t ss rsss rt - PowerPoint PPT Presentation

◗✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss❡s ❛♥❞ ❜✐rt❤ ❛♥❞ ❞❡❛t❤ ♣r♦❝❡ss❡s ♦♥ ♣❛rt✐t✐♦♥s ▼❛r❝✐♥ ➅✇✐❡❝❛ Pr♦❜❛❜✐❧✐t② ❛♥❞ ❆♥❛❧②s✐s ✶✺✳✵✺✳✷✵✶✼ ✲ ✶✾✳✵✺✳✷✵✶✼ ❇➛❞❧❡✇♦ ✶ ✴ ✷✺

❇❛s❡❞ ♦♥ ✿ ✶ ❲✳▼❛t②s✐❛❦✱ ▼✳ ➅✱ ❩♦♥❛❧ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss ✱ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss✳ ❆♣♣❧✳✱ ✷✵✶✺✳ ✷ ❲✳▼❛t②s✐❛❦✱ ▼✳ ➅✱ ❏♦r❞❛♥ ❛❧❣❡❜r❛s ❛♥❞ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss❡s ✱ ■♥t✳ ▼❛t❤✳ ❘❡s✳ ◆♦t✳ ■▼❘◆✱ ✷✵✶✻✳ ✷ ✴ ✷✺

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss ✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ R d ✳ ■♥❣r❡❞✐❡♥ts✿ • t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = C n × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇ ) , z , z ′ ∈ C n , w , w ′ ∈ R , ( z , w )( z ′ , w ′ ) = ( z + z ′ , w + w ′ + ■♠ z ′ | z � � ✷ || z || ✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿ • t❤❡ ❢✉♥❝t✐♦♥ ψ ( z , w ) = iw − ✶ ∀ t ≥ ✵✱ H ∋ g �→ ❡①♣ [ t ψ ( g )] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Q t : L ✶ ( H ) → L ✶ ( H ) Q t f ( g ) = ❡①♣ [ t ψ ( g )] f ( g ) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈ ∗ ( H ) ✳ ✸ ✴ ✷✺

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss ✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ R d ✳ ■♥❣r❡❞✐❡♥ts✿ • t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = C n × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇ ) , z , z ′ ∈ C n , w , w ′ ∈ R , ( z , w )( z ′ , w ′ ) = ( z + z ′ , w + w ′ + ■♠ z ′ | z � � ✷ || z || ✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿ • t❤❡ ❢✉♥❝t✐♦♥ ψ ( z , w ) = iw − ✶ ∀ t ≥ ✵✱ H ∋ g �→ ❡①♣ [ t ψ ( g )] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Q t : L ✶ ( H ) → L ✶ ( H ) Q t f ( g ) = ❡①♣ [ t ψ ( g )] f ( g ) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈ ∗ ( H ) ✳ ✸ ✴ ✷✺

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss ✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ R d ✳ ■♥❣r❡❞✐❡♥ts✿ • t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = C n × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇ ) , z , z ′ ∈ C n , w , w ′ ∈ R , ( z , w )( z ′ , w ′ ) = ( z + z ′ , w + w ′ + ■♠ z ′ | z � � ✷ || z || ✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿ • t❤❡ ❢✉♥❝t✐♦♥ ψ ( z , w ) = iw − ✶ ∀ t ≥ ✵✱ H ∋ g �→ ❡①♣ [ t ψ ( g )] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Q t : L ✶ ( H ) → L ✶ ( H ) Q t f ( g ) = ❡①♣ [ t ψ ( g )] f ( g ) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈ ∗ ( H ) ✳ ✸ ✴ ✷✺

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss ✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ R d ✳ ■♥❣r❡❞✐❡♥ts✿ • t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = C n × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇ ) , z , z ′ ∈ C n , w , w ′ ∈ R , ( z , w )( z ′ , w ′ ) = ( z + z ′ , w + w ′ + ■♠ z ′ | z � � ✷ || z || ✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿ • t❤❡ ❢✉♥❝t✐♦♥ ψ ( z , w ) = iw − ✶ ∀ t ≥ ✵✱ H ∋ g �→ ❡①♣ [ t ψ ( g )] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Q t : L ✶ ( H ) → L ✶ ( H ) Q t f ( g ) = ❡①♣ [ t ψ ( g )] f ( g ) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈ ∗ ( H ) ✳ ✸ ✴ ✷✺

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss ✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ R d ✳ ■♥❣r❡❞✐❡♥ts✿ • t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = C n × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇ ) , z , z ′ ∈ C n , w , w ′ ∈ R , ( z , w )( z ′ , w ′ ) = ( z + z ′ , w + w ′ + ■♠ z ′ | z � � ✷ || z || ✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿ • t❤❡ ❢✉♥❝t✐♦♥ ψ ( z , w ) = iw − ✶ ∀ t ≥ ✵✱ H ∋ g �→ ❡①♣ [ t ψ ( g )] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Q t : L ✶ ( H ) → L ✶ ( H ) Q t f ( g ) = ❡①♣ [ t ψ ( g )] f ( g ) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈ ∗ ( H ) ✳ ✸ ✴ ✷✺

❇✐❛♥❡✬s q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss P❤✐❧✐♣♣❡ ❇✐❛♥❡ ✭✶✾✾✻✮ ✲ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❇❡ss❡❧ ♣r♦❝❡ss ✭ q✉❛♥t✉♠ ❇❡ss❡❧ ♣r♦❝❡ss ✮✿ ❛ ▼❛r❦♦✈ ♣r♦❝❡ss ✭✐♥ ❛ ❝❧❛ss✐❝❛❧✱ ♥♦t ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s❡♥s❡✮ ❧✐✈✐♥❣ ♦♥ ❛ s✉❜s❡t ♦❢ R d ✳ ■♥❣r❡❞✐❡♥ts✿ • t❤❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣✿ H = C n × R ✇✐t❤ t❤❡ ❣r♦✉♣ ❧❛✇ ) , z , z ′ ∈ C n , w , w ′ ∈ R , ( z , w )( z ′ , w ′ ) = ( z + z ′ , w + w ′ + ■♠ z ′ | z � � ✷ || z || ✷ ♦♥ H ✇✐t❤ t❤❡ ♣r♦♣❡rt②✿ • t❤❡ ❢✉♥❝t✐♦♥ ψ ( z , w ) = iw − ✶ ∀ t ≥ ✵✱ H ∋ g �→ ❡①♣ [ t ψ ( g )] ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t Q t : L ✶ ( H ) → L ✶ ( H ) Q t f ( g ) = ❡①♣ [ t ψ ( g )] f ( g ) ❡①t❡♥❞s t♦ ❛ s❡♠✐❣r♦✉♣ ♦❢ ✭❝♦♠♣❧❡t❡❧②✮ ♣♦s✐t✐✈❡ ❝♦♥tr❛❝t✐♦♥s ♦♥ ❈ ∗ ( H ) ✳ ✸ ✴ ✷✺