✐t❤♠✿ ✳ ❈♦♠♣✉t✐♥❣ ❛♥ ❛❧♠♦st✲❜❛♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡✿ ✳ ❈♦♠♣✉t✐♥❣ ❛♥ ❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ▼❛tr✐① ❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡ ♠❛tr✐①✿ ❆ ≈ ( ■ + ❑ [ N ] ) − ✶ . ❚✇♦ ♣♦ss✐❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ✷✵✴✷✺
❈♦♠♣✉t✐♥❣ ❛♥ ❛❧♠♦st✲❜❛♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡✿ ✳ ❈♦♠♣✉t✐♥❣ ❛♥ ❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ▼❛tr✐① ❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡ ♠❛tr✐①✿ ❆ ≈ ( ■ + ❑ [ N ] ) − ✶ . ❚✇♦ ♣♦ss✐❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ❈♦♠♣✉t✐♥❣ t❤❡ ✭❞❡♥s❡✮ ✐♥✈❡rs❡✱ ✉s✐♥❣ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞✬s ❛❧❣♦r✐t❤♠✿ O ( n ✷ ( h + d )) ✳ � ✷✵✴✷✺
❈♦♠♣✉t✐♥❣ ❛♥ ❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ▼❛tr✐① ❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡ ♠❛tr✐①✿ ❆ ≈ ( ■ + ❑ [ N ] ) − ✶ . ❚✇♦ ♣♦ss✐❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ❈♦♠♣✉t✐♥❣ t❤❡ ✭❞❡♥s❡✮ ✐♥✈❡rs❡✱ ✉s✐♥❣ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞✬s ❛❧❣♦r✐t❤♠✿ O ( n ✷ ( h + d )) ✳ � ❈♦♠♣✉t✐♥❣ ❛♥ ( h ′ , d ′ ) ❛❧♠♦st✲❜❛♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡✿ O ( n ( h ′ + d ′ )( h + d )) ✳ ✷✵✴✷✺
❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ❢♦r ♦✉r ❊①❛♠♣❧❡ ✷✶✴✷✺
❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ❢♦r ♦✉r ❊①❛♠♣❧❡ ✷✶✴✷✺
❆❞❞✐t✐♦♥ ❛♥❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ tr✐✈✐❛❧❧② ❤❛♥❞❧❡❞✳ ❈♦♠♣✉t✐♥❣ × ✶ ✲♥♦r♠ ❂ ♠❛①✐♠✉♠ ♦❢ ✶✲♥♦r♠s ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❲✐t❤ ✲❛❧♠♦st✲❜❛♥❞❡❞ ✿ ✳ ❊①❛♠♣❧❡ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐s✿ ✸ ■ ❆ ■ ❑ ✶ ✺ ✶✵ × ✶ ❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✶✴✷✮ ❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦r ♥♦r♠✿ � ■ − ❆ ( ■ + ❑ ) � × ✶ ≤ � ■ − ❆ ( ■ + ❑ [ N ] ) � × ✶ + � ❆ ( ❑ − ❑ [ N ] ) � × ✶ . ✷✷✴✷✺
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