t t sts - PowerPoint PPT Presentation

❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ▼✐❦❡ P❊❘❊■❘❆ ✶ , ✷ ✱ ◆✐❝♦❧❛s ❉❊❙❆❙❙■❙ ✶ ✶ ●❡♦st❛t✐st✐❝s t❡❛♠✱ ▼■◆❊❙ P❛r✐s❚❡❝❤✱ P❙▲ ❘❡s❡❛r❝❤ ❯♥✐✈❡rs✐t② ✷ ❊❙❚■▼❆●❊❙ ✴ ❙❤♦❲❤❡r❡ ❙P❉❊✲■◆▲❆ ❲♦r❦s❤♦♣ ❘❊❙❙❚❊ ◆♦✈❡♠❜❡r ✽t❤✱ ✷✵✶✽

✐t ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ s♦❧✉t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ✭❘♦③❛♥♦✈✱ ✶✾✼✼✱ ❙✐♠♣s♦♥ ❡t ❛❧✳✱ ✷✵✶✷✮✿ ✶ ✷ ✶ ✷ ✐s t❤❡ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✇❤❡r❡ ✐s ❛ ●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡✱ ❛♥❞ ❞❡✜♥❡❞ ❛s ✿ ✶ ✷ ✶ ✷ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦♣❡r❛t♦r✳ ❈♦♥t✐♥✉♦✉s ▼❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞s P ♣♦❧②♥♦♠✐❛❧ t❛❦✐♥❣ str✐❝t❧② ♣♦s✐t✐✈❡ ✈❛❧✉❡ ♦♥ R + ✳ ■❢ Z ✐s ❛ ❝♦♥t✐♥✉♦✉s ▼❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞ ✜❡❧❞✱ ✐ts s♣❡❝tr❛❧ ❞❡♥s✐t② ✭❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥✮ ♦❢ t❤❡ ❢♦r♠ ✭❘♦③❛♥♦✈✱ ✶✾✼✼✮✿ g ( ω ) = ✶ / P ( � ω � ✷ ) P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✷ ✴ ✶✽

❈♦♥t✐♥✉♦✉s ▼❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞s P ♣♦❧②♥♦♠✐❛❧ t❛❦✐♥❣ str✐❝t❧② ♣♦s✐t✐✈❡ ✈❛❧✉❡ ♦♥ R + ✳ ■❢ Z ✐s ❛ ❝♦♥t✐♥✉♦✉s ▼❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞ ✜❡❧❞✱ ✐ts s♣❡❝tr❛❧ ❞❡♥s✐t② ✭❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥✮ ♦❢ t❤❡ ❢♦r♠ ✭❘♦③❛♥♦✈✱ ✶✾✼✼✮✿ g ( ω ) = ✶ / P ( � ω � ✷ ) ✐t ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ s♦❧✉t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ✭❘♦③❛♥♦✈✱ ✶✾✼✼✱ ❙✐♠♣s♦♥ ❡t ❛❧✳✱ ✷✵✶✷✮✿ P ( − ∆) ✶ / ✷ Z = W ✇❤❡r❡ W ✐s ❛ ●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡✱ ❛♥❞ P ( − ∆) ✶ / ✷ ✐s t❤❡ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ❞❡✜♥❡❞ ❛s ✿ P ( − ∆) ✶ / ✷ [ . ] = F − ✶ � � � P ( � ω � ✷ ) F [ . ]( ω ) ω �→ ✇❤❡r❡ F ❞❡♥♦t❡s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦♣❡r❛t♦r✳ P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✷ ✴ ✶✽

Pr♦♣♦s✐t✐♦♥ ✿ ▼❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞s ✭▲✐♥❞❣r❡♥ ❡t ❛❧✳✱ ✷✵✶✶✮ ❚❤❡ ♣r❡❝✐s✐♦♥ ♠❛tr✐① ♦❢ t❤❡ ✇❡✐❣❤ts ♦❢ t❤❡ ✜♥✐t❡ ❡❧❡♠❡♥t ✭❋❊✮ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ st❛t✐♦♥❛r② s♦❧✉t✐♦♥s ♦❢ ✭✶✮ ✐s✿ ❈ ✶ ✷ ❙ ❈ ✶ ✷ ◗ ✇❤❡r❡✿ ✶ ✷ ●❈ ✶ ✷ ❈ ❉✐❛❣ ✶ ● ❙ ❈ ❋♦r s♣❡❝tr❛❧ ❞❡♥s✐t✐❡s ♦❢ t❤❡ ❢♦r♠✿ ✶ ✷ ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ▼❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞s ❙✐♠✉❧❛t❡ ▼❛r❦♦✈✐❛♥ ✜❡❧❞s ❜② s♦❧✈✐♥❣ ♥✉♠❡r✐❝❛❧❧② t❤❡ ❙P❉❊✿ P ( − ∆) ✶ / ✷ Z = W ✭✶✮ ✉s✐♥❣ ✜♥✐t❡ ❡❧❡♠❡♥t ♠❡t❤♦❞ ✿ � Z ( s ) = z i ψ i ( s ) , s ∈ D ✇❤❡r❡ { ψ i } ❛r❡ ❜❛s✐s ❢✉♥❝t✐♦♥s ♦♥ ❛ tr✐❛♥❣✉❧❛t❡❞ ❞♦♠❛✐♥ D ✭❜♦✉♥❞❡❞ ♣♦❧②❣♦♥❛❧ ♦r ♠❛♥✐❢♦❧❞✮✱ ❛♥❞ { z i } ❛r❡ ●❛✉ss✐❛♥ ✇❡✐❣❤ts✳ P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✸ ✴ ✶✽

❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ▼❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞s ❙✐♠✉❧❛t❡ ▼❛r❦♦✈✐❛♥ ✜❡❧❞s ❜② s♦❧✈✐♥❣ ♥✉♠❡r✐❝❛❧❧② t❤❡ ❙P❉❊✿ P ( − ∆) ✶ / ✷ Z = W ✭✶✮ ✉s✐♥❣ ✜♥✐t❡ ❡❧❡♠❡♥t ♠❡t❤♦❞ ✿ � Z ( s ) = z i ψ i ( s ) , s ∈ D ✇❤❡r❡ { ψ i } ❛r❡ ❜❛s✐s ❢✉♥❝t✐♦♥s ♦♥ ❛ tr✐❛♥❣✉❧❛t❡❞ ❞♦♠❛✐♥ D ✭❜♦✉♥❞❡❞ ♣♦❧②❣♦♥❛❧ ♦r ♠❛♥✐❢♦❧❞✮✱ ❛♥❞ { z i } ❛r❡ ●❛✉ss✐❛♥ ✇❡✐❣❤ts✳ Pr♦♣♦s✐t✐♦♥ ✿ ▼❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞s ✭▲✐♥❞❣r❡♥ ❡t ❛❧✳✱ ✷✵✶✶✮ ❚❤❡ ♣r❡❝✐s✐♦♥ ♠❛tr✐① ♦❢ t❤❡ ✇❡✐❣❤ts { z i } ♦❢ t❤❡ ✜♥✐t❡ ❡❧❡♠❡♥t ✭❋❊✮ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ st❛t✐♦♥❛r② s♦❧✉t✐♦♥s ♦❢ ✭✶✮ ✐s✿ ◗ z = ❈ ✶ / ✷ P ( ❙ ) ❈ ✶ / ✷ ✇❤❡r❡✿ ❙ = ❈ − ✶ / ✷ ●❈ − ✶ / ✷ ❈ = ❉✐❛❣ ( � ψ i , ✶ � ) , ● = [ �∇ ψ i , ∇ ψ j � ] , ⇒ ❋♦r s♣❡❝tr❛❧ ❞❡♥s✐t✐❡s ♦❢ t❤❡ ❢♦r♠✿ ✶ g ( � ω � ) = P ( � ω � ✷ ) P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✸ ✴ ✶✽

❇② r❡♣r❡s❡♥t✐♥❣ ✉s✐♥❣ ❛ ✜♥✐t❡ ❡❧❡♠❡♥t ❛♣♣r♦❛❝❤✱ ✇❡ s❤♦✇❡❞✿ Pr♦♣♦s✐t✐♦♥ ✿ ❈♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ❋❊ ✇❡✐❣❤ts ✭P❡r❡✐r❛ ❛♥❞ ❉❡s❛ss✐s✱ ✷✵✶✽❜✮ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ✇❡✐❣❤ts ♦❢ t❤❡ ❋❊ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✭✷✮ ✐s✿ ✶ ✷ ✶ ✷ ❈ ❙ ❈ ✶ ✷ ●❈ ✶ ✷ ✱ ✇❤❡r❡✿ ❈ ❉✐❛❣ ✶ ✱ ● ✱ ❙ ❈ ✶ ✶ ✳✳✳ ✳✳✳ ❙ ❱ ❱ ❙ ❱ ❱ ❊①t❡♥s✐♦♥ t♦ ❣❡♥❡r❛❧✐③❡❞ r❛♥❞♦♠ ✜❡❧❞s Pr♦♣♦s✐t✐♦♥ ✿ ●❡♥❡r❛❧✐③❡❞ r❛♥❞♦♠ ✜❡❧❞ ✭▲❛♥❣ ❛♥❞ P♦tt❤♦✛✱ ✷✵✶✶✮ ❆ s❡❝♦♥❞✲♦r❞❡r st❛t✐♦♥❛r②✱ ✐s♦tr♦♣✐❝ ●❛✉ss✐❛♥ r❛♥❞♦♠ ✜❡❧❞ Z ✇✐t❤ s♣❡❝tr❛❧ ❞❡♥s✐t② g : R + �→ R + ♦♥ D ⊂ R d ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s✿ Z = L √ g W ✭✷✮ ✇❤❡r❡ L √ g [ . ] := F − ✶ � � � ω �→ g ( � ω � ✷ ) F [ . ]( ω ) ❛♥❞ W ✐s ❛ ●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡ ♦♥ D ✳ P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✹ ✴ ✶✽