r r sts t - PowerPoint PPT Presentation

●❛❜♦r ❋r❛♠❡ ❉❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ❊✈♦❧✉t✐♦♥ ❖♣❡r❛t♦rs ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ▼✐❝❤❡❧❡ ❇❡rr❛ ❯♥✐✈❡rs✐tà ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❚♦r✐♥♦ ❉✐♣❛rt✐♠❡♥t♦ ❞✐ ▼❛t❡♠❛t✐❝❛ ✏●✐✉s❡♣♣❡ P❡❛♥♦✑ ❳❳❳■■■ ❈♦♥✈❡❣♥♦ ◆❛③✐♦♥❛❧❡ ❞✐ ❆♥❛❧✐s✐ ❆r♠♦♥✐❝❛ ❆❧❜❛✱ ✶✼✲✷✵ ●✐✉❣♥♦ ✷✵✶✸

❖✉t❧✐♥❡ • ●❡❧❢❛♥❞✲❙❤✐❧♦✈ ❙♣❛❝❡s • ●❛❜♦r ❋r❛♠❡s • ●❛❜♦r ❋r❛♠❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs • ❍❡❛t ❊q✉❛t✐♦♥ • ●❡♥❡r❛❧✐③❡❞ ❍❡❛t ❊q✉❛t✐♦♥ • ❍❛r♠♦♥✐❝ ❘❡♣✉❧s♦r ✶ ✴ ✶✽

●❡❧❢❛♥❞✲❙❤✐❧♦✇ ❙♣❛❝❡s s, r ≥ 0 ✳ f ∈ S s r ( R d ) ✐❢ f ∈ S ( R d ) ❛♥❞ t❤❡r❡ ❡①✐st A, B > 0 s✉❝❤ t❤❛t x ∈ R d | x α ∂ β f ( x ) | � A | α | B | β | ( α !) r ( β !) s , α, β, ∈ Z d sup + . ❊q✉✐✈❛❧❡♥t❧② f ∈ S s r ( R d ) ✐✛ t❤❡r❡ ❡①✐st h, k > 0 s✉❝❤ t❤❛t � fe h | x | 1 /r � ∞ < ∞ fe k | ω | 1 /s � ∞ < ∞ . � ˆ ✭✶✮ ❛♥❞ r ( R d ) ⇔ ˆ ❚❤❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥t❡r❝❤❛♥❣❡s t❤❡ ✐♥❞❡①❡s✿ f ∈ S s f ∈ S r s ( R d ) . ■❢ 0 ≤ s 1 ≤ s 2 ❛♥❞ 0 ≤ r 1 ≤ r 2 t❤❡♥ S s 1 r 1 ( R d ) ⊆ S s 2 r 2 ( R d ) . S 1 / 2 1 / 2 ( R d ) ✐s t❤❡ s♠❛❧❧❡st ♥♦♥ tr✐✈✐❛❧ ●❡❧❢❛♥❞✲❙❤✐❧♦✇ s♣❛❝❡✳ ✷ ✴ ✶✽

●❛❜♦r ❋r❛♠❡s g ∈ S ( R d ) \{ 0 } ✱ Λ := α Z d × β Z d , α, β > 0 ✳ � � G ( g, α, β ) := g m,n = M n T m g, ( m, n ) ∈ Λ , ✇❤❡r❡ M n g ( x ) = e 2 πinx g ( x ) ❛♥❞ T m g ( x ) = g ( x − m ) ✳ G ( g, α, β ) ✐s ❛ ❢r❛♠❡ ❢♦r L 2 ( R d ) ✐✛ t❤❡r❡ ❡①✐st 0 < A ≤ B < + ∞ ✿ � |� f, g m,n �| 2 ≤ B � f � 2 A � f � 2 ∀ f ∈ L 2 ( R d ) . 2 ≤ 2 , ( m,n ) ∈ Λ ✸ ✴ ✶✽

■❢ G ( g, α, β ) ✐s ❛ ❢r❛♠❡✱ t❤❡♥ � � ∀ f ∈ L 2 ( R d ) , f = � f, g m,n � γ m,n = � f, γ m,n � g m,n , ( m,n ) ∈ Λ ( m,n ) ∈ Λ G ( γ, α, β ) ❞✉❛❧ ❢r❛♠❡✱ γ ❞✉❛❧ ✇✐♥❞♦✇✳ ❚❤❡♦r❡♠ ✭ ❙❡✐♣✲❲❛❧❧sté♥ ✶✾✾✷✱●rö❝❤❡♥✐❣✱ ▲②✉❜❛rs❦✐✐ ✷✵✵✽✮ • ▲❡t g = e − π | x | 2 , x ∈ R d ✳ ❚❤❡♥ G ( g, α, β ) ✐s ❛ ❢r❛♠❡ ✐✛ αβ < 1 ✳ • ▲❡t G ( g, α, β ) ❜❡ ❛s ❛❜♦✈❡✳ ❚❤❡♥ t❤❡ ❞✉❛❧ ✇✐♥❞♦✇ γ s❛t✐s❢②✿ γ ( x ) | ≤ Ce − π | x | 2 , ∀ x ∈ R d . | γ ( x ) | + | � ◆♦t✐❝❡ t❤❛t γ ∈ S 1 / 2 1 / 2 ✳ ❲❡ ✉s❡ ●❛✉ss✐❛♥ ●❛❜♦r ❢r❛♠❡s ✐♥ ♦✉r ♥✉♠❡r✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s ✇✐t❤ ❧❛tt✐❝❡ ♣❛r❛♠❡t❡rs α = 1 , β = 1 2 . ✹ ✴ ✶✽

●❛❜♦r ❋r❛♠❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs T ∈ L ( L 2 ( R d )) ✱ G ( g, α, β ) ●❛❜♦r ❢r❛♠❡ ✇✐t❤ ❞✉❛❧ ✇✐♥❞♦✇ γ ❛♥❞ f ∈ L 2 ( R d ) ✳   � �   = • Tf = T � f, γ m,n � g m,n � f, γ m,n � Tg m,n ( m,n ) ∈ Λ ( m,n ) ∈ Λ � • Tf = � Tf, g m ′ ,n ′ � γ m ′ ,n ′ ( m ′ ,n ′ ) ∈ Λ �� Tf = � Tg m,n , g m ′ ,n ′ � � f, γ m,n � γ m ′ ,n ′ , � �� � � �� � ( m,n ) , ( m ′ ,n ′ ) ∈ Λ T m,n,m ′ ,n ′ c ( f ) m,n T m,n,m ′ ,n ′ ✐s t❤❡ ●❛❜♦r ▼❛tr✐① ✺ ✴ ✶✽

❯s❡✿ • Tf = σ ( D ) f = F − 1 � � σ ˆ ❋♦✉r✐❡r ▼✉❧t✐♣❧✐❡r✳ f • Tf = µ ( A ) f ▼❡t❛♣❧❡❝t✐❝ ❖♣❡r❛t♦r✳ ❚❤❡♦r❡♠ ✭ ❈♦r❞❡r♦✱ ◆✐❝♦❧❛✱ ❘♦❞✐♥♦✱ ✷✵✶✷✮ ▲❡t s ≥ 1 2 , g ∈ S s s ( R d ) ❛♥❞ σ ∈ C ∞ ( R d ) ✳❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ • ❚❤❡ s②♠❜♦❧ σ s❛t✐s❢②❡s | ∂ α σ ( z ) | � C | α | ( α !) s , ∀ z ∈ R 2 d , ∀ α ∈ Z 2 d ✭✷✮ + • ❚❤❡r❡ ❡①✐sts ε > 0 s✉❝❤ t❤❛t 1 s , ∀ λ, µ ∈ α Z d × β Z d |� σπ ( λ ) g, π ( µ ) g ) �| � e − ε | λ − µ | ✭✸✮ ✇❤❡r❡ π ( λ ) g = M n T n g ✱ ✇✐t❤ λ = ( m, n ) ❛♥❞ µ = ( m ′ , n ′ ) ✳ ✻ ✴ ✶✽

❍❡❛t ❊q✉❛t✐♦♥ ❋♦r α > 0 t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠   ∂ t u − α ∆ u = 0  t ∈ R , x ∈ R d , u (0 , x ) = u 0 ( x ) , ❤❛s s♦❧✉t✐♦♥ � R d e 2 πix · ω σ α ( t, ω ) � u ( t, x ) = σ α ( t, D ) u 0 = u 0 ( ω )d ω, ✭✹✮ ✇✐t❤ σ α ( t, ω ) = e − 4 π 2 αt | ω | 2 ✳ ❚❤❡♦r❡♠ ▲❡t G ( g, 1 , 1 2 ) ●❛✉ss✐❛♥ ●❛❜♦r ❢r❛♠❡✱ t❤❡ ●❛❜♦r ▼❛tr✐① ❛ss♦❝✐❛t❡❞ t♦ ✭✹✮ ✐s |� σ α ( t, D ) g m,n , g m ′ ,n ′ �| = (2 + 4 παt ) − d 2 e − π [ n 2 + n ′ 2 + 2+4 παt ( ( m − m ′ ) 2 +( n + n ′ ) 2 )] 1 ✭✺✮ ( m, n ) ∈ Z d × 1 2 Z d . ✼ ✴ ✶✽

❈♦❡✣❝✐❡♥ts✬ ❉❡❝❛②✲ ❍❡❛t ❊q✉❛t✐♦♥ � � ❈♦❡✣❝✐❡♥ts✬ ❉❡❝❛② ✲ ❍❡❛t ❊q✉❛t✐♦♥ ❢♦r � � σ ( t, D ) g 0 , 0 , g m ′ ,n ′ � � ✱ ❞✐♠❡♥s✐♦♥ d = 2 ✳ ✽ ✴ ✶✽

❈♦♥t♦✉r ♣❧♦t � � ❈♦♥t♦✉r ♣❧♦t ♦❢ � � σ ( t, D ) g 0 , 0 , g m ′ ,n ′ � � ✳ ✾ ✴ ✶✽

●❡♥❡r❛❧✐③❡❞ ❍❡❛t ❊q✉❛t✐♦♥ ❋♦r k ∈ Z + ✱ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠  ∂ t u ( − ∆) k u = 0 ,   u (0 , x ) = u 0 ( x ) , t ∈ R , x ∈ R , ❤❛s s♦❧✉t✐♦♥ � R d e 2 πix · ω σ k ( t, ω ) � u ( t, x ) = σ k ( t, D ) u 0 = u 0 ( ω )d ω, ✭✻✮ ✇✐t❤ σ k ( t, ω ) = e − t (2 πω ) 2 k . ❚❤❡♦r❡♠ ▲❡t G ( g, α, β ) ❜❡ ❛ ●❛✉ss✐❛♥✳ ❚❤❡♥ ε t,k 2 − 1 1 s , s | ( m,n ) − ( m ′ ,n ′ ) | |� σ ( t, D ) g m,n , g m ′ ,n ′ �| ≤ C t,k e − ˜ � 2 k − 1 � � � 1 k − 1 k 2 k − 1 2 − 2 k 1 2 k − 1 ❛♥❞ C t,k = | 4 πkt | 2 k − 1 ✳ ✇✐t❤ s = 2 k − 1 ✱ ˜ ε k,t = 4 k 4 πkt ❈♦♥s✐st❡♥t ✇✐t❤ ✭✸✮✳ ✶✵ ✴ ✶✽

❖❜s❡r✈❡ |� σ k ( t, D ) g m,n , g m ′ ,n ′ �| ≤ e − π ( n 2 − n ′ 2 ) � � e − t (2 πω ) 2 k � � �� � � e − 2 πω 2 � � �� � � � � � F � ∗ � F � , θ m,n,m ′ ,n ′ θ m,n,m ′ ,n ′ ✇❤❡r❡ θ m,n,m ′ ,n ′ := ( m − m ′ ) + i ( n + n ′ ) ✳ � � e − t (2 πω ) 2 k �� � � ❲❡ ❤❛✈❡ t♦ ✜♥❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦r � F � . ■t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❝♦♠♣✉t❡ t❤❡ ●❛❜♦r ▼❛tr✐① ❡①♣❧✐❝✐t❧② s✐♥❝❡ t❤❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ♦❢ σ ( t, ω ) = e − t (2 πω ) 2 k ❝❛♥♥♦t ❜❡ ❝❛❧❝✉❧❛t❡❞ ❞✐r❡❝t❧②✳ ✶✶ ✴ ✶✽

❆s②♠♣t♦t✐❝ ■♥t❡❣r❛t✐♦♥ ❯s✐♥❣ t❤❡ ❆s②♠♣t♦t✐❝ ✐♥t❡❣r❛t✐♦♥✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✜♥❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞✦ ❚❤❡♦r❡♠ ▲❡t f ( x ) = e − αx 2 k ✱ ✇✐t❤ α > 0 ❛♥❞ k ≥ 1 ✳ ❚❤❡♥ ˆ f ( ω ) s❛t✐s✜❡s✿ 2 k 2 k − 1 , | ˆ f ( ω ) | ≤ C k,α e − ε k,α ω k − 1 2 k − 1 � 2 k − 1 � � � 1 2 k 2 k − 1 2 k − 1 ✳ | 2 kα | 1 ✇❤❡r❡ C k,α = ❛♥❞ ε k,α = (2 π ) 2 k ( k − 1) 2 k 2 kα 2 k − 1 (2 π ) ◆♦t✐❝❡ t❤❛t σ k ( x, t ) = e − t (2 πx ) 2 k ❢✉❧✜❧❧s ❝♦♥❞✐t✐♦♥ ✭✶✮ ✇❤✐t s = 2 k − 1 1 2 k , r = 2 k . ❚❤✉s σ ∈ S s r ✳ ■♥ ♣❛rt✐❝✉❧❛r σ ❢✉❧❧✜❧❧s ✭✷✮✳ ✶✷ ✴ ✶✽

❈♦❡✣❝✐❡♥ts✬ ❉❡❝❛②✲ ●❡♥❡r❛❧✐③❡❞ ❍❡❛t ❊q✉❛t✐♦♥ � � ❈♦❡✣❝✐❡♥ts✬ ❉❡❝❛② ✲ ●❡♥❡r❛❧✐③❡❞ ❍❡❛t ❊q✉❛t✐♦♥ ❢♦r � � σ 2 ( t, D ) g 0 , 0 , g m ′ ,n ′ � � ✱ ❞✐♠❡♥s✐♦♥ d = 1 ✳ ✶✸ ✴ ✶✽