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❋✐♥✐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 ▼❛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 ❛❧✳✱ ✷✵✶✷✮✿

✶ ✷

✇❤❡r❡ ✐s ❛ ●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡✱ ❛♥❞

✶ ✷ ✐s t❤❡ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r

❞❡✜♥❡❞ ❛s ✿

✶ ✷ ✶ ✷

✇❤❡r❡ ❞❡♥♦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 ✷ ✴ ✶✽

❈♦♥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 −✶ ω →

• P(ω✷)F[.](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 ✷ ✴ ✶✽

❘❡♣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) =

• ziψi(s) ,

s ∈ D ✇❤❡r❡ {ψi} ❛r❡ ❜❛s✐s ❢✉♥❝t✐♦♥s ♦♥ ❛ tr✐❛♥❣✉❧❛t❡❞ ❞♦♠❛✐♥ D ✭❜♦✉♥❞❡❞ ♣♦❧②❣♦♥❛❧ ♦r ♠❛♥✐❢♦❧❞✮✱ ❛♥❞ {zi} ❛r❡ ●❛✉ss✐❛♥ ✇❡✐❣❤ts✳ 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♠✿ ✶

✷

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) =

• ziψi(s) ,

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

• = [∇ψi, ∇ψj],

❙ = ❈ −✶/✷●❈ −✶/✷ ⇒ ❋♦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 ✸ ✴ ✶✽

❊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 ⊂ Rd ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s✿ Z = L√gW ✭✷✮ ✇❤❡r❡ L√g[.] := F −✶ ω →

• g(ω✷)F[.](ω)
• ❛♥❞ W ✐s ❛ ●❛✉ss✐❛♥ ✇❤✐t❡

♥♦✐s❡ ♦♥ D✳ ❇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❡✿ ❈ ❉✐❛❣ ✶ ✱ ● ✱ ❙ ❈

✶ ✷●❈ ✶ ✷✱

❙ ❱

✶

✳✳✳ ❱ ❙ ❱

✶

✳✳✳ ❱

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✹ ✴ ✶✽

❊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 ⊂ Rd ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s✿ Z = L√gW ✭✷✮ ✇❤❡r❡ L√g[.] := F −✶ ω →

• g(ω✷)F[.](ω)
• ❛♥❞ W ✐s ❛ ●❛✉ss✐❛♥ ✇❤✐t❡

♥♦✐s❡ ♦♥ D✳ ❇② r❡♣r❡s❡♥t✐♥❣ Z ✉s✐♥❣ ❛ ✜♥✐t❡ ❡❧❡♠❡♥t ❛♣♣r♦❛❝❤✱ ✇❡ s❤♦✇❡❞✿ Pr♦♣♦s✐t✐♦♥ ✿ ❈♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ❋❊ ✇❡✐❣❤ts ✭P❡r❡✐r❛ ❛♥❞ ❉❡s❛ss✐s✱ ✷✵✶✽❜✮ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ✇❡✐❣❤ts {zi} ♦❢ t❤❡ ❋❊ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✭✷✮ ✐s✿ Σz = ❈ −✶/✷g(❙)❈ −✶/✷ ✇❤❡r❡✿ ❈ = ❉✐❛❣(ψi, ✶)✱ ● = [∇ψi, ∇ψj]✱ ❙ = ❈ −✶/✷●❈ −✶/✷✱ ❙ = ❱ λ✶ ✳✳✳

λn

• ❱ T,

g(❙) = ❱ g(λ✶) ✳✳✳

g(λn)

• ❱ T

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✹ ✴ ✶✽

❚❤❡♦r❡t✐❝❛❧ ❢r❛♠❡✇♦r❦ ❢♦r t❤❡ ♣r♦♦❢

❙❛♠❡ ❢r❛♠❡✇♦r❦ ❛s ✐♥ ✭❇♦❧✐♥ ❡t ❛❧✳✱ ✷✵✶✼✮✳ L✷(D) ❂ ❍✐❧❜❡rt s♣❛❝❡ ♦❢ sq✉❛r❡✲✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ ♦♥ D ❚❤❡ ♥❡❣❛t✐✈❡ ▲❛♣❧❛❝✐❛♥ −∆ ✐s ❛ s❡❧❢✲❛❞❥♦✐♥t ♣♦s✐t✐✈❡ s❡♠✐✲❞❡✜♥✐t❡ ♦♣❡r❛t♦r ♦♥ L✷(D) ⇒ ❉✐❛❣♦♥❛❧✐③❛❜❧❡ ✿

❈♦✉♥t❛❜❧❡ ❡✐❣❡♥✈❛❧✉❡s ✿ ✵ ≤ µ✶ ≤ µ✷ ≤ · · · ≤ µj ≤ . . . , j ∈ N t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦❢ −∆ ❢♦r♠ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ L✷(D)

■t ❝❛♥ ❜❡ s❤♦✇❡❞ t❤❛t t❤❡♥✱ t❤❡ ●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ✿ ❢♦r ❛ ❢❛♠✐❧② ♦❢ ✐✳✐✳❞✳ st❛♥❞❛r❞ ●❛✉ss✐❛♥ ✇❡✐❣❤ts ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ r❛♥❞♦♠ ✜❡❧❞ ✐s ❣✐✈❡♥ ❜② ✿ ❚❤❡ ✜♥✐t❡ ❡❧❡♠❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ♦♥t♦ t❤❡ ❧✐♥❡❛r s♣❛♥ ♦❢ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥s s♣❛♥ ✶

✷

✳

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✺ ✴ ✶✽

❚❤❡♦r❡t✐❝❛❧ ❢r❛♠❡✇♦r❦ ❢♦r t❤❡ ♣r♦♦❢

❙❛♠❡ ❢r❛♠❡✇♦r❦ ❛s ✐♥ ✭❇♦❧✐♥ ❡t ❛❧✳✱ ✷✵✶✼✮✳ L✷(D) ❂ ❍✐❧❜❡rt s♣❛❝❡ ♦❢ sq✉❛r❡✲✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ ♦♥ D ❚❤❡ ♥❡❣❛t✐✈❡ ▲❛♣❧❛❝✐❛♥ −∆ ✐s ❛ s❡❧❢✲❛❞❥♦✐♥t ♣♦s✐t✐✈❡ s❡♠✐✲❞❡✜♥✐t❡ ♦♣❡r❛t♦r ♦♥ L✷(D) ⇒ ❉✐❛❣♦♥❛❧✐③❛❜❧❡ ✿

❈♦✉♥t❛❜❧❡ ❡✐❣❡♥✈❛❧✉❡s ✿ ✵ ≤ µ✶ ≤ µ✷ ≤ · · · ≤ µj ≤ . . . , j ∈ N t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦❢ −∆ ❢♦r♠ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ L✷(D)

■t ❝❛♥ ❜❡ s❤♦✇❡❞ t❤❛t t❤❡♥✱ t❤❡ ●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡ D ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ✿ W =

• j∈N

ξjej ❢♦r ❛ ❢❛♠✐❧② ♦❢ ✐✳✐✳❞✳ st❛♥❞❛r❞ ●❛✉ss✐❛♥ ✇❡✐❣❤ts {ξj}j∈N ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ r❛♥❞♦♠ ✜❡❧❞ Z ✐s ❣✐✈❡♥ ❜② ✿ Z = L√gW =

• j∈N
• g(µj)ξjej

❚❤❡ ✜♥✐t❡ ❡❧❡♠❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ ♦♥t♦ t❤❡ ❧✐♥❡❛r s♣❛♥ ♦❢ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥s s♣❛♥ ✶

✷

✳

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✺ ✴ ✶✽

❚❤❡♦r❡t✐❝❛❧ ❢r❛♠❡✇♦r❦ ❢♦r t❤❡ ♣r♦♦❢

❙❛♠❡ ❢r❛♠❡✇♦r❦ ❛s ✐♥ ✭❇♦❧✐♥ ❡t ❛❧✳✱ ✷✵✶✼✮✳ L✷(D) ❂ ❍✐❧❜❡rt s♣❛❝❡ ♦❢ sq✉❛r❡✲✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ ♦♥ D ❚❤❡ ♥❡❣❛t✐✈❡ ▲❛♣❧❛❝✐❛♥ −∆ ✐s ❛ s❡❧❢✲❛❞❥♦✐♥t ♣♦s✐t✐✈❡ s❡♠✐✲❞❡✜♥✐t❡ ♦♣❡r❛t♦r ♦♥ L✷(D) ⇒ ❉✐❛❣♦♥❛❧✐③❛❜❧❡ ✿

❈♦✉♥t❛❜❧❡ ❡✐❣❡♥✈❛❧✉❡s ✿ ✵ ≤ µ✶ ≤ µ✷ ≤ · · · ≤ µj ≤ . . . , j ∈ N t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦❢ −∆ ❢♦r♠ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ L✷(D)

■t ❝❛♥ ❜❡ s❤♦✇❡❞ t❤❛t t❤❡♥✱ t❤❡ ●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡ D ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ✿ W =

• j∈N

ξjej ❢♦r ❛ ❢❛♠✐❧② ♦❢ ✐✳✐✳❞✳ st❛♥❞❛r❞ ●❛✉ss✐❛♥ ✇❡✐❣❤ts {ξj}j∈N ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ r❛♥❞♦♠ ✜❡❧❞ Z ✐s ❣✐✈❡♥ ❜② ✿ Z = L√gW =

• j∈N
• g(µj)ξjej

❚❤❡ ✜♥✐t❡ ❡❧❡♠❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ Z ✐s ❞❡✜♥❡❞ ❛s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ Z ♦♥t♦ t❤❡ ❧✐♥❡❛r s♣❛♥ ♦❢ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥s s♣❛♥{ψk : k ∈ [ [✶, n] ]} ⊂ L✷(D)✳

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✺ ✴ ✶✽

• ❡♥❡r❛❧✐③❛t✐♦♥ t♦ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ■

❉❡✜♥✐t✐♦♥ ❆ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ M = (D, H) ♦❢ ❞✐♠✳ d ✐s ❛ ❝♦♠♣♦s❡❞ ❜②✿ ❛ ♠❛♥✐❢♦❧❞ D✱ ✐✳❡✳ ❛ ❞♦♠❛✐♥ t❤❛t ✧❜❡❤❛✈❡s✧ ❧♦❝❛❧❧② ❧✐❦❡ Rd ❛ ♠❡tr✐❝ H✱ ✐✳❡✳ ❛ s♠♦♦t❤ ❛♣♣❧✐❝❛t✐♦♥ t❤❛t ❛ss♦❝✐❛t❡s t♦ ❛♥② s ∈ D ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ t❤❡ t❛♥❣❡♥t s♣❛❝❡ ♦❢ D ❛t t❤❡ ♣♦✐♥t s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ H ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❢❛♠✐❧② ♦❢ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐❝❡s ♦❢ s✐③❡ d ✐♥❞❡①❡❞ ❜② t❤❡ ♣♦✐♥ts ♦❢ D ❚❤❡ ▲❛♣❧❛❝✐❛♥ ✭♦r ▲❛♣❧❛❝❡✲❇❡❧tr❛♠✐ ♦♣❡r❛t♦r✮ ♦♥ M ✐s ❞❡✜♥❡❞ ❜②✿ ∆Mf = ✶ √ ❞❡t H

d

• i=✶

∂i  √ ❞❡t H

d

• j=✶

[H−✶]ij∂jf   = ✶ √ ❞❡t H ❞✐✈ √ ❞❡t HH−✶∇f

• ⇒ ■t ✐s ❛ s❡❧❢✲❛❞❥♦✐♥t ♣♦s✐t✐✈❡ s❡♠✐✲❞❡✜♥✐t❡ ♦♣❡r❛t♦r ♦♥ L✷(M)✦

⇒ ●❡♥❡r❛❧✐③❡ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧t t♦ ✜❡❧❞s ♦♥ M ✭P❡r❡✐r❛ ❛♥❞ ❉❡s❛ss✐s✱ ✷✵✶✽❜✮

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✻ ✴ ✶✽

• ❡♥❡r❛❧✐③❛t✐♦♥ t♦ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ■■

▲❡t M = (D, H) ❜❡ ❛ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ❛♥❞ ❧❡t g : R+ → R+✳ ▲❡t Z ❜❡ t❤❡ ❣❡♥❡r❛❧✐③❡❞ r❛♥❞♦♠ ✜❡❧❞ ❞❡✜♥❡❞ ❜②✿ Z = L√gW :=

• j∈N
• g(µj)ξjej

✭✸✮ ✇❤❡r❡✿ {(µj, ej) : j ∈ N} ❛r❡ ❡✐❣❡♥♣❛✐rs ♦❢ t❤❡ ♥❡❣❛t✐✈❡ ▲❛♣❧❛❝✐❛♥ −∆M✱ ❢♦r♠✐♥❣ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ L✷(M) {ξj}j∈N ✐s ❛ s❡t ♦❢ ✐✳✐✳❞✳ st❛♥❞❛r❞ ●❛✉ss✐❛♥ ✇❡✐❣❤ts Pr♦♣♦s✐t✐♦♥ ✿ ❈♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ❋❊ ✇❡✐❣❤ts ✭P❡r❡✐r❛ ❛♥❞ ❉❡s❛ss✐s✱ ✷✵✶✽❜✮ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ✇❡✐❣❤ts {zi} ♦❢ t❤❡ ❋❊ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✭✸✮ ✐s✿ Σz = ❈ −✶/✷g(❙)❈ −✶/✷ ✇❤❡r❡✿ ❈ = ❉✐❛❣( √ ❞❡t Hψi, ✶),

• = [∇ψi,

√ ❞❡t HH−✶∇ψj] ❙ = ❈ −✶/✷●❈ −✶/✷

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✼ ✴ ✶✽

◆♦✇ ✇❤❛t❄

• ❡♥❡r❛❧ ❢♦r♠ ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ ✜♥✐t❡ ❡❧❡♠❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ♦❢
• ❛✉ss✐❛♥ ✜❡❧❞s ✿

Σz = ❈ −✶/✷g(❙)❈ −✶/✷ ✭✹✮ ✇❤❡r❡ ❈ ✐s ❛ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ✇✐t❤ str✐❝t❧② ♣♦s✐t✐✈❡ ❡❧❡♠❡♥ts✳ ❙ ✐s ❛ s②♠♠❡tr✐❝ ♣♦s✐t✐✈❡ s❡♠✐✲❞❡✜♥✐t❡ ♠❛tr✐① ✇❤♦s❡ ❡❧❡♠❡♥ts ❛r❡ ✐♥♥❡r ♣r♦❞✉❝ts ♦❢ ❣r❛❞✐❡♥ts ♦❢ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ⇒ ❍♦✇ t♦ s✐♠✉❧❛t❡ ✇❡✐❣❤ts ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ✭✹✮❄ Pr♦♣♦s✐t✐♦♥ ✿ ❙✐♠✉❧❛t✐♦♥ ♦❢ ❙P❉❊ ❋❊▼ s♦❧✉t✐♦♥s ❲❡✐❣❤ts ③ = (z✶, . . . , zn)T ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Σz ❣✐✈❡♥ ❜② ✭✹✮ ❝❛♥ ❜❡ s✐♠✉❧❛t❡❞ t❤r♦✉❣❤✿ ③ = ❈ −✶/✷√g(❙)ε ✇❤❡r❡ ε ✐s ❛ ●❛✉ss✐❛♥ ✈❡❝t♦r ✇✐t❤ ✐♥❞❡♣❡♥❞❡♥t st❛♥❞❛r❞ ❝♦♠♣♦♥❡♥ts ❛♥❞ √g : R+ → R s❛t✐s✜❡s √g ✷ = g✳

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✽ ✴ ✶✽

❲❡✐❣❤t s✐♠✉❧❛t✐♦♥

Pr♦❜❧❡♠ ❍♦✇ t♦ ❝♦♠♣✉t❡ √g(❙)ε❄ ❙ = ❱ λ✶ ✳✳✳

λn

• ❱ T ⇒ √g(❙)ε = ❱

√g(λ✶) ✳✳✳ √g(λn)

• ❱ Tε

⇒ ❉✐❛❣♦♥❛❧✐③❛t✐♦♥ ✰ ❙t♦r❛❣❡ ✿ ❊①♣❡♥s✐✈❡✦✦ ■❞❡❛ ✿ ✉s❡ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❝❛s❡ ✿ ❢♦r P(X) = akX k

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✾ ✴ ✶✽

❲❡✐❣❤t s✐♠✉❧❛t✐♦♥

Pr♦❜❧❡♠ ❍♦✇ t♦ ❝♦♠♣✉t❡ √g(❙)ε❄ ❙ = ❱ λ✶ ✳✳✳

λn

• ❱ T ⇒ √g(❙)ε = ❱

√g(λ✶) ✳✳✳ √g(λn)

• ❱ Tε

⇒ ❉✐❛❣♦♥❛❧✐③❛t✐♦♥ ✰ ❙t♦r❛❣❡ ✿ ❊①♣❡♥s✐✈❡✦✦ ■❞❡❛ ✿ ✉s❡ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❝❛s❡ ✿ ❢♦r P(X) = akX k P(❙)ε = ❱ P(λ✶) ✳✳✳

P(λn)

• ❱ Tε =
• ak❙kε

→ P(❙)ε ✐s ❝♦♠♣✉t❛❜❧❡ ✐t❡r❛t✐✈❡❧② ✇✐t❤♦✉t ❤❛✈✐♥❣ t♦ ❞✐❛❣♦♥❛❧✐③❡ ❙ ✿ ♦♥❧② ✐♥✈♦❧✈❡s ♠❛tr✐①✲✈❡❝t♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥s✦

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✾ ✴ ✶✽

❲❡✐❣❤t s✐♠✉❧❛t✐♦♥

Pr♦❜❧❡♠ ❍♦✇ t♦ ❝♦♠♣✉t❡ √g(❙)ε❄ ❙ = ❱ λ✶ ✳✳✳

λn

• ❱ T ⇒ √g(❙)ε = ❱

√g(λ✶) ✳✳✳ √g(λn)

• ❱ Tε

⇒ ❉✐❛❣♦♥❛❧✐③❛t✐♦♥ ✰ ❙t♦r❛❣❡ ✿ ❊①♣❡♥s✐✈❡✦✦ ■❞❡❛ ✿ ✉s❡ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❝❛s❡ ✿ ❢♦r P(X) = akX k P(❙)ε = ❱ P(λ✶) ✳✳✳

P(λn)

• ❱ Tε =
• ak❙kε

→ P(❙)ε ✐s ❝♦♠♣✉t❛❜❧❡ ✐t❡r❛t✐✈❡❧② ✇✐t❤♦✉t ❤❛✈✐♥❣ t♦ ❞✐❛❣♦♥❛❧✐③❡ ❙ ✿ ♦♥❧② ✐♥✈♦❧✈❡s ♠❛tr✐①✲✈❡❝t♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥s✦

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✾ ✴ ✶✽

❲❡✐❣❤t s✐♠✉❧❛t✐♦♥

Pr♦❜❧❡♠ ❍♦✇ t♦ ❝♦♠♣✉t❡ √g(❙)ε❄ ❙ = ❱ λ✶ ✳✳✳

λn

• ❱ T ⇒ √g(❙)ε = ❱

√g(λ✶) ✳✳✳ √g(λn)

• ❱ Tε

⇒ ❉✐❛❣♦♥❛❧✐③❛t✐♦♥ ✰ ❙t♦r❛❣❡ ✿ ❊①♣❡♥s✐✈❡✦✦ ■❞❡❛ ✿ ✉s❡ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❝❛s❡ ✿ ❢♦r P(X) = akX k P(❙)ε = ❱ P(λ✶) ✳✳✳

P(λn)

• ❱ Tε =
• ak❙kε

→ P(❙)ε ✐s ❝♦♠♣✉t❛❜❧❡ ✐t❡r❛t✐✈❡❧② ✇✐t❤♦✉t ❤❛✈✐♥❣ t♦ ❞✐❛❣♦♥❛❧✐③❡ ❙ ✿ ♦♥❧② ✐♥✈♦❧✈❡s ♠❛tr✐①✲✈❡❝t♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥s✦ → ■♥st❡❛❞ ♦❢ ❝♦♠♣✉t✐♥❣ √g(❙)ε✱ ❝♦♠♣✉t❡ P(❙)ε ✇❤❡r❡ P ✐s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ √g ♦✈❡r ❛♥ ✐♥t❡r✈❛❧ ❝♦♥t❛✐♥✐♥❣ {λ✶, . . . , λn} ⇒ P(❙)ε ≈ √g(❙)ε ❜❡❝❛✉s❡ P(λi) ≈ √g(λi) ∀i

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✾ ✴ ✶✽

❈❤❡❜②s❤❡✈ ❛❧❣♦r✐t❤♠ ❢♦r ✇❡✐❣❤t s✐♠✉❧❛t✐♦♥

❆❧❣♦r✐t❤♠ ✿ ❈❤❡❜②s❤❡✈ s✐♠✉❧❛t✐♦♥ ✭P❡r❡✐r❛ ❛♥❞ ❉❡s❛ss✐s✱ ✷✵✶✽❛✮ ❘❡q✉✐r❡✿ ❆♥ ♦r❞❡r ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ K ∈ N✳ ❖✉t♣✉t✿ ❆ ✈❡❝t♦r ③ ≈ ❈ −✶/✷√g(❙)ε✳ ✶✳ ❈♦♠♣✉t❡ ❛♥ ✐♥t❡r✈❛❧ [a, b] ❝♦♥t❛✐♥✐♥❣ ❛❧❧ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❙

❊①✿ [✵,

• ❚r❛❝❡(❙❙T)]✱ ●❡rs❤❣♦r✐♥ ❝✐r❝❧❡ t❤❡♦r❡♠

✷✳ ❈♦♠♣✉t❡ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✭❞❡♥♦t❡❞ P✮ ♦❢ √g ♦✈❡r [a, b] ❜② tr✉♥❝❛t✐♥❣ ✐ts ❈❤❡❜②s❤❡✈ s❡r✐❡s ❛t ♦r❞❡r K → ❈♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ ❈❤❡❜②s❤❡✈ ❜❛s✐s ♦❜t❛✐♥❡❞ ❜② ❋❋❚ ✸✳ ❈♦♠♣✉t❡ ✉ = P(❙)ε ✐t❡r❛t✐✈❡❧② ✭♦♥❧② r❡q✉✐r❡s ♠❛tr✐① t♦ ✈❡❝t♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥s✮ ✹✳ ❘❡t✉r♥ ③ = ❈ −✶/✷✉ ❈♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t②✿ O(Knnz) ♦♣❡r❛t✐♦♥s✱ nnz ♥✉♠❜❡r ♦❢ ♥♦♥✲③❡r♦ ❡♥tr✐❡s ♦❢ ❙ ◗✉❡st✐♦♥ ✿ ❍♦✇ t♦ ❝❤♦♦s❡ t❤❡ ♦r❞❡r ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ K t♦ ❣❡t ❛ ✧s❛t✐s❢②✐♥❣✧ ♦✉t♣✉t❄

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✵ ✴ ✶✽

❆♥ ❛♣♣r♦❛❝❤❡❞ ♦✉t♣✉t

■♥✐t✐❛❧ ●♦❛❧ ❙✐♠✉❧❛t❡ ❛ ③❡r♦✲♠❡❛♥ ●❛✉ss✐❛♥ ✈❡❝t♦r ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✿ Σ = ❈ −✶/✷g(❙)❈ −✶/✷ ❖✉t♣✉t ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ❆ ③❡r♦✲♠❡❛♥ ●❛✉ss✐❛♥ ✈❡❝t♦r ③s = ❈ −✶/✷PK(❙)ε ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✿ Σs = ❈ −✶/✷P✷

K(❙)❈ −✶/✷

→ ❲❤❡♥ ❝❛♥ t❤❡ s✐♠✉❧❛t❡❞ ♦✉t♣✉t ✧♣❛ss✧ ❛s t❤❡ t❛r❣❡t❡❞ ♦♥❡❄ ■❞❡❛✿ ❯s❡ st❛t✐st✐❝❛❧ t❡sts ♦♥ t❤❡ ♦✉t♣✉t ▲❡t ③ ✶ ③ ❜❡ ❛ ✲s❛♠♣❧❡ ♦❢ ✈❡❝t♦rs s✐♠✉❧❛t❡❞ ✉s✐♥❣ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❝

✶

❜❡ ❛r❜✐tr❛r② ❝♦❡✣❝✐❡♥ts✳ ■❢ t❤❡ ③ ❤❛✈❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ✱ t❤❡♥ ❝ ❝ ③ ✶ ❝ ③ ✐s ❛ ●❛✉ss✐❛♥ s❛♠♣❧❡ ✇✐t❤ ✈❛r✐❛♥❝❡ ❝ ❝✳ ❯s❡

✷ t❡st ♦❢ ✈❛r✐❛♥❝❡ ♦♥

❝ t♦ ❝❤❡❝❦ t❤❛t✳

• ✐✈❡♥ t❤❛t t❤❡ ❛❝t✉❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢

❝ ✐s ❦♥♦✇♥ ✭●❛✉ss✐❛♥ ✇✐t❤ ✈❛r✐❛♥❝❡ ❝ ❝✮✱ ✇❡ ❝❛♥ ❛♥t✐❝✐♣❛t❡ t❤❡ r❡s✉❧ts ✇✐t❤♦✉t ❛❝t✉❛❧❧② r❡❛❧✐s✐♥❣ ❛♥② t❡st✦

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✶ ✴ ✶✽

❆♥ ❛♣♣r♦❛❝❤❡❞ ♦✉t♣✉t

■♥✐t✐❛❧ ●♦❛❧ ❙✐♠✉❧❛t❡ ❛ ③❡r♦✲♠❡❛♥ ●❛✉ss✐❛♥ ✈❡❝t♦r ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✿ Σ = ❈ −✶/✷g(❙)❈ −✶/✷ ❖✉t♣✉t ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ❆ ③❡r♦✲♠❡❛♥ ●❛✉ss✐❛♥ ✈❡❝t♦r ③s = ❈ −✶/✷PK(❙)ε ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✿ Σs = ❈ −✶/✷P✷

K(❙)❈ −✶/✷

→ ❲❤❡♥ ❝❛♥ t❤❡ s✐♠✉❧❛t❡❞ ♦✉t♣✉t ✧♣❛ss✧ ❛s t❤❡ t❛r❣❡t❡❞ ♦♥❡❄ ■❞❡❛✿ ❯s❡ st❛t✐st✐❝❛❧ t❡sts ♦♥ t❤❡ ♦✉t♣✉t ▲❡t {③(✶)

s

, ..., ③(N)

s

} ❜❡ ❛ N✲s❛♠♣❧❡ ♦❢ ✈❡❝t♦rs s✐♠✉❧❛t❡❞ ✉s✐♥❣ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❝ = (c✶, ..., cn)T ∈ Rn ❜❡ ❛r❜✐tr❛r② ❝♦❡✣❝✐❡♥ts✳ ■❢ t❤❡ ③(i)

s

❤❛✈❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Σ✱ t❤❡♥ S(❝) = {❝T③(✶)

s

, ..., ❝T③(N)

s

} ✐s ❛ ●❛✉ss✐❛♥ s❛♠♣❧❡ ✇✐t❤ ✈❛r✐❛♥❝❡ ❝TΣ❝✳ ⇒ ❯s❡ χ✷ t❡st ♦❢ ✈❛r✐❛♥❝❡ ♦♥ S(❝) t♦ ❝❤❡❝❦ t❤❛t✳

• ✐✈❡♥ t❤❛t t❤❡ ❛❝t✉❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ S(❝) ✐s ❦♥♦✇♥ ✭●❛✉ss✐❛♥ ✇✐t❤ ✈❛r✐❛♥❝❡

❝TΣs❝✮✱ ✇❡ ❝❛♥ ❛♥t✐❝✐♣❛t❡ t❤❡ r❡s✉❧ts ✇✐t❤♦✉t ❛❝t✉❛❧❧② r❡❛❧✐s✐♥❣ ❛♥② t❡st✦

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✶ ✴ ✶✽

❈r✐t❡r✐♦♥ ♦♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♣r❡❝✐s✐♦♥

Pr♦♣♦s✐t✐♦♥ ✿ ❙t❛t✐st✐❝❛❧ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs ✭P❡r❡✐r❛ ❛♥❞ ❉❡s❛ss✐s✱ ✷✵✶✽❛✮ ▲❡t Rr❡❥❡❝t(❝) ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ χ✷ t❡st ✇✐t❤ s✐❣♥✐✜❝❛♥❝❡ α ♦♥ t❤❡ N✲s❛♠♣❧❡ S(❝) ✧❢❛✐❧s✧ ✭✐✳❡✳ ♥✉❧❧ ❤②♣♦t❤❡s✐s r❡❥❡❝t❡❞✮✳ ❚❤❡♥✱ ∀β > ✵✱ ∃ǫβ > ✵ s✉❝❤ t❤❛t ✿ ♠❛①

λ∈[λ♠✐♥,λ♠❛①]

• g(λ) − PK(λ)✷

PK(λ)✷

• ❊rr♦r ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥

≤ ǫβ ⇒ ∀❝, Rr❡❥❡❝t(❝) ≤ (✶ + β)α ǫβ ❝❛♥ ❜❡ ♥✉♠❡r✐❝❛❧❧② ❝♦♠♣✉t❡❞ ❛♥❞ ❞❡♣❡♥❞s ♦♥ α✱ N ❛♥❞ β > ✵✳

❚❛❜❧❡✿ ❊①❛♠♣❧❡s ♦❢ ǫN,β ✈❛❧✉❡s ♦❢ ❢♦r α = ✵.✵✺

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✷ ✴ ✶✽

❆♣♣❧✐❝❛t✐♦♥ ✿ ❙✐♠✉❧❛t✐♦♥ ♦❢ ✧❡①♦t✐❝✧ s♣❡❝tr❛❧ ❞❡♥s✐t✐❡s

❙✐♠✉❧❛t✐♦♥ ♦♥ ✶✵✵✵①✶✵✵✵ ❣r✐❞ Z = L√gW ✇✐t❤ g(ω✷) =

• κ✷ + ω✷−(π+✶)

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✸ ✴ ✶✽

❆♣♣❧✐❝❛t✐♦♥ ✿ ❙✐♠✉❧❛t✐♦♥ ♦❢ ✧❡①♦t✐❝✧ s♣❡❝tr❛❧ ❞❡♥s✐t✐❡s

❙✐♠✉❧❛t✐♦♥ ♦♥ ✶✵✵✵①✶✵✵✵ ❣r✐❞ Z = L√gW ✇✐t❤ g ✐s t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ♦ t❤❡ ❍②♣❡rs♣❤❡r✐❝❛❧ ❝♦✈❛r✐❛♥❝❡✳

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✹ ✴ ✶✽

❆♣♣❧✐❝❛t✐♦♥ ✿ ❙✐♠✉❧❛t✐♦♥ ♦♥ t❤❡ s♣❤❡r❡

❙✐♠✉❧❛t✐♦♥ ♦♥ ❛ tr✐❛♥❣✉❧❛t❡❞ s♣❤❡r❡ ♦❢ t❤❡ ✜❡❧❞ Z = LgW ✇✐t❤ g(ω✷) =

• κ✹ + ✷κ✷ ❝♦s(✷πθ)ω✷ + ω✹−✷

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✺ ✴ ✶✽

❆♣♣❧✐❝❛t✐♦♥ ✿ ❙✐♠✉❧❛t✐♦♥ ♦❢ ♥♦♥✲st❛t✐♦♥❛r② ✜❡❧❞s

❙✐♠✉❧❛t✐♦♥ ♦♥ ❛ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ♦❢ Z = LgW ✇✐t❤ g(ω✷) =

• κ✷ + ω✷−✶✳ ❚❤❡ ♠❡tr✐❝

t❡♥s♦r ✐s ❣✐✈❡♥ ❜② ✿ H−✶(s) = R(s)T R(s), R(s) = d✶(s) ✵ ✵ d✷(s) ❝♦s(θ(s)) − s✐♥(θ(s)) s✐♥(θ(s)) ❝♦s(θ(s))

• P❡r❡✐r❛✱ ❉❡s❛ss✐s

❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✻ ✴ ✶✽

❈♦♥❝❧✉s✐♦♥

▲❛r❣❡ ❝❧❛ss ♦❢ ●❛✉ss✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ✿ ❈❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡✐r s♣❡❝tr❛❧ ❞❡♥s✐t② ▲❛r❣❡ ❝❧❛ss ♦❢ ❞♦♠❛✐♥s ✿ ♠❛♥✐❢♦❧❞s ❛♥❞ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ❊①♣❧✐❝✐t ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ ❋❊ ✇❡✐❣❤ts ❊✣❝✐❡♥t ❛♣♣r♦①✐♠❛t❡ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ s❛♠♣❧❡s ♦❢ ✇❡✐❣❤ts ✿ ❧✐♥❡❛r ❝♦♠♣❧❡①✐t② ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r t♦❧❡r❛♥❝❡ s❡t t♦ r❡tr✐❡✈❡ st❛t✐st✐❝❛❧ ♣r♦♣❡rt✐❡s

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✼ ✴ ✶✽

❘❡❢❡r❡♥❝❡s

❇♦❧✐♥✱ ❉✳✱ ❑✐r❝❤♥❡r✱ ❑✳✱ ❛♥❞ ❑♦✈á❝s✱ ▼✳ ✭✷✵✶✼✮✳ ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧ ❡❧❧✐♣t✐❝ st♦❝❤❛st✐❝ ♣❞❡s ✇✐t❤ s♣❛t✐❛❧ ✇❤✐t❡ ♥♦✐s❡✳ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✶✼✵✺✳✵✻✺✻✺✳ ▲❛♥❣✱ ❆✳ ❛♥❞ P♦tt❤♦✛✱ ❏✳ ✭✷✵✶✶✮✳ ❋❛st s✐♠✉❧❛t✐♦♥ ♦❢ ❣❛✉ss✐❛♥ r❛♥❞♦♠ ✜❡❧❞s✳ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✶✼✭✸✮✿✶✾✺✕✷✶✹✳ ▲✐♥❞❣r❡♥✱ ❋✳✱ ❘✉❡✱ ❍✳✱ ❛♥❞ ▲✐♥❞strö♠✱ ❏✳ ✭✷✵✶✶✮✳ ❆♥ ❡①♣❧✐❝✐t ❧✐♥❦ ❜❡t✇❡❡♥ ❣❛✉ss✐❛♥ ✜❡❧❞s ✻✼✵ ❛♥❞ ❣❛✉ss✐❛♥ ♠❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞s✿ t❤❡ s♣❞❡ ❛♣♣r♦❛❝❤ ✭✇✐t❤ ❞✐s❝✉ss✐♦♥✮✳ ❏❘ ✻✼✶ ❙t❛t ❙♦❝✱ ❙❡r✐❡s ❇✱ ✼✸✿✹✷✸✕✹✾✽✳ P❡r❡✐r❛✱ ▼✳ ❛♥❞ ❉❡s❛ss✐s✱ ◆✳ ✭✷✵✶✽❛✮✳ ❊✣❝✐❡♥t s✐♠✉❧❛t✐♦♥ ♦❢ ●❛✉ss✐❛♥ ▼❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞s ❜② ❈❤❡❜②s❤❡✈ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❛r❳✐✈✿✶✽✵✺✳✵✼✹✷✸✳ P❡r❡✐r❛✱ ▼✳ ❛♥❞ ❉❡s❛ss✐s✱ ◆✳ ✭✷✵✶✽❜✮✳ ❋✐♥✐t❡ ❡❧❡♠❡♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s✳ ❛r❳✐✈✿✶✽✶✶✳✵✸✵✵✹✳ ❘♦③❛♥♦✈✱ ❏✳ ❆✳ ✭✶✾✼✼✮✳ ▼❛r❦♦✈ r❛♥❞♦♠ ✜❡❧❞s ❛♥❞ st♦❝❤❛st✐❝ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ▼❛t❤❡♠❛t✐❝s ♦❢ t❤❡ ❯❙❙❘✲❙❜♦r♥✐❦✱ ✸✷✭✹✮✿✺✶✺✳ ❙✐♠♣s♦♥✱ ❉✳✱ ▲✐♥❞❣r❡♥✱ ❋✳✱ ❛♥❞ ❘✉❡✱ ❍✳ ✭✷✵✶✷✮✳ ❚❤✐♥❦ ❝♦♥t✐♥✉♦✉s✿ ▼❛r❦♦✈✐❛♥ ❣❛✉ss✐❛♥ ♠♦❞❡❧s ✐♥ s♣❛t✐❛❧ st❛t✐st✐❝s✳ ❙♣❛t✐❛❧ ❙t❛t✐st✐❝s✱ ✶✿✶✻ ✕ ✷✾✳

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✽ ✴ ✶✽

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦ ◗✉❡st✐♦♥s❄

▲✐❦❡❧✐❤♦♦❞✲❜❛s❡❞ ♠❡t❤♦❞ ❢♦r ♠♦❞❡❧ ✐♥❢❡r❡♥❝❡ ■

❙✉♣♣♦s❡ t❤❛t g = gθ ❞❡♣❡♥❞s ♦♥ ❛ ✈❡❝t♦r ♦❢ ♣❛r❛♠❡t❡rs θ✳ ❚❤❡ ❧♦❣✲❧✐❦❡❧✐❤♦♦❞ ❛ss♦❝✐❛t❡❞ t♦ ③ ❛♥❞ θ ✐s ❣✐✈❡♥ ❜② ✿ L(③, θ) = −✶ ✷

• N ❧♦❣ ✷π + ❧♦❣ ❞❡t
• gθ(❙)
• + ③T❈ ✶/✷gθ(❙)−✶❈ ✶/✷③
• ■t ❝❛♥ ❜❡ ❡①♣❡♥s✐✈❡ t♦ ❝♦♠♣✉t❡✴♠❛①✐♠✐③❡✳✳✳

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✽ ✴ ✶✽

▲✐❦❡❧✐❤♦♦❞✲❜❛s❡❞ ♠❡t❤♦❞ ❢♦r ♠♦❞❡❧ ✐♥❢❡r❡♥❝❡ ■■

■♥✈❡rs❡ ❛♣♣r♦①✐♠❛t✐♦♥ ③T❈ ✶/✷gθ(❙)−✶❈ ✶/✷③ = ③T❈ ✶/✷❱ ✶/gθ(λ✶) ✳✳✳

✶/gθ(λn)

• ❱ T❈ ✶/✷③

= ③T❈ ✶/✷ ✶ gθ (❙)❈ ✶/✷③ ⇒ ❯s❡ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

✶ gθ

❉❡t❡r♠✐♥❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ❧♦❣ ❞❡t ❙

✶

❧♦❣

✵

❤✐st ❧♦❣ ✇❤❡r❡ ❤✐st ❈❛r❞ ✵ ✶ ✷ ✷ ✶

✷ ✷

❙

✷

P❡r❡✐r❛✱ ❉❡s❛ss✐s ❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✽ ✴ ✶✽

▲✐❦❡❧✐❤♦♦❞✲❜❛s❡❞ ♠❡t❤♦❞ ❢♦r ♠♦❞❡❧ ✐♥❢❡r❡♥❝❡ ■■

■♥✈❡rs❡ ❛♣♣r♦①✐♠❛t✐♦♥ ③T❈ ✶/✷gθ(❙)−✶❈ ✶/✷③ = ③T❈ ✶/✷❱ ✶/gθ(λ✶) ✳✳✳

✶/gθ(λn)

• ❱ T❈ ✶/✷③

= ③T❈ ✶/✷ ✶ gθ (❙)❈ ✶/✷③ ⇒ ❯s❡ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

✶ gθ

❉❡t❡r♠✐♥❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ❧♦❣ ❞❡t

• gθ(❙)
• =

n

• k=✶

❧♦❣(gθ(λk)) ≈

M

• m=✵

❤✐st(am) ❧♦❣(gθ(am)) ✇❤❡r❡ ❤✐st(am) := ❈❛r❞

• i ∈ [

[✵, N − ✶] ] : λi ∈]am − τ ✷, am + τ ✷]

• = E
• ||✶]am− τ

✷ ,am+ τ ✷ ](❙)ε||✷

• P❡r❡✐r❛✱ ❉❡s❛ss✐s

❋✐♥✐t❡ ❡❧❡♠❡♥t s✐♠✉❧❛t✐♦♥s ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥ r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✶✽ ✴ ✶✽