rr r qts - PowerPoint PPT Presentation

❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s s♦❧✈❛❜❧❡ ❜② ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ❇✐❛❧②st♦❦ ❆✳ ❉♦❜r♦❣♦✇s❦❛✱ ●✳ ❏❛❦✐♠♦✇✐❝③✱ ❋❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ ❛♣♣❧✐❡❞ t♦ t❤❡ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✱ ❆♣♣❧✳ ▼❛t❤✳ ▲❡tt✳✱ ✼✹✱ ✶✻✶✕✶✻✻✱ ✷✵✶✼✳ ❆✳ ❉♦❜r♦❣♦✇s❦❛✱ ▼✳ ◆✳ ❍♦✉♥❦♦♥♥♦✉✱ ❋❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ ❛♥❞ ❣❡♥❡r❛❧ s❡❝♦♥❞ ♦r❞❡r ❧✐♥❡❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✳ ✱ ❙♣r✐♥❣❡r Pr♦❝❡❡❞✐♥❣s ✐♥ ▼❛t❤❡♠❛t✐❝s ✫ ❙t❛t✐st✐❝s✱ ✷✸✵✱ ✷✵✶✽✳ ❆✳ ❉♦❜r♦❣♦✇s❦❛✱ ◆❡✇ ❝❧❛ss❡s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s s♦❧✈❛❜❧❡ ❜② ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ ✱ ❆♣♣❧✳ ▼❛t❤✳ ▲❡tt✳✱ ✾✽✱ ✸✵✵✕✸✵✺✱ ✷✵✶✾✳ ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❚❤❡ ♠❛✐♥ ♣✉r♣♦s❡ ♦❢ t❤✐s ♣r❡s❡♥t❛t✐♦♥ ✐s t♦ ❛♣♣❧② t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ t♦ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✳ ❚❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ ♦✛❡rs t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ✜♥❞✐♥❣ s♦❧✉t✐♦♥s ♦❢ ♥❡✇ ❝❧❛ss❡s ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✳ ❚❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ✐s ❛ ✇❡❧❧✲❦♥♦✇♥ ♠❡t❤♦❞ ♦❢ s♦❧✈✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❤❛✈✐♥❣ r♦♦ts ✐♥ t❤❡ ✇♦r❦s ♦❢ ●✳ ❉❛r❜♦✉①✱ ❊✳ ❙❝❤rö❞✐♥❣❡r✱ P✳❆✳▼✳ ❉✐r❛❝✱ ▲✳ ■♥❢❡❧❞ ❛♥❞ ❚✳❊✳ ❍✉❧❧✱ ❲✳❏r✳ ▼✐❧❧❡r✱ ❇✳ ▼✐❡❧♥✐❦✳ ▲❛t❡r t❤✐s ♠❡t❤♦❞ ✭❛♥❞ s♦♠❡ ✐ts ♠♦❞✐✜❝❛t✐♦♥✮ ✇❛s ❛♣♣❧✐❡❞ t♦ st✉❞② ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠s ✐♥ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s ❢♦r ✇❡❧❧✲❦♥♦✇♥ s♦❧✈❛❜❧❡ ♣♦t❡♥t✐❛❧s ❧✐❦❡✿ ❤❛r♠♦♥✐❝ ♦s❝✐❧❧❛t♦r✱ ✐s♦tr♦♣✐❝ ♦s❝✐❧❧❛t♦r✱ ▼♦rs❡✱ ❘♦s❡♥✲▼♦rs❡✱ ❊❝❦❛rt✱ Pös❝❤❧✲❚❡❧❧❡r✱ ❡t❝✳ ❆❞❞✐t✐♦♥❛❧❧②✱ t❤❡ ♠♦❞✐✜❡❞ ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ ✇❛s ❛♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ ♦t❤❡r t♦♣✐❝s ❛s✿ s✉♣❡rs②♠♠❡tr✐❝ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s✱ s❤❛♣❡✕✐♥✈❛r✐❛♥t ♣♦t❡♥t✐❛❧s✱ ✐♥✈❡rs❡ s❝❛tt❡r✐♥❣ ♠❡t❤♦❞✱ ❝♦❤❡r❡♥t st❛t❡s✱ ❡t❝✳ ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦r s❤✐❢t ♦♣❡r❛t♦rs ■♥ ♦✉r r❡s❡❛r❝❤ ✇❡ ❤❛✈❡ r❡str✐❝t❡❞ ♦✉r ❛tt❡♥t✐♦♥ t♦ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❞✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤✐s ♣r♦❜❧❡♠✳ ❲❡ ❞❡♥♦t❡ ❜② ℓ k ( Z , R ) ❛♥❞ ℓ k ( Z , C ) ✱ k ∈ N ∪ { 0 } ✱ t❤❡ s❡ts ♦❢ r❡❛❧✲✈❛❧✉❡❞ ❛♥❞ ❝♦♠♣❧❡①✲✈❛❧✉❡❞ s❡q✉❡♥❝❡s { x ( n ) } n ∈ Z ✱ r❡s♣❡❝t✐✈❡❧②✳ ❲❡ ❞❡✜♥❡ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♦♥ ℓ k ( Z , C ) ❛s ❢♦❧❧♦✇s✿ � x | y � k := � b n = a x ( n ) y ( n ) ρ k ( n ) , ✇❤❡r❡ a, b ∈ Z , ( a < b ) , ❛♥❞ ρ k ✐s ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥✳ ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

■♥ ♦✉r r❡s❡❛r❝❤ ✇❡ ❤❛✈❡ r❡str✐❝t❡❞ ♦✉r ❛tt❡♥t✐♦♥ t♦ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❞✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤✐s ♣r♦❜❧❡♠✳ ❲❡ ❞❡♥♦t❡ ❜② ℓ k ( Z , R ) ❛♥❞ ℓ k ( Z , C ) ✱ k ∈ N ∪ { 0 } ✱ t❤❡ s❡ts ♦❢ r❡❛❧✲✈❛❧✉❡❞ ❛♥❞ ❝♦♠♣❧❡①✲✈❛❧✉❡❞ s❡q✉❡♥❝❡s { x ( n ) } n ∈ Z ✱ r❡s♣❡❝t✐✈❡❧②✳ ❲❡ ❞❡✜♥❡ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♦♥ ℓ k ( Z , C ) ❛s ❢♦❧❧♦✇s✿ � x | y � k := � b n = a x ( n ) y ( n ) ρ k ( n ) , ✇❤❡r❡ a, b ∈ Z , ( a < b ) , ❛♥❞ ρ k ✐s ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥✳ ∆ x ( n ) := ( T + − 1 ) x ( n ) = x ( n + 1) − x ( n ) ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦r T ± x ( n ) := x ( n ± 1) s❤✐❢t ♦♣❡r❛t♦rs ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✇❡✐❣❤t s❡q✉❡♥❝❡ s❛t✐s✜❡s t❤❡ P❡❛rs♦♥ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❛♥❞ t❤❡ r❡❝✉rs✐♦♥ r❡❧❛t✐♦♥ ✇❤❡r❡ ❛♥❞ ❛r❡ s♦♠❡ r❡❛❧✲✈❛❧✉❡❞ s❡q✉❡♥❝❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ❢✉❧✜❧❧s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s � x | y � k = � b n = a x ( n ) y ( n ) ρ k ( n ) , ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❛♥❞ t❤❡ r❡❝✉rs✐♦♥ r❡❧❛t✐♦♥ ✇❤❡r❡ ❛♥❞ ❛r❡ s♦♠❡ r❡❛❧✲✈❛❧✉❡❞ s❡q✉❡♥❝❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ❢✉❧✜❧❧s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s � x | y � k = � b n = a x ( n ) y ( n ) ρ k ( n ) , ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✇❡✐❣❤t s❡q✉❡♥❝❡ s❛t✐s✜❡s t❤❡ P❡❛rs♦♥ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ∆ ( b k ( n ) ρ k ( n )) = ( c k ( n ) − b k ( n )) ρ k ( n ) , ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ❢✉❧✜❧❧s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s � x | y � k = � b n = a x ( n ) y ( n ) ρ k ( n ) , ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✇❡✐❣❤t s❡q✉❡♥❝❡ s❛t✐s✜❡s t❤❡ P❡❛rs♦♥ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ∆ ( b k ( n ) ρ k ( n )) = ( c k ( n ) − b k ( n )) ρ k ( n ) , ❛♥❞ t❤❡ r❡❝✉rs✐♦♥ r❡❧❛t✐♦♥ ρ k − 1 ( n ) = c k ( n ) ρ k ( n ) , ✇❤❡r❡ { b k } ❛♥❞ { c k } ❛r❡ s♦♠❡ r❡❛❧✲✈❛❧✉❡❞ s❡q✉❡♥❝❡s✳ ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

� x | y � k = � b n = a x ( n ) y ( n ) ρ k ( n ) , ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✇❡✐❣❤t s❡q✉❡♥❝❡ s❛t✐s✜❡s t❤❡ P❡❛rs♦♥ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ∆ ( b k ( n ) ρ k ( n )) = ( c k ( n ) − b k ( n )) ρ k ( n ) , ❛♥❞ t❤❡ r❡❝✉rs✐♦♥ r❡❧❛t✐♦♥ ρ k − 1 ( n ) = c k ( n ) ρ k ( n ) , ✇❤❡r❡ { b k } ❛♥❞ { c k } ❛r❡ s♦♠❡ r❡❛❧✲✈❛❧✉❡❞ s❡q✉❡♥❝❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ρ k ❢✉❧✜❧❧s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s b k ( a ) ρ k ( a ) = b k ( b + 1) ρ k ( b + 1) = 0 . ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❲❡ ❢❛❝t♦r✐③❡ t❤❡ ♦♣❡r❛t♦rs ❛s ❛ ♣r♦❞✉❝t ♦❢ t✇♦ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦rs ✐♥ t❤❡ ❢♦r♠ ❍ ✇❤❡r❡ ✳ ■♥ t❤✐s ✇❛②✱ ✇❡ ♦❜t❛✐♥ ❛ ❝❤❛✐♥ ♦❢ ❧❛❞❞❡r ♦♣❡r❛t♦rs ✭ ✕ ❧♦✇❡r✐♥❣ ♦♣❡r❛t♦rs ❛♥❞ ✕ r❛✐s✐♥❣ ♦♣❡r❛t♦rs✮ ❲❡ ❛♣♣❧② t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ t♦ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦rs H k : ℓ k ( Z , C ) − → ℓ k ( Z , C ) ❣✐✈❡♥ ❜② ❍ k = z k ( n ) T + + w k ( n ) T − + v k ( n ) , ✇❤❡r❡ { z k } , { w k } , { v k } ∈ ℓ ( Z , R ) ✱ k ∈ Z ✳ ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳