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❙❡❝♦♥❞ ♦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❡s❡❛r❝❤ ✇❡ ❤❛✈❡ r❡str✐❝t❡❞ ♦✉r ❛tt❡♥t✐♦♥ t♦ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❞✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤✐s ♣r♦❜❧❡♠✳ ❲❡ ❞❡♥♦t❡ ❜② ℓk(Z, R) ❛♥❞ ℓk(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 ♦♥ ℓk(Z, C) ❛s ❢♦❧❧♦✇s✿ x|yk := b

n=a x(n)y(n)ρk(n),

✇❤❡r❡ a, b ∈ Z, (a < b), ❛♥❞ ρk ✐s ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥✳ ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦r s❤✐❢t ♦♣❡r❛t♦rs

❆❧✐♥❛ ❉♦❜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 ℓk(Z, R) ❛♥❞ ℓk(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 ♦♥ ℓk(Z, C) ❛s ❢♦❧❧♦✇s✿ x|yk := b

n=a x(n)y(n)ρk(n),

✇❤❡r❡ a, b ∈ Z, (a < b), ❛♥❞ ρk ✐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 ✳✳✳

x|yk = b

n=a x(n)y(n)ρk(n),

❲❡ ❛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

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

x|yk = b

n=a x(n)y(n)ρk(n),

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✇❡✐❣❤t s❡q✉❡♥❝❡ s❛t✐s✜❡s t❤❡ P❡❛rs♦♥ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ∆ (bk(n)ρk(n)) = (ck(n) − bk(n)) ρk(n), ❛♥❞ t❤❡ r❡❝✉rs✐♦♥ r❡❧❛t✐♦♥ ✇❤❡r❡ ❛♥❞ ❛r❡ s♦♠❡ r❡❛❧✲✈❛❧✉❡❞ s❡q✉❡♥❝❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ❢✉❧✜❧❧s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

x|yk = b

n=a x(n)y(n)ρk(n),

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✇❡✐❣❤t s❡q✉❡♥❝❡ s❛t✐s✜❡s t❤❡ P❡❛rs♦♥ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ∆ (bk(n)ρk(n)) = (ck(n) − bk(n)) ρk(n), ❛♥❞ t❤❡ r❡❝✉rs✐♦♥ r❡❧❛t✐♦♥ ρk−1(n) = ck(n)ρk(n), ✇❤❡r❡ {bk} ❛♥❞ {ck} ❛r❡ s♦♠❡ r❡❛❧✲✈❛❧✉❡❞ s❡q✉❡♥❝❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ❢✉❧✜❧❧s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

x|yk = b

n=a x(n)y(n)ρk(n),

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✇❡✐❣❤t s❡q✉❡♥❝❡ s❛t✐s✜❡s t❤❡ P❡❛rs♦♥ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ∆ (bk(n)ρk(n)) = (ck(n) − bk(n)) ρk(n), ❛♥❞ t❤❡ r❡❝✉rs✐♦♥ r❡❧❛t✐♦♥ ρk−1(n) = ck(n)ρk(n), ✇❤❡r❡ {bk} ❛♥❞ {ck} ❛r❡ s♦♠❡ r❡❛❧✲✈❛❧✉❡❞ s❡q✉❡♥❝❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❝t✐♦♥ ρk ❢✉❧✜❧❧s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s bk(a)ρk(a) = bk(b + 1)ρk(b + 1) = 0.

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❲❡ ❛♣♣❧② t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ t♦ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦rs Hk : ℓk(Z, C) − → ℓk(Z, C) ❣✐✈❡♥ ❜② ❍k = zk(n)T+ + wk(n)T− + vk(n), ✇❤❡r❡ {zk}, {wk}, {vk} ∈ ℓ(Z, R)✱ k ∈ Z✳ ❲❡ ❢❛❝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✮

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❲❡ ❛♣♣❧② t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ t♦ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦rs Hk : ℓk(Z, C) − → ℓk(Z, C) ❣✐✈❡♥ ❜② ❍k = zk(n)T+ + wk(n)T− + vk(n), ✇❤❡r❡ {zk}, {wk}, {vk} ∈ ℓ(Z, R)✱ k ∈ Z✳ ❲❡ ❢❛❝t♦r✐③❡ t❤❡ ♦♣❡r❛t♦rs Hk ❛s ❛ ♣r♦❞✉❝t ♦❢ t✇♦ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦rs A±

k ✐♥ t❤❡ ❢♦r♠

❍k = A+

k A− k + αk = A− k+1A+ k+1 + αk+1,

✇❤❡r❡ αk ∈ R✳ ■♥ t❤✐s ✇❛②✱ ✇❡ ♦❜t❛✐♥ ❛ ❝❤❛✐♥ ♦❢ ❧❛❞❞❡r ♦♣❡r❛t♦rs ✭ ✕ ❧♦✇❡r✐♥❣ ♦♣❡r❛t♦rs ❛♥❞ ✕ r❛✐s✐♥❣ ♦♣❡r❛t♦rs✮

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❲❡ ❛♣♣❧② t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ t♦ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦rs Hk : ℓk(Z, C) − → ℓk(Z, C) ❣✐✈❡♥ ❜② ❍k = zk(n)T+ + wk(n)T− + vk(n), ✇❤❡r❡ {zk}, {wk}, {vk} ∈ ℓ(Z, R)✱ k ∈ Z✳ ❲❡ ❢❛❝t♦r✐③❡ t❤❡ ♦♣❡r❛t♦rs Hk ❛s ❛ ♣r♦❞✉❝t ♦❢ t✇♦ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦rs A±

k ✐♥ t❤❡ ❢♦r♠

❍k = A+

k A− k + αk = A− k+1A+ k+1 + αk+1,

✇❤❡r❡ αk ∈ R✳ ■♥ t❤✐s ✇❛②✱ ✇❡ ♦❜t❛✐♥ ❛ ❝❤❛✐♥ ♦❢ ❧❛❞❞❡r ♦♣❡r❛t♦rs ✭A−

k ✕ ❧♦✇❡r✐♥❣ ♦♣❡r❛t♦rs ❛♥❞ A+ k ✕ r❛✐s✐♥❣ ♦♣❡r❛t♦rs✮

. . . ℓk−1(Z, C)

Hk−1

• A+

k

ℓk(Z, C)

Hk

• A+

k+1

• A−

k

• ℓk+1(Z, C) . . .

. . . ,

Hk+1

• A−

k+1

• ❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛

❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t t❤❡s❡ ♦♣❡r❛t♦rs A±

k ❛r❡ r❡❛❧✐③❡❞ ❛s

A−

k = T+ + fk(n) − 1 = ∆ + fk(n),

❚❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤r❡❡ ❡q✉❛t✐♦♥s✿

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t t❤❡s❡ ♦♣❡r❛t♦rs A±

k ❛r❡ r❡❛❧✐③❡❞ ❛s

A−

k = T+ + fk(n) − 1 = ∆ + fk(n),

A+

k = (∆ + fk(n))∗ = bk(n)T− + (fk(n) − 1) ck(n).

❚❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤r❡❡ ❡q✉❛t✐♦♥s✿

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t t❤❡s❡ ♦♣❡r❛t♦rs A±

k ❛r❡ r❡❛❧✐③❡❞ ❛s

A−

k = T+ + fk(n) − 1 = ∆ + fk(n),

A+

k = (∆ + fk(n))∗ = bk(n)T− + (fk(n) − 1) ck(n).

❚❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥ Hk = A+

k A− k + αk = A− k+1A+ k+1 + αk+1 ✐s

❡q✉✐✈❛❧❡♥t t♦ t❤r❡❡ ❡q✉❛t✐♦♥s✿ 1. fk+1(n) − 1 = bk(n) bk+1(n) (fk(n − 1) − 1) , 2. ck+1(n) = bk+1(n) bk(n) ck(n − 1),

• 3. bk(n)−bk+1(n + 1) = αk+1−αk+ bk(n)

bk+1(n) ( fk(n − 1) − 1 )2ck(n − 1) − ( fk(n) − 1 )2ck(n).

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❇❛s✐❝ ❛ss✉♠♣t✐♦♥✿ t❤❡ s❡q✉❡♥❝❡ {bk} ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ♣❛r❛♠❡t❡rs k

❲❡ ❛ss✉♠❡ t❤❛t bk+1(n) = bk(n) =: b(n)✳ ❚❤❡ ❝♦♥❞✐t✐♦♥s ✭✶✳✮ ❛♥❞ ✭✷✳✮ ❣✐✈❡ ✉s t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❢♦r♠✉❧❛s ❢♦r t❤❡ s❡q✉❡♥❝❡s {fk} ❛♥❞ {ck} fk+1(n) = fk(n − 1) ck+1(n) = ck(n − 1) . ❯♥❞❡r t❤✐s ❤②♣♦t❤❡s✐s t❤❡ ❝♦♥❞✐t✐♦♥ ✭✸✳✮ ②✐❡❧❞s ❛ r❡❝✉rr❡♥❝❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ✐♥✐t✐❛❧ s❡q✉❡♥❝❡ {fk0}✱ ❜❡❝❛✉s❡ t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡ ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ♣❛r❛♠❡t❡r k✱ ( fk0(n) − 1 )2ck0(n)−( fk0( n−1 ) − 1 )2ck0( n−1 )= = b(n + 1) − b(n)+αk0+1−αk0.

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❈♦♠♣❛r✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts ✇❡ ✜♥❞ b(n) = β2n2 + β1n + β2, (fk0(n) − 1)2 ck0(n) = β2n2 + (2β2 + β1 + αk0+1 − αk0) n + γ. ❋✐♥❛❧❧② ✇❡ ♦❜t❛✐♥ t❤❡ ❡①♣r❡ss✐♦♥s ❢♦r ▼♦r❡♦✈❡r✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rr❡♥❝❡ r❡❧❛t✐♦♥ ❢♦r ❝♦♥st❛♥ts ♦r✱ ❛❢t❡r t❤❡ t❡❧❡s❝♦♣✐♥❣ s✉♠♠❛t✐♦♥

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❈♦♠♣❛r✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts ✇❡ ✜♥❞ b(n) = β2n2 + β1n + β2, (fk0(n) − 1)2 ck0(n) = β2n2 + (2β2 + β1 + αk0+1 − αk0) n + γ. ❋✐♥❛❧❧② ✇❡ ♦❜t❛✐♥ t❤❡ ❡①♣r❡ss✐♦♥s ❢♦r fk0 fk0(n) = 1 ±

• c−1

k0 (n)

• β2n2 + (2β2 + β1 + αk0+1 − αk0)n + γ,

▼♦r❡♦✈❡r✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rr❡♥❝❡ r❡❧❛t✐♦♥ ❢♦r ❝♦♥st❛♥ts ♦r✱ ❛❢t❡r t❤❡ t❡❧❡s❝♦♣✐♥❣ s✉♠♠❛t✐♦♥

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❈♦♠♣❛r✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts ✇❡ ✜♥❞ b(n) = β2n2 + β1n + β2, (fk0(n) − 1)2 ck0(n) = β2n2 + (2β2 + β1 + αk0+1 − αk0) n + γ. ❋✐♥❛❧❧② ✇❡ ♦❜t❛✐♥ t❤❡ ❡①♣r❡ss✐♦♥s ❢♦r fk0 fk0(n) = 1 ±

• c−1

k0 (n)

• β2n2 + (2β2 + β1 + αk0+1 − αk0)n + γ,

▼♦r❡♦✈❡r✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rr❡♥❝❡ r❡❧❛t✐♦♥ ❢♦r ❝♦♥st❛♥ts ak αk+1 = αk + αk0+1 − αk0 − 2β2(k − k0), ♦r✱ ❛❢t❡r t❤❡ t❡❧❡s❝♦♣✐♥❣ s✉♠♠❛t✐♦♥ αk = αk0+1 + (k − k0 − 1)(αk0+1 − αk0) − β2(k − k0)(k − k0 − 1).

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

■♥ ❝♦♥❝❧✉s✐♦♥ ✐❢ k0 = 0 t❤❡♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛s ❢♦r t❤❡ r❛✐s✐♥❣ ❛♥❞ ❧♦✇❡r✐♥❣ ♦♣❡r❛t♦rs s❛t✐s❢②✐♥❣ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❛r❡ ❣✐✈❡♥ ❜② A−

k = T+ ±

• c−1

0 (n − k)

• β2(n − k)2+(2β2+β1+α1−α0)(n − k)+γ,

A+

k =

• β2n2+β1n+β0
• T−±

±

• c0(n−k)
• β2(n−k)2+(2β2+β1+α1−α0)(n−k)+γ

❛♥❞ ❢r♦♠ t❤✐s t❤❡ ❡①♣❧✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r Hk ❝❛♥ ❜❡ ❞❡r✐✈❡❞✳ ❆❜♦✈❡ t❤❡ ♣❛r❛♠❡t❡rs✿ β2✱ β1✱ β0✱ γ✱ α1✱ α0 ❛r❡ ❛r❜✐tr❛r② ❛♥❞ {c0} ✐s ❛ ❢r❡❡ s❡q✉❡♥❝❡✳

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❇❛s✐❝ ❛ss✉♠♣t✐♦♥✿ bk(n) = λbk+1(n), λ ∈ R \ {0, 1}

❋r♦♠ ♥♦✇ ♦♥✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ s❡q✉❡♥❝❡ {ck} ✐s ❞❡✜♥❡❞ ❜② ck(n) := 1 fk(n) − 1. ❚❤❡ s②st❡♠ ♦❢ ❝♦♥❞✐t✐♦♥s ✐s r❡❞✉❝❡❞ ♥♦✇ t♦ fk+1(n) − 1 = bk(n) bk+1(n) (fk(n − 1) − 1) , bk(n)−bk+1(n + 1) = αk+1−αk+ bk(n) bk+1(n) ( fk(n − 1) − 1 ) − ( fk(n) − 1 ) .

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❈♦♠♣❛r✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts ✇❡ ✜♥❞ bk(n) = λ−kb0(n) = λ−k β2λ2n + β1λn + β0

• ,

fk(n) − 1 =λ2n−k+1β2 + λn

• f0(0)−λβ2−λ−1β0− 1− α1 − α0

1 − λ

• + λk

α1 − α0 1 − λ +λ−1β0

• ,

αk = λk α0 − α1 1 − λ − λ−k 1 + λ2k−1 β0 + α1 − λα0 1 − λ +

• 1 + λ−1

β0.

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❋✐♥❛❧❧②✱ ✇❡ ❤❛✈❡ t❤❡ ❡①♣❧✐❝✐t ❢♦r♠ ♦❢ ❝r❡❛t✐♦♥ ❛♥❞ ❛♥♥✐❤✐❧❛t✐♦♥ ♦♣❡r❛t♦rs

A−

k = T+ + λ2n−k+1β2 + λn

• f0(0) − λβ2 − λ−1β0 − 1 − α1 − α0

1 − λ

• + λk

α1 − α0 1 − λ + λ−1β0

• ,

A+

k =

• β2λ2n−k + β1λn−k + β0λ−k

T− + 1.

❚❤❡s❡ ♦♣❡r❛t♦rs ❢❛❝t♦r✐③❡ t❤❡ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦rs

Hk = T+ + λ−k β2λ2n + β1λn + β0 λ2n−k−1β2+ +λn−1

• f0(0) − λβ2 − λ−1β0 − 1 − α1 − α0

1 − λ

• +λk

α1 − α0 1 − λ + λ−1β0 T− +λ2n−k (1 + λ) β2+λn

• f0(0) + β1λ−k − λβ2 − λ−1β0 − 1 − α1 − α0

1 − λ

• +α1 − λα0

1 − λ + β0

• 1 + λ−1

.

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❙♦❧✉t✐♦♥s ♦❢ t❤❡ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s

■❢ t❤❡ ♦♣❡r❛t♦rs Hk ❛❞♠✐t t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ Hk = A+

k A− k + αk = A− k+1A+ k+1 + αk+1,

t❤❡♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠ Hkxl

k(n) = λl kxl k(n)

✭✶✮ ❝❛♥ ❜❡ s♦❧✈❡❞✱ {xl

k} ∈ ℓ(Z, R)✱ k, l ∈ Z✱ λl k ∈ R✳ ❋♦r ❛ ❧❛r❣❡ ❝❧❛ss

♦❢ s✉❝❤ ❤♦♠♦❣❡♥❡♦✉s ❧✐♥❡❛r s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♠❡t❤♦❞ ❡♥❛❜❧❡s ✉s t♦ ✜♥❞ ✐♠♠❡❞✐❛t❡❧② t❤❡ ❡✐❣❡♥✈❛❧✉❡s λk ❛♥❞ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✭s♦❧✉t✐♦♥s✮ {xk} ❜② ♠❛♥✉❢❛❝t✉r✐♥❣ ♣r♦❝❡ss✳

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❚❤❡ ✉♥❞❡r❧②✐♥❣ ✐❞❡❛ ✐s t♦ ❝♦♥s✐❞❡r ❛ ♣❛✐r ♦❢ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s A−

k x0 k(n) = 0

♦r A+

k+1x−1 k (n) = 0.

■♥ t❤❡ ❝❛s❡ λ0

k = αk ♦r λ−1 k

= αk+1✱ ❛♥② ♥♦♥③❡r♦ s♦❧✉t✐♦♥ ♦❢ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ❛❧s♦ ❛ s♦❧✉t✐♦♥ ♦❢ ❡q✉❛t✐♦♥ ✭✶✮✳ ▼♦r❡♦✈❡r x1

k+1(n) = A+ k+1x0 k(n)

♦r x−2

k−1(n) = A− k x−1 k (n)

❛r❡ s♦❧✉t✐♦♥s ♦❢ ❡q✉❛t✐♦♥s ✭✶✮ ❢♦r k + 1 ❛♥❞ λ1

k+1 = αk ♦r ❢♦r k − 1

❛♥❞ λ−2

k−1 = αk+1✱ r❡s♣❡❝t✐✈❡❧②✳ ❘❡♣❡❛t✐♥❣ t❤✐s ♣r♦❝❡❞✉r❡ ✇❡ ♠❛②

❣❡t ❛ s♦❧✉t✐♦♥ ♦❢ ✭✶✮ xp−k

p

(n) = A+

p A+ p−1 . . . A+ k+2A+ k+1x0 k(n)

❢♦r p ❛♥❞ λp−k

p

= αk ✭❛❢t❡r ❝❤❛♥❣✐♥❣ k → p✮✱ p > k✱ ♦r xr−k

r

(n) = A−

r+1A− r+2 . . . A− k−1A− k x−1 k (n)

❢♦r r ❛♥❞ λr−k

r

= αk ✭❛❢t❡r ❝❤❛♥❣✐♥❣ k → r✮✱ r < k✳

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❋✐♥❛❧❧②✱ ✇❡ ♣r❡s❡♥t t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ r❛✐s✐♥❣ ♦♣❡r❛t♦rs ♦♥ t❤❡ ❞✐❛❣r❛♠✿ l {x3

0}, α−3

{x3

1}, α−2

{x3

2}, α−1

{x2

−1}, α−3

A+

• {x2

0}, α−2

A+

1

• {x2

1}, α−1

A+

2

• {x2

2}, a0

{x1

−1}, α−2

A+

• {x1

0}, α−1

A+

1

• {x1

1}, α0

A+

2

• {x1

2}, α1

{x0

−1}, a−1

• A+
• {x0

0}, α0

A+

1

• A−
• {x0

1}, α1

A+

2

• A−

1

• {x0

2}, α2

A−

2

• k

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❲❡ ♣r❡s❡♥t t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❧♦✇❡r✐♥❣ ♦♣❡r❛t♦rs ♦♥ t❤❡ ❞✐❛❣r❛♠✿ l

• k

{x−1

−1}, a−4

A+

• {x−1

0 }, α0

A−

• A+

1

• {x−1

1 }, α1

A−

1

• A+

2

• {x−1

2 }, α2

A−

2

• {x−2

−1}, α0

{x−2

0 }, α1

A−

• {x−2

1 }, α2

A−

1

• {x−2

2 }, α3

A−

2

• {x−3

−1}, α1

{x−3

0 }, α2

A−

• {x−3

1 }, α3

A−

1

• {x−3

2 }, α4

A−

2

• {x−4

−1}, α2

{x−4

0 }, α3

• {x−4

1 }, α4

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❊①❛♠♣❧❡ ✶

▲❡t ✉s ❝♦♥s✐❞❡r ❛ ❝❛s❡ ✇❤❡♥ β2 = β1 = γ = 0✱ β0 = 1 ❛♥❞ fk(n) = 0 ❢♦r k ∈ Z✱ t❤❡♥ c0(n) = (α1 − α0)n✳ ❚❤❡ r❛✐s✐♥❣ ❛♥❞ ❧♦✇❡r✐♥❣ ♦♣❡r❛t♦rs ❛r❡ ❣✐✈❡♥ ❜② A−

k = T+ − 1,

A+

k = T− − (α1 − α0)(n − k).

❚❤❡ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✭✶✮ ♣❛r❛♠❡tr✐③❡❞ ❜② ♦♥❡ ♣❛r❛♠❡t❡r l ∈ Z ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ♥♦✇ ✐♥ t❤❡ ❢♦r♠

• −(α1 −α0)(n−k)T+ −T− +(α1 −α0)(n−k +l)+1
• xl

k(n) = 0.

❚❤❡ s♦❧✉t✐♦♥s ❢♦r l ∈ N ∪ {0} ✐♥ t❤✐s ❝❛s❡ ❛r❡ ♦❢ t❤❡ ❢♦r♠

l = 0 x0

k(n) = 1,

l = 1 x1

k(n) = −(α1 − α0)(n − k) + 1,

l = 2 x2

k(n) = (α1 − α0)(n − k)2 − 2(α1 − α0)(n − k) + (α1 − α0)2 + 1,

. . .

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❛♥❞ s✐♠✐❧❛r❧② ❢♦r l < 0

l = −1 x−1

k (n) =

λ (α1 − α0)n(n − k − 1)!, l = −2 x−2

k (n) =

λ (α1 − α0)n+1(n − k)! − λ (α1 − α0)n(n − k − 1)!, ✳ ✳ ✳ l = −m − 1 x−m−1

k

(n) =

m

• i=0

m i

• (−1)iλ

(α1 − α0)n+m(n − k + m − 1 − i)!.

✇❤❡r❡ λ ∈ R✳

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❊①❛♠♣❧❡ ✷

▲❡t ✉s ❝♦♥s✐❞❡r ❛ ❝❛s❡ ✇❤❡♥ fk(n) ≡ 0. ❚❤❡♥ bk+1(n) = bk(n) =: b(n), ck(n) = c0(n − k), b(n) − b(n + 1) = αk+1 − αk + c0(n − k − 1) − c0(n − k). ❚❤❡♥✱ t❤❡ r❡❧❛t✐♦♥ Hkxl

k(n) = λl kxl k(n) ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡

❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♦❢ ❤②♣❡r❣❡♦♠❡tr✐❝ t②♣❡✿

• − b(n)∇ + b(n) − c0(n − k)
• ∆xl

k(n) = (λl k − αk)xl k(n).

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❲❡ tr❛♥s❢♦r♠ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐♥t♦ t❤❡ st❛♥❞❛r❞ ❢♦r♠ σ(n)∆∇xl

k(n) + τ(n)∆xl k(n) + λxl k(n) = 0,

✇❤❡r❡ σ(n) = −b(n), τ(n) = b(n) − c0(n − k), λ = αk − λl

k = αk − αk−l = −l

• τ ′(n) + l − 1

2 σ′′(n)

• .

■t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❞❡s❝r✐❜❡s ❝❧❛ss✐❝❛❧ ♦rt❤♦❣♦♥❛❧ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ s✉❝❤ ❛s t❤❡ ❈❤❛r❧✐❡r✱ ▼❡✐①♥❡r✱ ❑r❛✈❝❤✉❦✱ ❍❛❤♥ ♣♦❧②♥♦♠✐❛❧s✳

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳

❊①❛♠♣❧❡ ✸

▲❡t ✉s ❝♦♥s✐❞❡r ❛ ❝❛s❡ ✇❤❡♥ λ = 1

2✱ β2 = β1 = 0✱ β0 = 1✱ α1 = 1

❛♥❞ α0 = 2✳ ❚❤✐s ❣✐✈❡s ✉s ❢♦r♠✉❧❛s ❢♦r t❤❡ ❝r❡❛t✐♦♥ ❛♥❞ ❛♥♥✐❤✐❧❛t✐♦♥ ♦♣❡r❛t♦rs A−

k = T+ + 2−n(f0(0) − 1),

A+

k = 2kT− + 1.

❲❡ ♦❜t❛✐♥ t❤❡ ❝❤❛✐♥ ♦❢ ♦♣❡r❛t♦rs ❢♦r k ∈ Z ♦❢ t❤❡ ❢♦r♠ Hk = T+ + 2k−n+1(f0(0) − 1)T− + 2−n(f0(0) − 1) + 3. ❚❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦❢ t❤❡s❡ ♦♣❡r❛t♦rs ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ ❢♦r♠ ψl

k(n) =

• 2kT− + 1
• . . .
• 2k−l+1T− + 1
• ψ0

k−l(n)

❢♦r ❡✐❣❡♥✈❛❧✉❡ λl

k = αk−l = 3 − 2k−l✱ ✇❤❡r❡ t❤❡ ❣r♦✉♥❞ st❛t❡s ❛r❡

❣✐✈❡♥ ❜② ψ0

k(n) = (1 − f0(0))n2− n(n−1)

n

ψ0

k(0).

❆❧✐♥❛ ❉♦❜r♦❣♦✇s❦❛ ❙❡❝♦♥❞ ♦r❞❡r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✳✳✳