A review on harmonic wavelets and their fractional extension

❱❖▲❯▼❊✿ ✷

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

❆ ❘❡✈✐❡✇ ♦♥ ❍❛r♠♦♥✐❝ ❲❛✈❡❧❡ts ❛♥❞ ❚❤❡✐r
❋r❛❝t✐♦♥❛❧ ❊①t❡♥s✐♦♥
❈❛r❧♦ ❈❆❚❚❆◆■ 1,2,∗
1

❊♥❣✐♥❡❡r✐♥❣ ❙❝❤♦♦❧✱ ❉❊■▼✱ ❚✉s❝✐❛ ❯♥✐✈❡rs✐t②✱ ❱✐t❡r❜♦✱ ■t❛❧②

2

❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✱ ❱✐❡t♥❛♠

✯❈♦rr❡s♣♦♥❞✐♥❣ ❆✉t❤♦r✿ ❈❛r❧♦ ❈❆❚❚❆◆■ ✭❡♠❛✐❧✿ ❝❛tt❛♥✐❅✉♥✐t✉s✳✐t✮
✭❘❡❝❡✐✈❡❞✿ ✷✵✲❉❡❝❡♠❜❡r✲✷✵✶✽❀ ❛❝❝❡♣t❡❞✿ ✷✻✲❉❡❝❡♠❜❡r✲✷✵✶✽❀ ♣✉❜❧✐s❤❡❞✿ ✸✶✲❉❡❝❡♠❜❡r✲✷✵✶✽✮
❉❖■✿ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✷✺✵✼✸✴❥❛❡❝✳✷✵✶✽✷✹✳✷✷✺

❆❜str❛❝t✳ ■♥ t❤✐s ♣❛♣❡r ❛ r❡✈✐❡✇ ♦♥ ❤❛r✲
♠♦♥✐❝ ✇❛✈❡❧❡ts ❛♥❞ t❤❡✐r ❢r❛❝t✐♦♥❛❧ ❣❡♥❡r❛❧✐③❛✲
t✐♦♥✱ ✇✐t❤✐♥ t❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s✱ ✇✐❧❧
❜❡ ❞✐s❝✉ss❡❞✳ ❚❤❡ ♠❛✐♥ ♣r♦♣❡rt✐❡s ♦❢ ❤❛r♠♦♥✐❝
✇❛✈❡❧❡ts ❛♥❞ ❢r❛❝t✐♦♥❛❧ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ✇✐❧❧
❜❡ ❣✐✈❡♥✱ ❜② t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t ♦❢ t❤❡✐r ❝❤❛r❛❝✲
t❡r✐st✐❝ ❢❡❛t✉r❡s ✐♥ t❤❡ ❋♦✉r✐❡r ❞♦♠❛✐♥✳ ■t ✇✐❧❧
❜❡ s❤♦✇♥ t❤❛t t❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢
❢r❛❝t✐♦♥❛❧ ✇❛✈❡❧❡ts ❤❛✈❡ ❛ ✈❡r② s✐♠♣❧❡ ❡①♣r❡s✲
s✐♦♥ t❤✉s ♦♣❡♥✐♥❣ ♥❡✇ ❢r♦♥t✐❡rs ✐♥ t❤❡ s♦❧✉t✐♦♥
♦❢ ❢r❛❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ♣r♦❜❧❡♠s✳

❧✉t✐♦♥ ❜② s♦♠❡ ✇❛✈❡❧❡t s❡r✐❡s ❛♥❞ t❤❡♥ ❜② ❝♦♠✲
♣✉t✐♥❣ t❤❡ ✐♥t❡❣r❛❧s ✭♦r ❞❡r✐✈❛t✐✈❡s✮ ♦❢ t❤❡ ❜❛s✐❝
✇❛✈❡❧❡t ❢✉♥❝t✐♦♥s✱ t♦ ❝♦♥✈❡rt t❤❡ st❛rt✐♥❣ ❞✐✛❡r✲
❡♥t✐❛❧ ♣r♦❜❧❡♠ ✐♥t♦ ❛♥ ❛❧❣❡❜r❛✐❝ s②st❡♠ ❢♦r t❤❡
✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ✭s❡❡ ❡✳❣✳ ❬✷✻✕✸✵❪✮✳
❲❛✈❡❧❡ts ❛r❡ s♦♠❡ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭s❡❡ ❡✳❣✳
❬✺✱ ✾✱ ✷✹❪✮ ✇❤✐❝❤ ❞❡♣❡♥❞ ♦♥ t✇♦ ♣❛r❛♠❡t❡rs✱ t❤❡
s❝❛❧❡ ♣❛r❛♠❡t❡r ✭❛❧s♦ ❝❛❧❧❡❞ r❡✜♥❡♠❡♥t✱ ❝♦♠✲
♣r❡ss✐♦♥✱ ♦r ❞✐❧❛t✐♦♥ ♣❛r❛♠❡t❡r✮ ❛♥❞ ❛ t❤❡ ❧♦❝❛❧✲
✐③❛t✐♦♥ ✭tr❛♥s❧❛t✐♦♥✮ ♣❛r❛♠❡t❡r✳ ❚❤❡s❡ ❢✉♥❝t✐♦♥s
❢✉❧✜❧❧ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❛①✐♦♠s ♦❢ ♠✉❧t✐r❡s♦❧✉t✐♦♥
❛♥❛❧②s✐s s♦ t❤❛t ❜② ❛ s✉✐t❛❜❧❡ ❝❤♦✐❝❡ ♦❢ t❤❡ s❝❛❧❡
❛♥❞ tr❛♥s❧❛t✐♦♥ ♣❛r❛♠❡t❡r ♦♥❡ ✐s ❛❜❧❡ t♦ ❡❛s✐❧②

❑❡②✇♦r❞s
❍❛r♠♦♥✐❝

❛♥❞ q✉✐❝❦❧② ❛♣♣r♦①✐♠❛t❡ ✭❛❧♠♦st✮ ❛❧❧ ❢✉♥❝t✐♦♥s

✇❛✈❡❧❡ts✱

❧♦❝❛❧

❢r❛❝t✐♦♥❛❧

✭❡✈❡♥ t❛❜✉❧❛r✮ ✇✐t❤ ❞❡❝❛② t♦ ✐♥✜♥✐t②✳
❚❤❡r❡❢♦r❡ ✇❛✈❡❧❡ts s❡❡♠s t♦ ❜❡ t❤❡ ♠♦r❡ ❡①✲

❞❡r✐✈❛t✐✈❡✱ ✇❛✈❡❧❡t s❡r✐❡s✳

♣❡❞✐❡♥t t♦♦❧ ❢♦r st✉❞②✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ♣r♦❜❧❡♠s
✇❤✐❝❤ ❛r❡ ❧♦❝❛❧✐③❡❞ ✭✐♥ t✐♠❡ ♦r ✐♥ ❢r❡q✉❡♥❝②✮✳
❚❤❡r❡ ❡①✐sts ❛ ✈❡r② ❧❛r❣❡ ❧✐t❡r❛t✉r❡ ❞❡✈♦t❡❞ t♦

✶✳

✇❛✈❡❧❡t s♦❧✉t✐♦♥ ♦❢ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❛♥❞ ✐♥t❡✲

■♥tr♦❞✉❝t✐♦♥

❣r❛❧ ❡q✉❛t✐♦♥s ✭s❡❡ ❡✳❣✳ t❤❡ ♣✐♦♥❡r✐st✐❝ ✇♦r❦s ❬✶✵✱
✶✸✱ ✷✺✱ ✸✺❪✮ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✭s❡❡ ❡✳❣✳ ❬✶✶✱ ✷✸✱ ✸✹❪

❍❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ❛r❡ s♦♠❡ ❦✐♥❞ ♦❢ ❝♦♠♣❧❡①

❛♥❞ ♠♦r❡ ❣❡♥❡r❛❧ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

✇❛✈❡❧❡ts ❬✶✕✾❪ ✇❤✐❝❤ ❛r❡ ❛♥❛❧✐t✐❝❛❧❧② ❞❡✜♥❡❞✱ ✐♥✲

❛♥❞ ♦♣❡r❛t♦rs ✭s❡❡ ❡✳❣✳ ❬✷✻✕✸✵❪✮✳

✜♥✐t❡❧② ❞✐✛❡r❡♥t✐❛❜❧❡✱ ❛♥❞ ❜❛♥❞✲❧✐♠✐t❡❞ ✐♥ t❤❡
❋♦✉r✐❡r ❞♦♠❛✐♥✳

❆❧t❤♦✉❣❤ t❤❡ s❧♦✇ ❞❡❝❛② ✐♥

t❤❡ s♣❛❝❡ ❞♦♠❛✐♥✱ t❤❡✐r s❤❛r♣ ❧♦❝❛❧✐③❛t✐♦♥ ✐♥ ❢r❡✲
q✉❡♥❝②✱ ✐s ❛ ❣♦♦❞ ♣r♦♣❡rt② ❡s♣❡❝✐❛❧❧② ❢♦r t❤❡
❛♥❛❧②s✐s ♦❢ ✇❛✈❡ ❡✈♦❧✉t✐♦♥ ♣r♦❜❧❡♠s ✭s❡❡ ❡✳❣✳
❬✶✕✸✱ ✶✵✱ ✶✸✱ ✶✺✱ ✶✻✱ ✷✺✱ ✸✷✱ ✸✸❪✳ ■♥ t❤❡ s❡❛r❝❤ ❢♦r ♥✉✲

♠❡r✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♣r♦❜❧❡♠s✱

❇② ✉s✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡s ✭♦r ✐♥t❡❣r❛❧s✮ ♦❢ t❤❡
✇❛✈❡❧❡t ❜❛s✐s t❤❡ P❉❊ ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ tr❛♥s✲
❢♦r♠❡❞ ✐♥t♦ ❛♥ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s②st❡♠ ♦❢ ♦r✲
❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❇② ✜①✐♥❣ t❤❡ s❝❛❧❡
♦❢ ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡ ♣r♦❥❡❝t✐♦♥ ❝♦rr❡s♣♦♥❞ t♦
t❤❡ ❝❤♦✐❝❡ ♦❢ ❛ ✜♥✐t❡ s❡t ♦❢ ✇❛✈❡❧❡t s♣❛❝❡s✱ t❤✉s

t❤❡ ♠❛✐♥ ✐❞❡❛ ✐s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ✉♥❦♥♦✇♥ s♦✲

✷✷✹

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮


❱❖▲❯▼❊✿ ✷

♦❜t❛✐♥✐♥❣ t❤❡ ♥✉♠❡r✐❝❛❧ ✭✇❛✈❡❧❡t✮ ❛♣♣r♦①✐♠❛✲
t✐♦♥✳

❚❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✿

✐♥ s❡❝✲

t✐♦♥ ✷ s♦♠❡ ♣r❡❧✐♠✐♥❛r② ❞❡✜♥✐t✐♦♥s ❛❜♦✉t ❤❛r✲

❇② ✉s✐♥❣ t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ♦❢ t❤❡ ✇❛✈❡❧❡t ❜❛✲
s✐s ❛♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦❢
t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ♦r ✐♥✲

t❡❣r❛❧s ✭♦♣❡r❛t✐♦♥❛❧ ♠❛tr✐①✱ ❛❧s♦ ❝❛❧❧❡❞ ❝♦♥♥❡❝✲
t✐♦♥ ❝♦❡✣❝✐❡♥ts✮✱ ✇❡ ❝❛♥ ❝♦♥✈❡rt t❤❡ ❞✐✛❡r❡♥t✐❛❧
♣r♦❜❧❡♠ ✐♥t♦ ❛♥ ❛❧❣❡❜r❛✐❝ s②st❡♠ ❛♥❞ t❤✉s ✇❡
❝❛♥ ❡❛s✐❧② ❞❡r✐✈❡ t❤❡ ✇❛✈❡❧❡t ❛♣♣r♦①✐♠❛t❡ s♦❧✉✲
t✐♦♥✳

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ✜①❡❞

s❝❛❧❡ ✭♦❢ ❛♣♣r♦①✐♠❛t✐♦♥✮ ❛♥❞ ♦♥ t❤❡ ♥✉♠❜❡r ♦❢
❞✐❧❛t❡❞ ❛♥❞ tr❛♥s❧❛t❡❞ ✐♥st❛♥❝❡s ♦❢ t❤❡ ✇❛✈❡❧❡ts✳
❍♦✇❡✈❡r✱ ❞✉❡ t♦ t❤❡✐r ❧♦❝❛❧✐③❛t✐♦♥ ♣r♦♣❡rt② ❥✉st
❛ ❢❡✇ ✐♥st❛♥❝❡s ❛r❡ ❛❜❧❡ t♦ ❝❛♣t✉r❡ t❤❡ ♠❛✐♥ ❢❡❛✲

♠♦♥✐❝ ✭❝♦♠♣❧❡① ✇❛✈❡❧❡ts✮ t♦❣❡t❤❡r ✇✐t❤ t❤❡✐r
❢r❛❝t✐♦♥❛❧ ❝♦✉♥t❡r♣❛rts ❛r❡ ❣✐✈❡♥✳ ❚❤❡ ❤❛r♠♦♥✐❝
✇❛✈❡❧❡t r❡❝♦♥str✉❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ✐s ❞❡s❝r✐❜❡❞
✐♥ s❡❝t✐♦♥ ✸✳

■♥ t❤❡ s❛♠❡ s❡❝t✐♦♥✱ t❤❡ ❤❛r✲

♠♦♥✐❝ ✇❛✈❡❧❡t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧
❤❛r♠♦♥✐❝ ❢✉♥❝t✐♦♥s ✇✐❧❧ ❜❡ ❛❧s♦ ❣✐✈❡♥✳

❙❡❝✲

t✐♦♥ ✹ s❤♦✇s s♦♠❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢❡❛t✉r❡s ♦❢ ❤❛r✲
♠♦♥✐❝ ✇❛✈❡❧❡ts✳ ■♥ s❡❝t✐♦♥ ✺ t❤❡ ❜❛s✐❝ ❞❡✜♥✐t✐♦♥s
❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s ❛r❡

❣✐✈❡♥ ❛♥❞ t❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡
❢r❛❝t✐♦♥❛❧ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ✇✐❧❧ ❜❡ ❡①♣❧✐❝✐t❧②
❝♦♠♣✉t❡❞✳

t✉r❡ ♦❢ t❤❡ s✐❣♥❛❧✱ ❛♥❞ ❢♦r t❤✐s r❡❛s♦♥ ✐t ✐s ❡♥♦✉❣❤
t♦ ❝♦♠♣✉t❡ ❛ ❢❡✇ ♥✉♠❜❡r ♦❢ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts
t♦ q✉✐❝❦❧② ❣❡t ❛ q✉✐t❡ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡

✷✳

s♦❧✉t✐♦♥✳

❍❛r♠♦♥✐❝ ✭◆❡✇❧❛♥❞✮

■♥ r❡❝❡♥t ②❡❛rs t❤❡r❡ ❤❛s ❜❡❡♥ ❛ ❢❛st r✐s✐♥❣

❲❛✈❡❧❡ts

✐♥t❡r❡st ❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ♣r♦❜❧❡♠s✳
■♥❞❡❡❞ t❤❡ ✐❞❡❛ ♦❢ ❢r❛❝t✐♦♥❛❧ ♦r❞❡r ❞❡r✐✈❛t✐✈❡
✐s ❞❡❡♣❧② r♦♦t❡❞ ✐♥ t❤❡ ❤✐st♦r② ♦❢ ♠❛t❤❡♠❛t✲

❍❛r♠♦♥✐❝

✐❝s✱ s✐♥❝❡ ❛❧r❡❛❞② ❈❛✉❝❤② ✇❛s ✇♦♥❞❡r✐♥❣ ❛❜♦✉t

✇❛✈❡❧❡ts ❬✶✱ ✸✱ ✺✱ ✼✱ ✽❪ ❛r❡ ❝♦♠♣❧❡① ♦rt❤♦♥♦r♠❛❧

t❤❡ ♣♦ss✐❜❧❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♦r❞✐♥❛r② ❞✐✛❡r❡♥✲

✇❛✈❡❧❡ts t❤❛t ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡ s❤❛r♣❧②

t✐❛❧ ♦♣❡r❛t♦rs t♦ ❢r❛❝t✐♦♥❛❧ ♦r❞❡r ❞✐✛❡r❡♥t✐❛❧ ♦♣✲

❜♦✉♥❞❡❞ ❢r❡q✉❡♥❝② ❛♥❞ s❧♦✇ ❞❡❝❛② ✐♥ t❤❡ s♣❛❝❡

❡r❛t♦rs✳

♦❢ ✈❛r✐❛❜❧❡✳ ▲✐❦❡ ❛♥② ♦t❤❡r ✇❛✈❡❧❡t t❤❡② ❞❡♣❡♥❞

❚❤❡ ♠❛✐♥ ❛❞✈❛♥t❛❣❡ ♦❢ ❢r❛❝t✐♦♥❛❧ ♦r✲

✇❛✈❡❧❡ts

❛❧s♦

❦♥♦✇♥

❛s

◆❡✇❧❛♥❞

❞❡r ❞❡r✐✈❛t✐✈❡ ✐s t♦ ❤❛✈❡ ❛♥ ❛❞❞✐t✐♦♥❛❧ ♣❛r❛♠❡t❡r

❜♦t❤ ♦♥ t❤❡ s❝❛❧❡ ♣❛r❛♠❡t❡r

✭t❤❡ ♦r❞❡r ♦❢ ❞❡r✐✈❛t✐✈❡✮ t♦ ❜❡ ✉s❡ ✐♥ t❤❡ ❛♥❛❧②s✐s

❞❡❣r❡❡ ♦❢ r❡✜♥❡♠❡♥t✱ ❝♦♠♣r❡ss✐♦♥✱ ♦r ❞✐❧❛t✐♦♥

♦❢ ❞✐✛❡r❡♥t✐❛❧ ♣r♦❜❧❡♠s✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ t❤❡

❛♥❞ ♦♥ ❛ s❡❝♦♥❞ ♣❛r❛♠❡t❡r

♠❛✐♥ ❞r❛✇❜❛❝❦ ❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ♦♣✲

t♦ t❤❡ s♣❛❝❡ ❧♦❝❛❧✐③❛t✐♦♥✳

❡r❛t♦rs ✐s t❤❛t t❤✐s ❞❡r✐✈❛t✐✈❡ ✐s ♥♦t ✉♥✐✈♦❝❛❧❧②

♠♦♥✐❝ ✇❛✈❡❧❡ts ❢✉❧✜❧❧ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❛①✐♦♠s ♦❢

❞❡✜♥❡❞ ✭s❡❡ ❡✳❣✳ ❬✶✾✕✷✷❪ ❛♥❞ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✮✳

♠✉❧t✐r❡s♦❧✉t✐♦♥ ❛♥❛❧②s✐s ✭s❡❡ ❡✳❣✳ ❬✷✹❪✮✱ ❜✉t t❤❡②

❲❡ ✇✐❧❧ ♥♦t ❣♦ ❞❡❡♣❧② ✐♥t♦ t❤✐s s✉❜❥❡❝t✱ s✐♥❝❡

❛❧s♦ ❡♥❥♦② s♦♠❡ ♠♦r❡ s♣❡❝✐❛❧ ❢❡❛t✉r❡s ❡s♣❡❝✐❛❧❧②

✇❡ ✇✐❧❧ ❢♦❝✉s ♦♥❧② ♦♥ ❛ s♣❡❝✐❛❧ ❢r❛❝t✐♦♥❛❧ ♦♣❡r✲

✐♥ t❤❡ ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥✳

n
k

✇❤✐❝❤ ❞❡✜♥❡ t❤❡
✇❤✐❝❤ ✐s r❡❧❛t❡❞

❆s ✇❡ ✇✐❧❧ s❡❡✱ ❤❛r✲

❛t♦r✱ t❤❡ s♦✲❝❛❧❧❡❞ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡✱ ❛s
❞❡✜♥❡❞ ❜② ❨❛♥❣ ❬✶✷✱ ✸✶✱ ✸✻✱ ✸✼❪✳
❚❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ✇❤❡♥ ❛♣♣❧✐❡❞ t♦
t❤❡ ♠♦st ♣♦♣✉❧❛r ❢✉♥❝t✐♦♥s ❣✐✈❡ ❛ ♥❛t✉r❛❧ ❣❡♥❡r✲

✷✳✶✳

❍❛r♠♦♥✐❝ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥

❛❧✐③❛t✐♦♥ ♦❢ ❦♥♦✇♥ r❡s✉❧ts ❛♥❞ ❢✉❧✜❧❧s t❤❡ ❜❛s✐❝❛
❛①✐♦♠s ♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s✳

❚❤❡ ❤❛r♠♦♥✐❝ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❢t❡r r❡✈✐❡✇✐♥❣ ♦♥ t❤❡ ❝❧❛ss✐✲
❝❛❧ ❍❛r♠♦♥✐❝ ✇❛✈❡❧❡t✱ t❤❡ ❢r❛❝t✐♦♥❛❧ ❤❛r♠♦♥✐❝
✇❛✈❡❧❡ts ✇✐❧❧ ❜❡ ❞❡✜♥❡❞✳

❞❡❢

▼♦r❡♦✈❡r t❤❡✐r ❧♦✲

ϕ(x) =

❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s ✇✐❧❧ ❜❡ ❡①♣❧✐❝✐t❧② ❝♦♠✲
♣✉t❡❞✳

■t

✇✐❧❧

❜❡

s❤♦✇♥

t❤❛t

t❤❡s❡

e2πix − 1
2πix

✭✶✮

❢r❛❝✲

t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s✱ ❛r❡ s♦♠❡ ❦✐♥❞ ♦❢ ❣❡♥❡r❛❧✐③❛✲

t❤❛t ✐s

t✐♦♥ ❛❧r❡❛❞② ♦❜t❛✐♥❡❞ ❢♦r t❤❡ s♦ ❝❛❧❧❡❞ ❙❤❛♥✲
♥♦♥ ✇❛✈❡❧❡ts ❬✶✼✱ ✶✽❪ ❛♥❞ t❤❡ s✐♥❝✲❞❡r✐✈❛t✐✈❡
❬✶✾✱ ✷✵✱ ✷✷❪

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮

ϕ(x) =

sin(2πx)
1 − cos(2πx)
+i
2πx
2πx

✷✷✺


❱❖▲❯▼❊✿ ✷

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

1
Re(φ )

0.6

Im (φ )

π

π

-

1
2

0.25

2

- 0.7

0.2

-0.2

❋✐❣✳ ✶✿ P❧♦t ♦❢ t❤❡ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡

❋✐❣✳ ✷✿ P❧♦t ♦❢ t❤❡ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡

(0 ≤ x ≤ 4)✳

(0 ≤ x ≤ 4)✳

t❤❡r❡ ❢♦❧❧♦✇ t❤❡ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rt ♦❢ t❤❡

❞❡❢

sin(2πx)
[ϕ(x)] =
,
2πx

❞❡❢

1 − cos(2πx)
.

[ϕ(x)] =
2πx

ϕr (x) =

r❡❛❧

❚❤❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

ϕr (x)

❛♥❞

✷✳✷✳

ϕi (x)}

♦❢ t❤❡ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ r❡❛❧ ♣❧❛♥❡

{ϕr (x), ϕi (x)} ♦❢ t❤❡
ϕ(x) ✐s s❤♦✇♥ ✐♥ ❋✐❣✳

♣❛rt

✭✺✮

❋r❛❝t✐♦♥❛❧ ♣r♦❧✉♥❣❛t✐♦♥ ♦❢
t❤❡ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥

✱

❚❤❡ ♣❛r❛♠❡tr✐❝ ♣❧♦t

❝♦♠♣❧❡① s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥

❚❤❡ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥ ✭✶✮ ✐s t❤❡ ♣♦✇❡r s❡r✐❡s✱ ✇✐t❤
❝♦♠♣❧❡① ❝♦❡✣❝✐❡♥ts✱

✷✳

ϕ(x) =

■t ❝❛♥ ❜❡ ❡❛s✐❧② s❡❡♥ t❤❛t

e2πix − 1
=
2πix

∞

k=0

(2πi)k k
x
(k + 1)!

✭✻✮

▲❡t ✉s s❧✐❣❤t❧② ♠♦❞✐❢② t❤❡ ❤❛r♠♦♥✐❝ s❝❛❧✐♥❣

lim ϕr (x) = lim ϕi (x) = 0

x→∞

1 − e−2πix
.
2πix

✭✷✮

♦❢

❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳

✐♠❛❣✐♥❛r②

ϕ(x) =

P❧♦ts

x→∞

❢✉♥❝t✐♦♥ ❜② ✉s✐♥❣ t❤❡ ▼✐tt❛❣✲▲❡✤❡r ❢✉♥❝t✐♦♥✱
✐♥st❡❛❞ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧✳ ❙♦ t❤❛t ✇❡ ❤❛✈❡

❛♥❞

lim ϕr (x) = 1,

x→0

lim ϕi (x) = 0

❞❡❢

x→0

ϕα (x) =

▼♦r❡♦✈❡r✱ s✐♥❝❡

eπin =


 1,
 −1,

Eα (2απix) − 1
,
2πix

❜❡✐♥❣

❞❡❢

k∈Z

n = 2k,
n = 2k + 1,

k∈Z

ϕ(n) = 0,

k=0

(0 ≤ α ≤ 1)

xαk
.
Γ(αk + 1)

✭✼✮

✭✽✮

t❤❡ ▼✐tt❛❣✲▲❡✤❡r ❢✉♥❝t✐♦♥✳
❲❤❡♥

n ∈ Z.

∞

Eα (x) =

✭✸✮

✐t ✐s✱ ✐♥ ♣❛rt✐❝✉❧❛r✱

✷✷✻

ϕ(x) ✐s

t❤❡ ❢✉♥❝t✐♦♥

s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥

ϕi (x) =

1

✭✹✮

α = 1✱

♥❛♠❡❧② ✇❡ ❤❛✈❡

ϕ1 (x) → ϕ(x)

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮


❱❖▲❯▼❊✿ ✷

✇❤✐❧❡ ❢♦r

α = 0✱

✷✳✹✳

✐t ✐s

ϕ0 (x) → δ(x)
✇❤❡r❡

δ(x)

0,
1,

❞❡❢

x=0
x=0

❇② ❛ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ✇❡ ❤❛✈❡ t❤❡

s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥

❍❛r♠♦♥✐❝ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥

❚❤❡♦r❡♠ ✶✳ ❚❤❡ ❤❛r♠♦♥✐❝ ✭◆❡✇❧❛♥❞✮ ✇❛✈❡❧❡t
❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s ❬✸✱ ✹✱ ✼✱ ✽❪

✐s t❤❡ ❉✐r❛❝ ❞❡❧t❛

δ(x) =

ψ(x) =

e4πix − e2πix
= e2πix ϕ(x)
2πix

❞❡❢

E 2παix − 1
=
2παix

✭✶✸✮

❢r❛❝t✐♦♥❛❧ ❛♥❞ ✐ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s
ψ(ω) =

ϕα (x) =

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

1
χ(ω)
2π

✭✶✹✮

∞

(2πi)k
xk ,
αΓ(k + α + 1)

k=0
(0 ≤ α ≤ 1)

Pr♦♦❢✿

❙t❛rt✐♥❣ ❢r♦♠

ϕ(x)

✇❡ ❤❛✈❡ t♦ ❞❡✜♥❡

❛ ✜❧t❡r ❛♥❞ t♦ ❞❡r✐✈❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✇❛✈❡❧❡t
❢✉♥❝t✐♦♥ ✭s❡❡ ❡✳❣✳ ❬✼❪✮✳ ❋r♦♠ ✭✶✵✮ ✇❡ ❤❛✈❡

✭✾✮

ω
1
χ(2π + ω)χ(2π + )
2π
2
ω
= χ(2π + ω)ϕ(
ˆ )
2

ϕ (ω) =
✷✳✸✳

❙❝❛❧✐♥❣ ❢✉♥❝t✐♦♥ ✐♥ ❋♦✉r✐❡r

❞♦♠❛✐♥

s♦ t❤❛t✱

ϕ (ω) = H

❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥ ✭✶✮
✐s ❞❡✜♥❡❞ ❛s

1
ϕ(ω) = ϕ(x) =
2π
❞❡❢

✇✐t❤

∞
−∞

❙♦ t❤❛t✱ ✐♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥✱ ✐✳❡✳ ✇✐t❤ r❡✲
s♣❡❝t t♦ t❤❡ ✈❛r✐❛❜❧❡

ω

ω
2

H

ϕ(x)e−iωx ❞x.

t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s

❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ ❝♦♠♣❛❝t s✉♣♣♦rt ✭✐✳❡✳ ✇✐t❤ ❛

■♥ ♦r❞❡r t♦ ❤❛✈❡ ❛ ♠✉❧t✐r❡s♦❧✉t✐♦♥ ❛♥❛❧②s✐s ❬✸✱
✺✱ ✼✱ ✷✹❪ t❤❡ ✇❛✈❡❧❡t ❢✉♥❝t✐♦♥ ♠✉st ❜❡ ❞❡✜♥❡❞ ❛s
✭s❡❡ ❡✳❣✳ ❬✷✹❪✮

ω
ω
± 2π ϕ
2
2

ψ (ω) = H
1
χ(2π + ω)
2π

✭✶✵✮

χ(ω) ❜❡✐♥❣ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛s
❞❡❢

χ(ω) =

1, 2π ≤ ω ≤ 4π,
0, elsewhere.

✇❤❡r❡ t❤❡ ❜❛r st❛♥❞s ❢♦r ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t✐♦♥✳
❲✐t❤ t❤❡ ✜❧t❡r

❢✉♥❝t✐♦♥ t❤✉s ❜❡✐♥❣ ❞❡✜♥❡❞ ✐♥ ❛ s❤❛r♣ ❞♦♠❛✐♥
✇✐t❤ s❧♦✇ ❞❡❝❛② ✐♥ ❢r❡q✉❡♥❝②✳
✇❤✐❧❡ ✇✐t❤

❤❛✈❡ ❛t t❤❡ ✜rst ❛♣♣r♦①✐♠❛t✐♦♥

ψ (ω) =
ϕ(ω) =

2π
δ(ω)
αΓ(1 + α)

✭✶✷✮

ω
− 2π = χ(ω)
2

✇❡ ❤❛✈❡

ω
ω
− 2π ϕˆ
2
2
ω

1
χ 2π +
= χ (ω)
2π
2
1
=
χ (ω)
2π

❚❤❡ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥ ✐♥ ❋♦✉r✐❡r ❞♦♠❛✐♥ ✐s ❜♦①✲

❢✉♥❝t✐♦♥ ✭✾✮ ❝❛♥ ❜❡ ❛❧s♦ ❝♦♠♣✉t❡❞ s♦ t❤❛t ✇❡

H

ψ (ω) = H

✭✶✶✮

❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧ s❝❛❧✐♥❣

ω
ω
ϕ
2
2

= χ(2π + ω).

❜♦✉♥❞❡❞ ❢r❡q✉❡♥❝②✮

ϕ(ω) =

✭✶✺✮

H

ω
+ 2π
2

✇❡ ♦❜t❛✐♥

1
ω
χ (4π + ω) χ(2π + ) = 0
2π
2

∀ω

❢r♦♠ ✇❤❡r❡ t❤❡r❡ ❢♦❧❧♦✇s ✭✶✹✮✳

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮

✷✷✼


❱❖▲❯▼❊✿ ✷

❇② t❤❡ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ ✭✶✹✮ ✇❡
❣❡t

❢❛♠✐❧② ♦❢ ❢✉♥❝t✐♦♥s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ s❝❛❧✐♥❣ ♣❛✲
r❛♠❡t❡r

∞
−∞

1
1
χ(ω)eiωx ❞ω =
2π
2π

eiωx ❞ω,

❚❤❡ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ ✭✶✸✮ ❛r❡✿

■♥ ♣❛rt✐❝✉❧❛r✱ ❛❝❝♦r❞✐♥❣ t♦ ✭✸✮✱ ✭✹✮✱ ✭✶✸✮ ✐t ✐s


2πi (2n x−k)
−1

❞❡❢ n/2 e
n



 ϕk (x) = 2
2πi(2n x − k)
n
n

e4πi(2 x−k) − e2πi(2 x−k)

❞❡❢

 ψkn (x) = 2n/2
2πi(2n x − k)

✭✶✾✮

✇✐t❤ n, k ∈ Z✳
❋♦r

n ∈ Z.

❡❛❝❤

✭✶✾✮✱ ✐t

lim

ψ(n) = 0,

n,k,x→∞

❢✉♥❝t✐♦♥

❚❤❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

t❤❡

✇❛✈❡❧❡t

sin π (2n x − k)
n
✐s |ψk (x)| =
π (2n x − k)
|ψkn (x)| = 0✳

❢❛♠✐❧②
s♦ t❤❛t

ψ(x)

t❤❡ ♣❛r❛♠❡t❡r ❞❡♣❡♥❞✐♥❣ ✐♥st❛♥❝❡s ✭✶✾✮✱ ❜② ✉s✲
✐♥❣ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠✳
✐s ❦♥♦✇♥ t❤❛t ✐❢

e−2πix − e−4πix
.
2πix

✭✶✻✮

f (x)

❋r❛❝t✐♦♥❛❧ ♣r♦❧✉♥❣❛t✐♦♥ ♦❢
t❤❡ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡t

■t

f (ω) ✐s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢

t❤❡♥

f (ax ± b) =
✷✳✺✳

♦❢

▲❡t ✉s ♥♦✇ ❝♦♠♣✉t❡ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢

✐s t❤❡ ❢✉♥❝t✐♦♥

ψ(x) =

k✳

✭✶✮✱ ✭✶✸✮ t❤❡r❡ ✐♠♠❡❞✐❛t❡❧② ❢♦❧❧♦✇s

❚❤❡♦r❡♠ ✷✳ ❚❤❡ ❞✐❧❛t❡❞ ❛♥❞ tr❛♥s❧❛t❡❞ ✐♥✲
st❛♥❝❡s ♦❢ t❤❡ ❤❛r♠♦♥✐❝ s❝❛❧✐♥❣ ❛♥❞ ✇❛✈❡❧❡t
❢✉♥❝t✐♦♥ ❛r❡


e4πix − e2πix + e−2πix − e−4πix



(ψ(x)) =



4πix




sin
4πx
sin
2πx


=
−
,

2πx
2πx

−e4πix + e2πix + e−2πix − e−4πix



(ψ(x)) =



4πx




cos 4πx
cos 2πx


=−
+
.
2πx
2πx

sin πx
,
πx

❛♥❞ ♦♥ t❤❡ tr❛♥s❧❛t✐♦♥ ♣❛r❡♠❛t❡r

✭s❡❡ ❡✳❣✳ ❬✶✱ ✸✱ ✼✱ ✽❪✮✱

2π

✇❡ ❣❡t t❤❡ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡t ✭✶✸✮✳

|ψ(x)| = |ϕ(x)| =

n

❋r♦♠ ❊qs✳

4π

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

1 ±iωb/a
e
f (ω/a) ,
a

✭✷✵✮

s♦ t❤❛t ✇❡ ❝❛♥ ❡❛s✐❧② ♦❜t❛✐♥ t❤❡ ❞✐❧❛t❡❞ ❛♥❞
tr❛♥s❧❛t❡❞ ✐♥st❛♥❝❡s ♦❢ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢
✭✶✾✮✱ ✭s❡❡ ❡✳❣✳ ❬✸❪✮✿

❋r♦♠ ❊qs✳ ✭✶✸✮✱ ✭✽✮ ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❢r❛❝t✐♦♥❛❧
♣r♦❧✉♥❣❛t✐♦♥ ♦❢ t❤❡ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡t ❛s

❞❡❢ 2παix

ψα (x) = e

ϕα (x)

✭✶✼✮

❛♥❞ ✐ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s

ψα (ω) =

✷✳✻✳

2π
δ(2α2 π − ω)
αΓ(1 + α)


2−n/2 −iωk/2n


e
χ(2π + ω/2n )
 ϕnk (ω) =
2π
−n/2

n

 ψ n (ω) = 2
e−iωk/2 χ(ω/2n )
k
2π
✭✷✶✮

✭✶✽✮

✸✳

❉✐❧❛t❡❞ ❛♥❞ tr❛♥s❧❛t❡❞

▼✉❧t✐s❝❛❧❡ ❤❛r♠♦♥✐❝
✇❛✈❡❧❡t r❡❝♦♥str✉❝t✐♦♥

✐♥st❛♥❝❡s
■♥ ♦r❞❡r t♦ ❤❛✈❡ ❛ ❢❛♠✐❧② ♦❢ ✭❤❛r♠♦♥✐❝✮ ✇❛✈❡❧❡t

♦❢ ❢✉♥❝t✐♦♥s

❢✉♥❝t✐♦♥s ✇❡ ❤❛✈❡ t♦ ❞❡✜♥❡ t❤❡ ❞✐❧❛t❡❞ ✭❝♦♠✲
♣r❡ss❡❞✮ ❛♥❞ tr❛♥s❧❛t❡❞ ✐♥st❛♥❝❡s ♦❢ t❤❡ ❢✉♥❞❛✲

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❣✐✈❡ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t s♣❛❝❡

♠❡♥t❛❧ ❢✉♥❝t✐♦♥s ✭✶✮✱ ✭✶✸✮✱ s♦ t❤❛t t❤❡r❡ ✇✐❧❧ ❜❡ ❛

str✉❝t✉r❡ t♦ t❤❡ ❢❛♠✐❧② ♦❢ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts

✷✷✽

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮


❱❖▲❯▼❊✿ ✷

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

✭✶✾✮ ❛♥❞ t❤❡ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡t r❡❝♦♥str✉❝t✐♦♥ ♦❢

▼♦r❡♦✈❡r✱ ❛❝❝♦r❞✐♥❣ t♦ ✭✶✶✮✱ ❜② t❤❡ ❝❤❛♥❣❡ ♦❢

❢✉♥❝t✐♦♥s✳

✈❛r✐❛❜❧❡

ξ = ω/2n
4π

✸✳✶✳

ψkn

❍✐❧❜❡rt s♣❛❝❡ str✉❝t✉r❡

(x) , ψhn

1
(x) =
2π

e−i(h−k)ξ ❞ξ.
2π

▲❡t

f (x), g(x)

❜❡ ❣✐✈❡♥ t✇♦ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s✱

t❤❡ ✐♥♥❡r ✭♦r s❝❛❧❛r ♦r ❞♦t✮ ♣r♦❞✉❝t✱ ♦❢ t❤❡s❡
❢✉♥❝t✐♦♥s ✐s

h = k ✭❛♥❞ n = m✮✱ tr✐✈✐❛❧❧②
ψkn (x) , ψkn (x) = 1 , ✇❤✐❧❡ ❢♦r h = k ✱

❋♦r

∞

❞❡❢

f, g

=

♦♥❡ ❤❛s✿
✐t ✐s

4π

e−i(h−k)ξ ❞ξ

f (x) g (x)❞x
−∞

2π

∞

P ars.

=

f (ω) g (ω)❞ω = 2π f , g ,

= 2π

i
e−4iπ(h−k) − e−2iπ(h−k) .
(h − k)

−∞
✭✷✷✮
✇❤❡r❡ ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ P❛rs❡✈❛❧ ✐❞❡♥t✐t② ❢♦r t❤❡
❡q✉✐✈❛❧❡♥t ✐♥♥❡r ♣r♦❞✉❝t ✐♥ t❤❡ ❋♦✉r✐❡r ❞♦♠❛✐♥✳

❛♥❞ s✐♥❝❡✱ ❛❝❝♦r❞✐♥❣ t♦ ✭✸✮✱

e±4iπ(h−k) = e±2iπ(h−k) = 1,

(h − k ∈ Z),
✭✷✹✮

❲✐t❤ r❡s♣❡❝t t♦ t❤❡ ❢❛♠✐❧② ♦❢ t❤❡ ❢✉♥❞❛♠❡♥t❛❧
t❤❡ ♣r♦♦❢ ❡❛s✐❧② ❢♦❧❧♦✇s✳

❢✉♥❝t✐♦♥s ✭✶✾✮✱ ✐t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t

❚❤❡♦r❡♠ ✸✳ ❍❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ❛r❡ ♦rt❤♦♥♦r✲
♠❛❧ ❢✉♥❝t✐♦♥s✱ s✉❝❤ t❤❛t
ψkn

✇❤❡r❡ δ

nm

Pr♦♦❢✿

(x) , ψhm

(x) = δ

nm

δhk ,

✭✷✸✮

✭δhk ✮ ✐s t❤❡ ❑r♦♥❡❝❦❡r s②♠❜♦❧✳

■t ✐s ✭❢♦r ❛♥ ❛❧t❡r♥❛t✐✈❡ ♣r♦♦❢ s❡❡ ❛❧s♦

❬✼❪✮

ψkn (x) , ψhm (x)
∞

= 2π

2−n/2 −iωk/2n
2−m/2
e
χ(ω/2n )
2π
2π

−∞

❆♥❛❧♦❣♦✉s❧② ✐t ❝❛♥ ❜❡ ❡❛s✐❧② s❤♦✇♥ t❤❛t


nm
ϕnk (x) , ϕm
δkh ,

h (x) = δ





ϕnk (x) , ϕm

h (x)





n

ϕk (x) , ϕm

h (x)


nm
ψ nk (x) , ψ m
δkh ,
h (x) = δ



n

ψk (x) , ψ m

h (x)





ϕnk (x) , ψ m

h (x) = 0,





ϕnk (x) , ψ m
h (x)

= δ nm δkh ,
= 0,

= 0,

= 0.
✭✷✺✮

▼♦r❡♦✈❡r✱ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❢✉♥❝t✐♦♥s ✭✶✮✱ ✭✶✸✮
m

× eiωh/2 χ(ω/2m )❞ω
2−(n+m)/2
=
2π

❢✉❧✜❧❧s t❤❡ ❜❛s✐❝ ✭❡✈❡♥✲♦❞❞✮ ♣r♦♣❡rt✐❡s ♦❢ s❝❛❧✐♥❣
❛♥❞ ✇❛✈❡❧❡t✱ t❤❛t ✐s

∞

e

−iωk/2n

n

χ(ω/2 )

[ϕ(x)] = [ϕ(−x)],

[ϕ(x)] = − [ϕ(−x)]

[ψ(x)] = − [ψ(−x)],

[ψ(x)] = [ψ(−x)]

−∞
m

× eiωh/2 χ(ω/2m )❞ω
✇❤✐❝❤ ✐s ③❡r♦ ❢♦r

n = m✳

❋♦r

n=m

✐t ✐s

❚❤❡♦r❡♠ ✹✳ ❚❤❡ ❤❛r♠♦♥✐❝ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥
❛♥❞ t❤❡ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ❢✉❧✜❧❧ t❤❡ ❝♦♥❞✐t✐♦♥s

ψkn (x) , ψhn (x)

2−n
=
2π

❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣

∞
n

e−iω(h−k)/2 χ(ω/2n )❞ω.
−∞

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮

∞
−∞

ϕ(x)❞x = 1,

∞
−∞

ψkn (x)❞x = 0.

✷✷✾


❱❖▲❯▼❊✿ ✷

Pr♦♦❢✿

❆❝❝♦r❞✐♥❣ t♦ ✭✶✵✮✲✭✷✷✮ ♦♥❡ ❤❛s

t❤✉s ❜❡✐♥❣

∞



αk



















αk∗














ϕ(x)❞x
−∞

= 1, ϕ(x) = 2π 1, ϕ(ω)
∞

δ(ω)

= 2π
−∞
2π

1
χ(2π + ω)❞ω
2π

δ(ω)❞ω = 1,

=

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

= 2π f (x), ϕ0k (x)
∞

2π

f (ω)eiωk ❞ω

f (ω)ϕ0k (ω)❞ω =

=
−∞

0

= 2π f (x), ϕ0k (x)
2π

= ... =

f (ω)e−iωk ❞ω

0

0
✇❤❡r❡

δ(ω)

✐s t❤❡ ❉✐r❛❝ ❞❡❧t❛ ❢✉♥❝t✐♦♥✳

❆♥❛❧♦❣♦✉s❧②✱ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t ✭✷✶✮✲✭✷✷✮✱

∞

ψkn (x)❞x
−∞

=

1, ψkn (x)

= 2π

∞

= 2π

δ(ω)
−∞

1, ψkn (ω)

2−n/2 −iωk/2n
e
χ(ω/2n )❞ω

2π

2n+2 π

n

δ(ω)e−iωk/2

=

❞ω



βkn






















β ∗n
k










= 2π f (x), ψkn (x)
= . . . = 2−n/2

✇❤❡r❡

B

= . . . = 2−n/2

n, k ✱

2n+2 π

❞ω,

2n+1 π

✭✷✼✮

✇❤❡r❡ t❤❡ ❤❛t st❛♥❞s ❢♦r t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠✳
■t ❝❛♥ ❜❡ ❡❛s✐❧② s❡❡♥ ✭s❡❡ ❡✳❣✳ ❬✶✹❪✮ t❤❛t

▲❡t

✐s t❤❡ s♣❛❝❡ ♦❢ ❝♦♠♣❧❡①

t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧s✱ ✇❤✐❝❤ ❞❡✜♥❡

t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts✱ ❡①✐st ❛♥❞ ❤❛✈❡ ✜♥✐t❡ ✈❛❧✲

❍❛r♠♦♥✐❝ ✇❛✈❡❧❡t s❡r✐❡s

f (x) ∈ B

❜❡ ❛ ❝♦♠♣❧❡① ❢✉♥t✐♦♥ ✇✐t❤ ✜♥✐t❡

✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ✭✷✻✮✱ ✭✷✼✮✳ ❇② t❛❦✐♥❣ ✐♥t♦ ❛❝✲
❝♦✉♥t t❤❡ ♦rt❤♦♥♦r♠❛❧✐t② ♦❢ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥s
✭✷✸✮✱ ✭✷✺✮ t❤❡ ❢✉♥❝t✐♦♥

f (x)

❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s

❛ ✇❛✈❡❧❡t ✭❝♦♥✈❡r❣❡♥t✮ s❡r✐❡s ✭s❡❡ ❡✳❣✳ ❬✼❪✮✳

■♥

❢❛❝t✱ ✐❢ ✇❡ ♣✉t

✉❡s

∞




αk = f (x), ϕ0k (x) =









0

∗



 αk = f (x), ϕk (x) =

n

f (ω)e−iωk/2

f (x) = f (−ω).

❢✉♥❝t✐♦♥s✱ s✉❝❤ t❤❛t ❢♦r ❛♥② ✈❛❧✉❡ ♦❢ t❤❡ ♣❛r❛♠✲
❡t❡rs

❞ω

n

= 0.

❲❛✈❡❧❡t r❡❝♦♥str✉❝t✐♦♥

f (x) ∈ B ✱

n

= f (x), ψ k (x)

✸✳✸✳

▲❡t

f (ω)eiωk/2

2n+1 π

2n+1 π

✸✳✷✳

2n+2 π

∞

∞

βkn ψkn (x)
n=0 k=−∞

k=−∞

f (x)ϕ0k (x)❞x

∞

∞

−∞

k=−∞

f (x)ϕ0k (x)❞x

∞

αk∗ ϕ0k (x) +

+

∞

∞

αk ϕ0k (x) +

f (x) =

n
β ∗n
k ψ k (x)
n=0 k=−∞

✭✷✽✮

−∞
∞





βkn = f (x), ψkn (x) =









n


 β ∗nk = f (x), ψ k (x) =

f (x)ψ nk (x)❞x
−∞

t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ❝❛♥ ❜❡ ❡❛s✐❧② ❝♦♠♣✉t❡❞
❜② ✉s✐♥❣ t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ♦❢ t❤❡ ❜❛s✐s ❛♥❞ ✐ts

∞

f (x)ψkn (x)❞x.

❝♦♥❥✉❣❛t❡✳
■♥ ❬✼❪ ✭s❡❡ ❛❧s♦ ❬✷✹❪✮ ✐t ✇❛s s❤♦✇♥ t❤❛t✱ ✉♥✲

−∞
✭✷✻✮

❞❡r s✉✐t❛❜❧❡ ❛♥❞ q✉✐t❡ ❣❡♥❡r❛❧ ❤②♣♦t❤❡s❡s ♦♥ t❤❡

❆❝❝♦r❞✐♥❣ t♦ ✭✷✶✮✱✭✷✷✮✱ t❤❡s❡ ❝♦❡✣❝✐❡♥ts ❝❛♥ ❜❡

❢✉♥❝t✐♦♥

❡q✉✐✈❛❧❡♥t❧② ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ❋♦✉r✐❡r ❞♦♠❛✐♥✱

t♦

✷✸✵

f (x)✱

t❤❡ ✇❛✈❡❧❡t s❡r✐❡s ✭✷✽✮ ❝♦♥✈❡r❣❡s

f (x)✳

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮


❱❖▲❯▼❊✿ ✷

❚❤❡ ❝♦♥❥✉❣❛t❡ ♦❢ t❤❡ r❡❝♦♥str✉❝t✐♦♥ ✭✷✽✮ ✐t ✐s
∞

∞

n

βn
k ψ k (x)

❡r❢ (x)

n=0 k=−∞

k=−∞
∞

∞

∞

α∗k ϕ0k (x) +

+

β

∞

∞

α∗k ϕ0k (x)

∞

f (x) ∼
=

n=0 k=−∞

k=−∞

∞

∞

αk ϕ0k (x)

+

1
2
+

∞
n
βn
k ψ k (x)

+

❡r❢ (π

1
2

√

σ) ϕ00 (x) + ϕ00 (x)

❡r❢ (2π

√

σ) −

❡r❢ (π

√

σ)

0

× ψ00 (x) + ψ 0 (x) ,

n=0 k=−∞

k=−∞

e−u du
0

t❤❡ ●❛✉ss✐❛♥

∗
n
β n
k ψk (x)

+

x

2
= √
π

❞❡❢

❚❤❡r❡ ❢♦❧❧♦✇s t❤❡ ③❡r♦ ♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

∗n n
k ψk (x)

n=0 k=−∞

k=−∞

=

❜❡✐♥❣ t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❛s

∞

αk ϕ0k (x) +

f (x) =

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

❛♥❞ s✐♥❝❡
❚❤❡ ✇❛✈❡❧❡t ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ♦❜t❛✐♥❡❞ ❜② ✜①✲

ϕ00 (x) + ϕ00 (x) =

✐♥❣ ❛♥ ✉♣♣❡r ❧✐♠✐t ✐♥ t❤❡ s❡r✐❡s ❡①♣❛♥s✐♦♥ ✭✷✽✮✱
s♦ t❤❛t ✇✐t❤

N < ∞, M < ∞

M

N

f (x) ∼
=

✇❡ ❤❛✈❡

βkn ψkn (x)

0

ψ00 (x) + ψ 0 (x) =

n=0 k=−M
N

M

M

2

e−x

n=0 k=−M

k=0

sin 4πx − sin 2πx
πx

✇❡ ❤❛✈❡

n
β ∗n
k ψ k (x) .

αk∗ ϕ0k (x) +

+

❛♥❞

M

αk ϕ0k (x) +
k=0

sin 2πx
x

✭✷✾✮

/σ

1
∼
=
2

√

sin 2πx
x
√
❡r❢ (2π σ) −

❡r❢ (π

σ)

1
2
sin 4πx − sin 2πx
×
πx
+

❙✐♥❝❡ ✇❛✈❡❧❡ts ❛r❡ ❧♦❝❛❧✐③❡❞✱ t❤❡② ❝❛♥ ❝❛♣t✉r❡
✇✐t❤ ❢❡✇ t❡r♠s t❤❡ ♠❛✐♥ ❢❡❛t✉r❡s ♦❢ ❢✉♥❝t✐♦♥s
❞❡✜♥❡❞ ✐♥ ❛ s❤♦rt r❛♥❣❡ ✐♥t❡r✈❛❧✳

❡r❢ (π

√

σ)

❋♦r ✐♥st❛♥❝❡✱ t❤❡ s❡❝♦♥❞ s❝❛❧❡ ❛♣♣r♦①✐♠❛t✐♦♥

✶✮

N = 2, M = 0
2
e−(16x) ✐s ✭s❡❡ ❋✐❣✳

❊①❛♠♣❧❡s ♦❢ ❍❛r♠♦♥✐❝ ✇❛✈❡❧❡t
r❡❝♦♥str✉❝t✐♦♥

2
sin 2πx
e−(16x) ∼
=
2πx

π
π

− ❡r❢
16
8
π
π
− 2 cos 2πx ❡r❢
− ❡r❢
16
8
π
π
− 2 cos 6πx ❡r❢ − ❡r❢
8
4
− (cos 10πx + cos 14πx)

▲❡t ✉s ❣✐✈❡ ❛ ❝♦✉♣❧❡ ♦❢ ❡①❛♠♣❧❡s t♦ s❤♦✇ t❤❡
♣♦✇❡r❢✉❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❜t❛✐♥❡❞ ❜② t❤❡ ❤❛r✲
♠♦♥✐❝ ✇❛✈❡❧❡ts✳
▲❡t ✉s ✜rst ❝♦♥s✐❞❡r t❤❡ r❡❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡
●❛✉ss✐❛♥ ❢✉♥❝t✐♦♥✿

f (x) = e−x

2

/σ

.
×

❚❤❡

tr✉♥❝❛t❡❞

0 ,M = 0

✇❛✈❡❧❡t

s❡r✐❡s

✇✐t❤

N

=

✐s

s♦ t❤❛t ✐❢ ✇❡ ❝♦♠♣✉t❡ t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts
❜② ✉s✐♥❣ t❤❡ ❊qs✳ ✭✷✻✮ ✭♦r ✭✷✼✮✮

N

❡r❢

π
−
4

❡r❢

π
2

✇❡ ✇✐❧❧ ❣❡t ❛ ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥✳

✷✮

✇❡ ❣❡t

√
1
2 ❡r❢ (π σ),
√
β ∗00 = 12 [ ❡r❢ (2π σ)

α0 = α0∗ =
β00 =

❡r❢

2

❆s ❡①♣❡❝t❡❞✱ ❜② ✐♥❝r❡❛s✐♥❣ t❤❡ s❝❛❧✐♥❣ ♣❛r❛♠❡t❡r

f (x) ∼
= α0 ϕ00 (x) + α0∗ ϕ00 (x) + β00 ψ00 + β ∗00 ψ 00 ,
α0 , α0∗ , β00 , β ∗00

❢♦r t❤❡ ●❛✉ss✐❛♥ ❢✉♥❝t✐♦♥
✸✮

−

❡r❢ (π

√

❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✇❛✈❡❧❡t
❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ ❋♦✉r✐❡r ❞♦♠❛✐♥

❆❝❝♦r❞✐♥❣ t♦ ✭✷✼✮ t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥t ❛r❡ ♦❜✲

σ)]

t❛✐♥❡❞ ❜② ❋♦✉r✐❡r tr❛♥s❢♦r♠✳

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮

✷✸✶


❱❖▲❯▼❊✿ ✷

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

✐✳❡✳

1

f (ω) ∼
=

1
χ(2π + ω)
2π

M

N

αk e−iωk +
k=0

2−n/2
2π
n=0

M
n

βkn e−iωk/2

n

× χ(2π + ω/2 )
N =2 , M =0

k=−M

1
χ(2π + ω)
+
2π

0.5

M

N

αk∗ eiωk
k=0

N =0 , M =0

+

2−n/2
2π
n=0

M

β ∗nk eiωk/2

× χ(2π + ω/2n )

n

k=−M
❛♥❞ ❢♦r ❛ r❡❛❧ ❢✉♥❝t✐♦♥
- 0.2

- 0.1

0.1

1
χ(2π + ω)
f (ω) ∼
=
2π

0.2

❋✐❣✳ ✸✿ ❍❛r♠♦♥✐❝ ✇❛✈❡❧❡t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

N

2

f (x) = e−(16x) ❛♥❞ t❤❡ ✵✲s❝❛❧❡ N = 0, M = 0
❛♥❞ ✷✲s❝❛❧❡ N = 2, M = 0 ❛♣♣r♦①✐♠❛t✐♦♥✳

M

αk e−iωk + eiωk
k=0

−n/2

2

+
n=0

2π

χ(2π + ω/2n )

M
n

n

βkn e−iωk/2 + eiωk/2

×
k=−M
■❢ ✇❡ ❛♣♣❧② t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ t♦ ✭✷✾✮✱ ✇❡
❣❡t

t❤❛t ✐s✱

1
χ(π + ω)
f (ω) ∼
=

2π
M

N

f (ω) ∼
=

M

+

n=−N k=−M

k=0
M

N

M
n
β ∗n
k ψ k (ω)

αk∗ ϕ0k (ω) +

+

2−n/2
χ(2π + ω/2n )

π
n=0
M

βkn cos(ωk/2n )

×

n=−N k=−M

k=0

αk cos(ωk)
k=0

N

βkn ψkn (ω)

αk ϕ0k (ω) +

M

k=−M
❙♦ t❤❛t t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥t ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞

❛♥❞✱ ❛❝❝♦r❞✐♥❣ t♦ ✭✷✶✮✱

❜② t❤❡ ❢❛st ❋♦✉r✐❡r tr❛♥s❢♦r♠✳ ■♥ ❬✼❪ ✐t ✇❛s ❣✐✈❡♥
❛ s✐♠♣❧❡ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡s❡

1
f (ω) ∼
=
2π

M

N

αk e

−iωk

k=0

2−n/2
χ(2π + ω) +
2π
n=0

M
n

βkn e−iωk/2 χ(2π + ω/2n )

×
k=−M

+

1
2π

M

✸✮

❍❛r♠♦♥✐❝ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ♦❢
t❤❡ ❢r❛❝t✐♦♥❛❧ ❤❛r♠♦♥✐❝ s❝❛❧✐♥❣ ❛♥❞
✇❛✈❡❧❡t

N

αk∗ eiωk χ(2π + ω) +
k=0

2−n/2
2π
n=0

n

β ∗nk eiωk/2 χ(2π + ω/2n )

×
k=−M

❚❤❡ ❢r❛❝t✐♦♥❛❧ ❤❛r♠♦♥✐❝ s❝❛❧✐♥❣ ❛♥❞ ✇❛✈❡❧❡t
❢✉♥❝t✐♦♥s ✭✾✮✱ ✭✶✼✮ ✐♥ ❣❡♥❡r❛❧ ❛r❡ ♥♦t ♦rt❤♦❣♦✲

♥❛❧ ❛s ❝❛♥ ❜❡ ❝❤❡❝❦❡❞ ❜② ❛ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥

M

✷✸✷

❝♦❡✣❝✐❡♥ts t❤r♦✉❣❤ t❤❡ ❢❛st ❋♦✉r✐❡r tr❛♥s❢♦r♠✳

♦❢ t❤❡✐r ✐♥♥❡r ♣r♦❞✉❝t✳ ❍♦✇❡✈❡r✱ t❤❡② ❝❛♥ ❜❡ ❡①✲
♣r❡ss❡❞✱ ❜② t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ✇✐t❤ r❡s♣❡❝t

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮


❱❖▲❯▼❊✿ ✷

t♦ t❤❡ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡t ❜❛s✐s✳

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

❇② t❛❦✐♥❣ ✐♥t♦

❛❝❝♦✉♥t t❤❡ s✐♠♣❧❡ ❢♦r♠ ♦❢ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠
♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ✭✶✷✮✱ ✭✶✽✮

ϕα (x) =

2π
ϕα (ω) =
δ(ω),

αΓ(1 + α)
2π
ψα (ω) =
δ(2α2 π − ω)
αΓ(1 + α)
✇❡ ❤❛✈❡ ❢♦r t❤❡ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥

2π
αΓ(1 + α)

∞

ϕ0k (x) + ϕ0k (x)
k=−∞
✭✸✸✮

✭✸✵✮

❛♥❞

❛♥❛❧♦❣♦✉s❧②

❢♦r

t❤❡

❢r❛❝t✐♦♥❛❧

❤❛r♠♦♥✐❝

✇❛✈❡❧❡t

∞

ψα (x) =

ϕα (x)

2π
αΓ(1 + α) n=0

∞
2

e2πiα

k

k=−∞

× ψkn (x) + ψ nk (x) .




αk












αk∗











n

 βk

















β ∗nk












2π

ϕα (ω)eiωk ❞ω =

=
0

❇② t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t ❊qs✳✭✶✮✱ ✭✺✮✱ t❤❡ ❜❛s✐❝
❢✉♥❝t✐♦♥s ♦♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞

2π

ϕα (ω)e−iωk ❞ω =

=

✭✸✹✮

2π
αΓ(1 + α)

0
n+2

2

π

−n/2

=2

ϕα (ω)e

2π
αΓ(1 + α)

iωk/2n

t❤✉s ❣✐✈✐♥❣

ϕα (x) =
❞ω

4π
αΓ(1 + α)

=

t♦ t❤❡ s✐♥❝✲❢r❛❝t✐♦♥❛❧ ♦♣❡r❛t♦r ✭s❡❡ ❡✳❣✳ ❬✷✷❪✮ ❛♥❞
❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ ✇❛✈❡❧❡t✱ ❢r♦♠ ✭✶✸✮✱ ✭✶✻✮✱ ❛♥❛❧✲

2n+2 π

= 2−n/2

n

ϕα (ω)e−iωk/2

♦❣♦✉s❧② ✇❡ ❣❡t
❞ω

2n+1 π

ϕα (x) =

2π
,

=
αΓ(1 + α)

2π

f (ω)eiωk ❞ω =

=
0
2π

αk∗

=

2π
αΓ(1 + α)
×



















β ∗n

k











= 2−n/2

2π
e2πiα
αΓ(1 + α)

2n+2 π

f (ω)eiωk/2

2

n

=

k=−∞

sin 4π(x − k) sin 2π(x − k)
−
π(x − k)
π(x − k)

t♦ t❤❡ ❙❤❛♥♥♦♥ ✇❛✈❡❧❡t ❛♥❞ t❤❡ s✐♥❝✲❢r❛❝t✐♦♥❛❧
✇❛✈❡❧❡ts ❬✷✷❪✳
k

✹✳

❞ω

f (ω)e−iωk/2

k

❆❧s♦ t❤❡ ❢r❛❝t✐♦♥❛❧ ✇❛✈❡❧❡t ✐s ❝❧♦s❡❧② r❡❧❛t❡❞

❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡
❍❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ✐♥

2
2π
=
e2πiα k
αΓ(1 + α)

2n+2 π

2

e2πiα

✭✸✻✮

2n+1 π

= 2−n/2

∞

ψα (x)

2
2π
e2πiα k
αΓ(1 + α)

f (ω)e−iωk ❞ω =

0

βkn

✭✸✺✮

s♦ t❤❛t t❤❡ ❢r❛❝t✐♦♥❛❧ s❝❛❧✐♥❣ ✐s ❝❧♦s❡❧② r❡❧❛t❡❞

2π
αΓ(1 + α)

❆♥❛❧♦❣♦✉s❧② ❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ ✇❛✈❡❧❡t

αk

k=−∞

sin 2π(x − k)
2π(x − k)

2n+1 π

✭✸✶✮
































∞

❋♦✉r✐❡r ❞♦♠❛✐♥
n

■t ✐s ❝❧❡❛r ❢r♦♠ ✭✷✼✮ t❤❛t t❤❡ r❡❝♦♥str✉❝t✐♦♥ ♦❢

❞ω

2n+1 π

❛ ❢✉♥❝t✐♦♥

2
2π
e2πiα k ,
αΓ(1 + α)

tr❛♥s❢♦r♠

f (x) ✐t ✐s ✐♠♣♦ss✐❜❧❡ ✇❤❡♥ ✐ts ❋♦✉r✐❡r
f (ω) ✐s ♥♦t ❞❡✜♥❡❞✳ ▼♦r❡♦✈❡r✱ t❤❡

❢✉♥❝t✐♦♥ ✭t♦ ❜❡ r❡❝♦♥str✉❝t❡❞✮ ♠✉st ❜❡ ❝♦♥✲

✭✸✷✮

❝❡♥tr❛t❡❞ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥ ✭❧✐❦❡ ❛ ♣✉❧s❡✮ ❛♥❞

❙♦ t❤❛t ❛❝❝♦r❞✐♥❣ t♦ ✭✷✽✮ ✇❡ ❣❡t t❤❡ ❢r❛❝t✐♦♥❛❧

s❤♦✉❧❞ r❛♣✐❞❧② ❞❡❝❛② t♦ ③❡r♦✳

s❝❛❧✐♥❣ ❛s ❛ ✇❛✈❡❧❡t s❡r✐❡s

t✐♦♥ ❝❛♥ ❜❡ ❞♦♥❡ ❛❧s♦ ❢♦r ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥s✱ ♦r

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮

❚❤❡ r❡❝♦♥str✉❝✲

✷✸✸


❱❖▲❯▼❊✿ ✷

❢✉♥❝t✐♦♥s ❧♦❝❛❧✐③❡❞ ✐♥ ❛ ♣♦✐♥t ❞✐✛❡r❡♥t ❢r♦♠ ③❡r♦✿

x0 = 0 ✱

s♦ t❤❛t

ϕ0k (2πh) = 0,

❜② ✉s✐♥❣ t❤❡ s♦✲❝❛❧❧❡❞ ♣❡r✐♦❞✐③❡❞ ❤❛r✲

♠♦♥✐❝ ✇❛✈❡❧❡ts ❬✶✱ ✼✱ ✽❪✮✳

❚❤❡r❡ ❢♦❧❧♦✇s t❤❛t

f (x)

❆♠♦♥❣ ❛❧❧ ❢✉♥❝t✐♦♥s

s♦♠❡ ♦❢ t❤❡♠ ❛r❡

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

∀h = 0.

αh = 0✱ ❛s ✇❡❧❧ ❛s t❤❡ r❡♠❛✐♥✲
cos(2kπx) ✭✇✐t❤ k ∈ Z

✐♥❣ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ♦❢

k = 0✮

❝♦♥st❛♥t ✉♥❞❡r ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡t ♠❛♣ ✭✷✽✮✳ ■♥

❛♥❞

❢❛❝t✱ ✇❡ ❤❛✈❡ t❤❛t✱

✐t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t ❛❧❧ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ♦❢

❚❤❡♦r❡♠ ✺✳ ❋♦r ❛ ♥♦♥ tr✐✈✐❛❧ ❢✉♥❝t✐♦♥ f (x) =
0 t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ✭✷✼✮✱ ✐♥
❣❡♥❡r❛❧✱ ✈❛♥✐s❤ ✇❤❡♥ ❡✐t❤❡r
f (ω) = 0, ∀k

or

f (ω) = Cnst., k = 0.

❛r❡ tr✐✈✐❛❧❧② ✈❛♥✐s❤✐♥❣✳ ❆♥❛❧♦❣♦✉s❧②✱

cos(2kπx) ✭∀k ∈ Z✮

❛r❡ ③❡r♦✳

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥

f (x)✱

❢♦r

✇❤✐❝❤ t❤❡ ❝♦❡✣❝✐❡♥ts ✭✷✻✮ ❛r❡ ❞❡✜♥❡❞✱ ❛❞♠✐ts
t❤❡ s❛♠❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ♦❢

∞

■♥ ♣❛rt✐❝✉❧❛r✱ ✐t ❝❛♥ ❜❡ s❡❡♥ t❤❛t t❤❡ ✇❛✈❡❧❡t
❝♦❡✣❝✐❡♥ts ✭✷✼✮ tr✐✈✐❛❧❧② ✈❛♥✐s❤ ✇❤❡♥

[Ah sin(2hπx) + Bh cos(2hπx)] − B0 ,

f (x) +
h=0

✭✸✽✮
♦r ✭❜② ❛ s✐♠♣❧❡ tr❛♥❢♦r♠❛t✐♦♥✮ ✐♥ t❡r♠s ♦❢ ❝♦♠✲



 f (x) = sin(2kπx),

k∈Z




k∈Z

f (x) = cos(2kπx),

♣❧❡① ❡①♣♦♥❡♥t✐❛❧s✱
✭✸✼✮

∞

Ch e2ihπx ,

f (x) − C0 +

(k = 0)

✭✸✾✮

h=−∞

Pr♦♦❢✿

❋♦r ✐♥st❛♥❝❡ ❢r♦♠

(26)1 ✱

❢♦r

cos(2kπx)

✐t ✐s

f (x)

❛r❡ ❞❡✲

✜♥❡❞ ✉♥❧❡ss ❛♥ ❛❞❞✐t✐♦♥❛❧ tr✐❣♦♥♦♠❡tr✐❝ s❡r✐❡s
✭t❤❡ ❝♦❡✣❝✐❡♥ts

∞

αk

s♦ t❤❛t t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ♦❢

cos(2kπx)ϕ0k (x)❞x

=

Ah , Bh , Ch

❜❡✐♥❣ ❝♦♥st❛♥t✮ ❛s

✐♥ ✭✸✽✮✳

−∞

=

=

1
2
1
2

∞

e−2ihπx + e2ihπx ϕ0k (x)❞x

✺✳

−∞

▲♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s

∞

e−2ihπx ϕ0k (x)❞x

■♥ ♦r❞❡r t♦ ❣❡t s♦♠❡ ❛❞✈❛♥t❛❣❡s ❢r♦♠ t❤❡ ❞❡✜♥✐✲

−∞

t✐♦♥ ♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ✇❡ ❣✐✈❡

∞

e2ihπx ϕ0k (x)❞x

+
−∞

✐♥ t❤✐s s❡❝t✐♦♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥
❞❡r✐✈❛t✐✈❡✱ ❛♥❞ t❤❡♥ ✇❡ ❛♣♣❧② t❤✐s ♦♣❡r❛t♦r t♦
t❤❡ ❢r❛❝t✐♦♥❛❧ ✇❛✈❡❧❡ts ✭✾✮✱ ✭✶✼✮✳ ❇② t❛❦✐♥❣ ✐♥t♦

❢r♦♠ ✇❤❡r❡ ❜② t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡

2πx = ξ

❛♥❞ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t ✭✷✵✮ t❤❡r❡ ❢♦❧❧♦✇s

❛❝❝♦✉♥t t❤❛t ✇❛✈❡❧❡ts ❛r❡ ❧♦❝❛❧✐③❡❞ ❢✉♥❝t✐♦♥s✱ ✇❡
♥❡❡❞ t♦ ❞❡✜♥❡ ❛ s✉✐t❛❜❧❡ ❧♦❝❛❧ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r✲
❛t♦r ❛s t❤❡ ♦♥❡s ♣r♦♣♦s❡❞ ❜② ❨❛♥❣ ❬✸✻✕✸✾❪✿

1 0
αk =
ϕ (x) + ϕ0k (x)
2 k

.
x=2πh

✺✳✶✳

❆❝❝♦r❞✐♥❣ t♦ ✭✷✶✮ ✐t ✐s

1 −i2πhk
e
χ(2π + 2πh)
2
(3) 1
= χ(2π + 2πh)
2

ϕ0k (2πh) =

χ(2π + 2πh) = 1,

✷✸✹

❉❡✜♥✐t✐♦♥ ✶✳ ❚❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢
f (x) ♦❢ ♦r❞❡r α ❛t x = x0 ✐s t❤❡ ♦♣❡r❛t♦r
dα f
dxα

❛♥❞✱ ❜❡❝❛✉s❡ ♦❢ ✭✶✶✮

0
▲♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

= lim
x=x0

x→x0

∆α (f (x) − f (x0 ))
,
(xα − xα
0)
0<α≤1

✭✹✵✮

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮


❱❖▲❯▼❊✿ ✷

❜❡✐♥❣

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

t♦ t❤❡ ♠♦st s✐❣♥✐✜❝❛♥t ❢✉♥❝t✐♦♥s✳

❇② ❛ ❞✐r❡❝t

❝♦♠♣✉t❛t✐♦♥ ✐t ❝❛♥ ❜❡ ❡❛s✐❧② s❤♦✇♥ t❤❛t✱ st❛rt✲
✐♥❣ ❢r♦♠ t❤❡ ♣♦✇❡r s❡r✐❡s ❬✸✶✱ ✸✻✕✸✾❪✿

∆α (f (x) − f (x0 )) ∼
= Γ(1 + α) [(f (x) − f (x0 ))] .
✭✹✶✮

+∞

Eα (xα ) =

❚❤❡r❡ ❢♦❧❧♦✇s t❤❛t

0 < α ≤ 1,
✭✹✸✮

α

α α

d x
dxα

xmα
,
Γ(1 + mα)
m=0

= lim
x=x0

x→x0

∆ (f (x) − f (x0 ))
(xα − xα
0)

+∞

(xα − xα
0)
∼
= Γ(1 + α) lim
x→x0 (xα − xα )
0

sinα (xα ) =

(−1)m
m=0

= Γ(1 + α)

x(2m+1)α
,
Γ(1 + (2m + 1)α)
0<α≤1
✭✹✹✮

✐✳❡✳✱

dα xα = Γ(1 + α)dxα
❋♦r ❛♥②

x

+∞

✐♥ ❛ s✉✐t❛❜❧❡ ✐♥t❡r✈❛❧ ❝❡♥t❡r❡❞ ✐♥

x0 ✱

cosα (xα ) =

m=0

✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

❞❡❢

Dxα f (x) =

✺✳✷✳

dα
f (x),
dxα

x ∈ (x0 − δ, x0 + δ)

▲♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ✐♥t❡❣r❛❧

❉❡✜♥✐t✐♦♥ ✷✳ ❚❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ✐♥t❡❣r❛❧ ♦❢
f (x) ♦❢ ❢r❛❝t✐♦♥❛❧ ♦r❞❡r α ✐♥ t❤❡ ✐♥t❡r✈❛❧ (a, b)
✐s ❞❡✜♥❡❞ ❛s ✭ ❬✸✻✱ ✸✼❪✮
(α)
a Ib f (x)

=
=

1
Γ(1 + α)

dα xmα
Γ(1 + mα)
=
x(m−1)α .
dxα
Γ(1 + (m − 1)α)

✭✹✺✮

✭✹✻✮

✇❡ ❝❛♥ ❡❛s✐❧② s❤♦✇ t❤❛t

f (u)(du)α
a
N −1

f (uj )(∆uj )α ,
j=0
✭✹✷✮

✇❡ ❤❛✈❡ ∆uj = uj+1 − uj , ∆u =
max {∆u0 , ∆u1 , ∆u2 , · · · } ❛♥❞ [uj , uj+1 ] , u0 =

a, uN = b, ✐s ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✐♥t❡r✈❛❧ [a, b]✳
❋♦r ❛♥② x ∈ (a, b)✱ ✇❡ ❝❛♥ ❛❧s♦ ❞❡✜♥❡ t❤❡ ✐♥t❡✲
(α)
❣r❛❧ ♦♣❡r❛t♦r a Ix f (x),

x2mα
,
Γ(1 + 2mα)
0<α≤1

❛♥❞ ❜② t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❛t ❬✸✻✱ ✸✼❪

b

1
lim
Γ(1 + α) ∆u−→0

(−1)m

dα
Eα (xα ) = Eα (xα ).
dxα

✭✹✼✮

dα
sinα (xα ) = cosα (xα ).
dxα

✭✹✽✮

dα
cosα (xα ) = − sinα (xα ).
dxα

✭✹✾✮

✇❤❡r❡

(α)
0 Ix

✺✳✹✳
✺✳✸✳

xmα
x(m+1)α
=
.
Γ(1 + mα)
Γ(1 + (m + 1)α)

✭✺✵✮

▲♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡

♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧ ❍❛r♠♦♥✐❝

❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ♦♣❡r❛t♦rs

✇❛✈❡❧❡ts

❚❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ♦♣❡r❛t♦rs✱ ♣r❡✈✐♦✉s❧② ❞❡✲

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❣✐✈❡ t❤❡ ❡①♣❧✐❝✐t ❡①♣r❡ss✐♦♥

✜♥❡❞✱ ❤❛✈❡ s♦♠❡ s♣❡❝✐❛❧ ❢❡❛t✉r❡s ✇❤❡♥ ❛♣♣❧✐❡❞

♦❢ t❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❤❛r♠♦♥✐❝

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮

✷✸✺


❱❖▲❯▼❊✿ ✷

❢r❛❝t✐♦♥❛❧ s❝❛❧✐♥❣ ✭✾✮ ❛♥❞ ✇❛✈❡❧❡t ✭✶✼✮✱ ♥❛♠❡❧②

∞

ϕα (x) =
k=0

(0 ≤ α ≤ 1)

ψα (x) = E(2παix)ϕα (x)

✭✺✶✮

❆❝❝♦r❞✐♥❣ t♦ ❊qs✳ ✭✹✻✮✱ ✭✹✼✮ ✐t ✐s

d
ϕα (x) =
dxα

♥✐q✉❡ ✐♥ t❤❡ ❞✐s♣❡rs✐✈❡ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥✳ ■♥✲

(2πi)
d k
x
αΓ(k + α + 1) dxα

t❡r♥❛t✐♦♥❛❧ ❆♣♣❧✐❡❞ ▼❡❝❤❛♥✐❝s✱ ✸✾✭✹✮✱ ✶✸✷✲
✶✹✵✳

(2πi)k
Γ(1 + k) k−1
x
,
αΓ(k + α + 1) Γ(k)

=
k=1

(0 ≤ α ≤ 1)

❬✷❪ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✵✸✮✳ ❍❛r♠♦♥✐❝ ❲❛✈❡❧❡t ❙♦❧✉✲
t✐♦♥s ♦❢ t❤❡ ❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥✳ ■♥t❡r♥❛✲
t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❋❧✉✐❞ ▼❡❝❤❛♥✐❝s ❘❡s❡❛r❝❤✱
✺✱ ✶✲✶✵✳

d
ψα (x) = Eα (2παix)
dxα

❬✸❪ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✵✺✮✳ ❍❛r♠♦♥✐❝ ❲❛✈❡❧❡ts t♦✲

d
× 2παiϕα (x) + α ϕα (x)
dx

✇❛r❞s ❙♦❧✉t✐♦♥ ♦❢ ◆♦♥❧✐♥❡❛r P❉❊✳ ❈♦♠✲
♣✉t❡rs ❛♥❞ ▼❛t❤❡♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱
✭✺✷✮

✺✵✭✽✲✾✮✱ ✶✶✾✶✲✶✷✶✵✳
❬✹❪ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✵✻✮✳ ❈♦♥♥❡❝t✐♦♥ ❈♦❡✣❝✐❡♥ts

❛♥❞ s②♠♣❧✐❢②✐♥❣

d
ϕα (x) =
dxα

❘❡❢❡r❡♥❝❡s
❬✶❪ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✵✸✮✳ ❚❤❡ ✇❛✈❡❧❡ts ❜❛s❡❞ t❡❝❤✲

k

k=0
∞

❚❤❡ ❆✉t❤♦r ✐s ❣r❛t❡❢✉❧ t♦ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✲
✈❡rs✐t② ❢♦r ♣❛rt✐❛❧❧② s✉♣♣♦rt✐♥❣ t❤✐s ✇♦r❦✳

(2πi)k
xk ,
αΓ(k + α + 1)

∞

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

♦❢ ❙❤❛♥♥♦♥ ❲❛✈❡❧❡ts✳ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦❞✲

∞

k=1

k(2πi)k
xk−1 ,
αΓ(k + α + 1)

❡❧❧✐♥❣ ❛♥❞ ❆♥❛❧②s✐s✱ ✶✶✭✷✮✱ ✶✲✶✻✳
❬✺❪ ❈❛tt❛♥✐✱ ❈✳✱ ✫ ❘✉s❤❝❤✐ts❦②✱ ❏✳ ❏✳ ✭✷✵✵✼✮✳

(0 ≤ α ≤ 1)
d
ψα (x) = Eα (2παix)
dxα
∞

+
k=1

∞

k=0

(2πi)k+1
xk
Γ(k + α + 1)

k

k(2πi)
xk−1
αΓ(k + α + 1)

✭✺✸✮

❲❛✈❡❧❡t ❛♥❞ ❲❛✈❡ ❆♥❛❧②s✐s ❛s ❛♣♣❧✐❡❞ t♦
▼❛t❡r✐❛❧s ✇✐t❤ ▼✐❝r♦ ♦r ◆❛♥♦str✉❝t✉r❡✳ ❙❡✲
r✐❡s ♦♥ ❆❞✈❛♥❝❡s ✐♥ ▼❛t❤❡♠❛t✐❝s ❢♦r ❆♣♣❧✐❡❞
❙❝✐❡♥❝❡s✱ ❲♦r❧❞ ❙❝✐❡♥t✐✜❝✱ ❙✐♥❣❛♣♦r❡✱ ✼✹✳
❬✻❪ ▼♦✉r✐✱ ❍✳✱ ✫ ❑✉❜♦t❛♥✐✱ ❍✳ ✭✶✾✾✺✮✳

❘❡❛❧✲

✈❛❧✉❡❞ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts✳ P❤②s✳▲❡tt✳ ❆✱ ✷✵✶✱
✺✸✲✻✵✳

❚❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛✲
t✐✈❡ ♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ❝❛♥ ❜❡

❬✼❪ ◆❡✇❧❛♥❞✱ ❉✳ ❊✳ ✭✶✾✾✸✮✳

❍❛r♠♦♥✐❝ ✇❛✈❡❧❡t

❛♥❛❧②s✐s✳ Pr♦❝✳❘✳❙♦❝✳▲♦♥❞✳ ❆✱ ✹✹✸✱ ✷✵✸✲✷✷✷✳

❛ ❢✉♥❞❛♠❡♥t❛❧ t♦♦❧ ✐♥ t❤❡ s❡❛r❝❤ ❢♦r ♥✉♠❡r✐❝❛❧
❬✽❪ ▼✉♥✐❛♥❞②✱ ❙✳ ❱✳✱ ✫ ▼♦r♦③✱ ■✳ ▼✳ ✭✶✾✾✼✮✳

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

●❛❧❡r❦✐♥ ♠♦❞❡❧❧✐♥❣ ♦❢ t❤❡ ❇✉r❣❡rs ❡q✉❛t✐♦♥
✉s✐♥❣ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts✳ P❤②s✳▲❡tt✳ ❆✱ ✷✸✺✱
✸✺✷✲✸✺✻✳

❈♦♥❝❧✉s✐♦♥

❬✾❪ ❲♦❥t❛s③❝③②❦✱ P✳ ❆✳ ✭✷✵✵✸✮✳ ❆ ▼❛t❤❡♠❛t✐✲
■♥ t❤✐s ♣❛♣❡r t❤❡ ♠❛✐♥ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❝♦♠✲
♣❧❡① ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ❛r❡ ❣✐✈❡♥✳ ▼♦r❡♦✈❡r t❤❡
❢r❛❝t✐♦♥❛❧ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ✇❡r❡ ❞❡✜♥❡❞ ❛♥❞

t❤❡✐r ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s ❡①♣❧✐❝✐t❧② ❝♦♠✲
♣✉t❡❞✳

❝❛❧ ■♥tr♦❞✉❝t✐♦♥ t♦ ❲❛✈❡❧❡ts✳ ▲♦♥❞♦♥ ▼❛t❤✲
❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ❙t✉❞❡♥t ❚❡①ts ✸✼✱ ❈❛♠✲
❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡ ✭✶✾✾✼✱

2nd

❡❞✳✮✳

❚❤❡s❡ ❢r❛❝t✐♦♥❛❧ ❤❛r♠♦♥✐❝ ✇❛✈❡❧❡ts ❛r❡

❬✶✵❪ ❇❛❝r②✱ ❊✳✱ ▼❛❧❧❛t✱ ❙✳ ✫ P❛♣❛♥✐❝♦❧❛♦✉✱ ●✳

t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❢✉♥❝t✐♦♥s t♦ ❜✉✐❧❞ ❛ ♠♦❞❡❧ ❢♦r

✭✷✵✵✹✮✳ ❆ ✇❛✈❡❧❡t ❜❛s❡❞ s♣❛❝❡✲t✐♠❡ ♥✉♠❡r✐✲

t❤❡ s♦❧✉t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ♣r♦❜❧❡♠s✳

❝❛❧ ♠❡t❤♦❞ ❢♦r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳

❆❈❑◆❖❲▲❊❉●❊▼❊◆❚
✷✸✻

▼❛t❤❡♠❛t✐❝❛❧ ▼♦❞❡❧s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s✱
✷✻ ✹✶✼✲✹✸✽✳

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮

❱❖▲❯▼❊✿ ✷

❬✶✶❪ ❇❛❝❝♦✉✱ ❏✳✱ ✫ ▲✐❛♥❞r❛t✱ ❏✳ ✭✷✵✵✺✮✳ ❉❡✜♥✐✲

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

❬✷✸❪ ❈❤❡♥✱ ❲✳✲❙✳✱ ✫ ▲✐♥✱ ❲✳ ✭✷✵✵✶✮✳

●❛❧❡r❦✐♥

t✐♦♥ ❛♥❞ ❛♥❛❧②s✐s ♦❢ ❛ ✇❛✈❡❧❡t✴✜❝t✐t✐♦✉s ❞♦✲

tr✐❣♦♥♦♠❡tr✐❝ ✇❛✈❡❧❡t ♠❡t❤♦❞s ❢♦r t❤❡ ♥❛t✲

♠❛✐♥ s♦❧✈❡r ❢♦r t❤❡ ✷❉✲❤❡❛t ❡q✉❛t✐♦♥ ♦♥ ❛

✉r❛❧ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✳

❣❡♥❡r❛❧ ❞♦♠❛✐♥✳ ❚♦ ❛♣♣❡❛r ♦♥

▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ✶✷✶✱ ✼✺✲✾✷✳

▼❛t❤❡♠❛t✐✲

❆♣♣❧✐❡❞

❝❛❧ ▼♦❞❡❧s ❛♥❞ ▼❡t❤♦❞s ✐♥ ❆♣♣❧✐❡❞ ❙❝✐❡♥❝❡s✳
❬✶✷❪ ❇❛❧❡❛♥✉✱ ❉✳✱ ❉✐❡t❤❡❧♠✱ ❑✳✱ ❙❝❛❧❛s✱ ❊✳✱ ✫

❚r✉❥✐❧❧♦✱ ❏✳ ❏✳ ✭✷✵✶✷✮✳ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s✿
▼♦❞❡❧s ❛♥❞ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞s ✭❙❡r✐❡s ♦♥
❈♦♠♣❧❡①✐t②✱ ◆♦♥❧✐♥❡❛r✐t② ❛♥❞ ❈❤❛♦s✮✳ ❲♦r❧❞
❙❝✐❡♥t✐✜❝✳

❬✷✹❪ ❉❛✉❜❡❝❤✐❡s✱ ■✳ ✭✶✾✾✷✮✳

❚❡♥ ▲❡❝t✉r❡s ♦♥

✇❛✈❡❧❡ts✳ ❙■❆▼✱ P❤✐❧❛❞❡❧♣❤✐❛✱ P❆✳
❬✷✺❪ ●♦❡❞❡❝❦❡r✱ ❙✳✱ ✫ ■✈❛♥♦✈✱ ❖✳ ✭✶✾✾✽✮✳ ❙♦❧✉✲
t✐♦♥ ♦❢ ▼✉❧t✐s❝❛❧❡ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛✲
t✐♦♥s ❯s✐♥❣ ❲❛✈❡❧❡ts✳ ❈♦♠♣✉t❡rs ✐♥ P❤②s✐❝s✱

❬✶✸❪ ❇❡♥❤❛✐❞✱ ❨✳ ✭✷✵✵✼✮✳ ❈♦♠♣❛❝t❧② s✉♣♣♦rt❡❞

✶✷✭✻✮✱ ✺✹✽✲✺✺✺✳

✇❛✈❡❧❡t ❛♥❞ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡
❱❧❛s♦✈ ❡q✉❛t✐♦♥✳ ❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ ▼❛t❤✲
❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t✐♥❣✱ ✷✹ ✭✶✲✷✮✱ ✶✼✲✸✵✳
❬✶✹❪ ❇r❛❝❡✇❡❧❧✱ ❘✳ ◆✳ ✭✶✾✼✽✮✳ ❚❤❡ ❋♦✉r✐❡r ❚r❛♥s✲
❢♦r♠ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✳ ▼❝●r❛✇✲❍✐❧❧✱ ✷♥❞
❡❞✳

❬✷✻❪ ❍❡②❞❛r✐✱ ▼✳ ❍✳✱ ❍♦♦s❤♠❛♥❞❛s❧✱ ▼✳ ❘✳✱ ❈❛t✲
t❛♥✐✱ ❈✳✱ ✫ ▼❛❛❧❡❦ ●❤❛✐♥✐✱ ❋✳▼✳ ✭✷✵✶✺✮✳
❆♥ ❡✣❝✐❡♥t ❝♦♠♣✉t❛t✐♦♥❛❧ ♠❡t❤♦❞ ❢♦r s♦❧✈✲
✐♥❣ ♥♦♥❧✐♥❡❛r st♦❝❤❛st✐❝ ■tÔ ✐♥t❡❣r❛❧ ❡q✉❛✲
t✐♦♥s✿ ❆♣♣❧✐❝❛t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ♣r♦❜❧❡♠s ✐♥

♣❤②s✐❝s✳ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ P❤②s✐❝s✱

❬✶✺❪ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✵✸✮✳ ▼✉❧t✐s❝❛❧❡ ❆♥❛❧②s✐s ♦❢

✷✽✸✱ ✶✹✽✲✶✻✽✳

❲❛✈❡ Pr♦♣❛❣❛t✐♦♥ ✐♥ ❈♦♠♣♦s✐t❡ ▼❛t❡r✐❛❧s✳
▼❛t❤❡♠❛t✐❝❛❧ ▼♦❞❡❧❧✐♥❣ ❛♥❞ ❆♥❛❧②s✐s✱ ✽✭✹✮✱

❬✷✼❪ ❍❡②❞❛r✐✱ ▼✳ ❍✳ ✱ ❍♦♦s❤♠❛♥❞❛s❧✱ ▼✳ ❘✳
✱

✷✻✼✲✷✽✷✳
❬✶✻❪ ❈❛tt❛♥✐✱ ❈✳✱ ✫ ❘✉s❤❝❤✐ts❦②✱ ❏✳ ❏✳ ✭✷✵✵✸✮✳
❙♦❧✐t❛r② ❊❧❛st✐❝ ❲❛✈❡s ❛♥❞ ❊❧❛st✐❝ ❲❛✈❡❧❡ts✳
■♥t❡r♥❛t✐♦♥❛❧ ❆♣♣❧✐❡❞ ▼❡❝❤❛♥✐❝s✱ ✸✾✭✻✮ ✼✹✶✲
✼✺✷✳
❬✶✼❪ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✵✽✮✳ ❙❤❛♥♥♦♥ ❲❛✈❡❧❡ts ❚❤❡✲

▼❛❛❧❡❦ ●❤❛✐♥✐✱

❋✳ ▼✳✱

✫ ❈❛tt❛♥✐✱

❈✳

✭✷✵✶✺✮✳ ❲❛✈❡❧❡ts ♠❡t❤♦❞ ❢♦r t❤❡ t✐♠❡ ❢r❛❝✲
t✐♦♥❛❧ ❞✐✛✉s✐♦♥✲✇❛✈❡ ❡q✉❛t✐♦♥✳ P❤②s✐❝s ▲❡t✲
t❡rs✱ ❙❡❝t✐♦♥ ❆✿ ●❡♥❡r❛❧✱ ❆t♦♠✐❝ ❛♥❞ ❙♦❧✐❞

❙t❛t❡ P❤②s✐❝s✱ ✸✼✾✭✸✮✱ ✼✶✲✼✻✳
❬✷✽❪ ❍❡②❞❛r✐✱

▼✳ ❍✳✱

▲♦❣❤♠❛♥✐✱

❍♦♦s❤♠❛♥❞❛s❧✱
●✳✱

✫

▼✳ ❘✳✱

♦r②✳ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❜❧❡♠s ✐♥ ❊♥❣✐♥❡❡r✐♥❣✱

❇❛r✐❞

❈❛tt❛♥✐✱

✷✵✵✽✱ ✶✲✷✹✳

✭✷✵✶✻✮✳ ❲❛✈❡❧❡ts ●❛❧❡r❦✐♥ ♠❡t❤♦❞ ❢♦r s♦❧✈✲

❈✳

✐♥❣ st♦❝❤❛st✐❝ ❤❡❛t ❡q✉❛t✐♦♥✳ ■♥t❡r♥❛t✐♦♥❛❧
❬✶✽❪ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✶✵✮✳ ❙❤❛♥♥♦♥ ❲❛✈❡❧❡ts ❢♦r
t❤❡

❙♦❧✉t✐♦♥

♦❢

■♥t❡❣r♦❞✐✛❡r❡♥t✐❛❧

❊q✉❛✲

❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❡r ▼❛t❤❡♠❛t✐❝s✱ ✾✸✭✾✮✱
✶✺✼✾✲✶✺✾✻✳

t✐♦♥s✳ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❜❧❡♠s ✐♥ ❊♥❣✐♥❡❡r✲
✐♥❣✱ ✷✵✶✵✱ ✶✲✷✹✳

❬✷✾❪ ❍❡②❞❛r✐✱

❬✶✾❪ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✶✷✮✳ ❋r❛❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s ❛♥❞
❙❤❛♥♥♦♥ ❲❛✈❡❧❡t✳ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❜❧❡♠s
✐♥ ❊♥❣✐♥❡❡r✐♥❣✱ ✷✵✶✷✳

▼✳ ❍✳✱

❍♦♦s❤♠❛♥❞❛s❧✱

▼✳ ❘✳✱

▼❛❛❧❡❦ ●❤❛✐♥✐✱ ❋✳ ▼✳ ✫ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✶✻✮✳
❲❛✈❡❧❡ts ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ❢r❛❝t✐♦♥❛❧ ♦♣t✐✲
♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s✳ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s
❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ✷✽✻✱ ✶✸✾✲✶✺✹✳

❬✷✵❪ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✶✺✮✳ ▲♦❝❛❧ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉✲
❧✉s ♦♥ ❙❤❛♥♥♦♥ ❲❛✈❡❧❡t ❇❛s✐s✳ ■♥ ❋r❛❝t✐♦♥❛❧

❬✸✵❪ ❍❡②❞❛r✐✱

▼✳ ❍✳✱

❍♦♦s❤♠❛♥❞❛s❧✱

▼✳ ❘✳✱

❉②♥❛♠✐❝s✱ ❈✳ ❈❛tt❛♥✐✱ ❍✳ ❙r✐✈❛st❛✈❛✱ ❳✳❏✳

❙❤❛❦✐❜❛✱

❨❛♥❣ ✭❊❞s✳✮✱ ❉❡ ●r✉②t❡r✱ ❑r❛❦♦✇✱ ❈❤♣✳ ✶✳

✇❛✈❡❧❡ts ●❛❧❡r❦✐♥ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ♥♦♥❧✐♥✲

❬✷✶❪ ❈❛tt❛♥✐✱

❈✳✱

❙r✐✈❛st❛✈❛✱

❍✳✱

✫

❳✐❛♦✲

❏✉♥ ❨❛♥❣ ✭✷✵✶✺✮✳ ❋r❛❝t✐♦♥❛❧ ❉②♥❛♠✐❝s✳ ❉❡
●r✉②t❡r ❖♣❡♥✳

❆✳✱

✫

❈❛tt❛♥✐✱

❈✳✱

✏▲❡❣❡♥❞r❡

❡❛r st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✑✱ ◆♦♥❧✐♥❡❛r
❉②♥❛♠✐❝s✱ ✈♦❧✳ ✽✺✱ ♥✳✷ ✭✷✵✶✻✮✱ ✶✶✽✺✕✶✷✵✷✳
❬✸✶❪ ❍✉✱ ▼✳ ❙✳✱ ❆❣❛r✇❛❧✱ ❘✳ P✳✱ ✫ ❨❛♥❣✱ ❳✳ ❏✳

❬✷✷❪ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✶✽✮✳ ❙✐♥❝✲❋r❛❝t✐♦♥❛❧ ♦♣❡r❛✲

▲♦❝❛❧ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❙❡r✐❡s ✇✐t❤ ❆♣♣❧✐❝❛✲

t♦r ♦♥ ❙❤❛♥♥♦♥ ❲❛✈❡❧❡t ❙♣❛❝❡✳ ❋r♦♥t✐❡rs ✐♥

t✐♦♥ t♦ ❲❛✈❡ ❊q✉❛t✐♦♥ ✐♥ ❋r❛❝t❛❧ ❱✐❜r❛t✐♥❣

P❤②s✐❝s✱ ✻✭✶✶✽✮✱ ✶✲✶✻✳

❙tr✐♥❣✳ ❆❜s✳ ❛♥❞ ❆♣♣❧✳ ❆♥❛❧✳✱ ✷✵✶✷✱ ✶✲✶✺✳

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮

✷✸✼


❱❖▲❯▼❊✿ ✷

❬✸✷❪ ◗✐❛♥✱ ❙✳✱ ✫ ❲❡✐ss✱ ❏✳ ✭✶✾✾✸✮✳

| ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r

❲❛✈❡❧❡ts

❬✸✽❪ ❨❛♥❣✱ ❳✳✲❏✳✱ ▼❛❝❤❛❞♦✱ ❏✳ ❆✳ ❚✳✱ ❈❛tt❛♥✐✱

❛♥❞ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ♣❛rt✐❛❧ ❞✐✛❡r✲

❈✳✱ ✫ ●❛♦✱ ❋✳ ✭✷✵✶✼✮✳ ❖♥ ❛ ❢r❛❝t❛❧ ▲❈✲

❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳

❡❧❡❝tr✐❝ ❝✐r❝✉✐t ♠♦❞❡❧❡❞ ❜② ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧

❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧

P❤②s✐❝s✱ ✶✵✻✱ ✶✺✺✲✶✼✺✳

❝❛❧❝✉❧✉s✳ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ◆♦♥❧✐♥❡❛r ❙❝✐✲

❬✸✸❪ ❘❡str❡♣♦✱ ❏✳ ✫ ▲❡❛❢✱ ●✳ ❑✳ ✭✶✾✾✺✮✳ ❲❛✈❡❧❡t✲
●❛❧❡r❦✐♥ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ ❤②♣❡r❜♦❧✐❝ ❡q✉❛✲
t✐♦♥s✳

❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ P❤②s✐❝s✱

✶✷✷✱ ✶✶✽✲✶✷✽✳

❡♥❝❡ ❛♥❞ ◆✉♠❡r✐❝❛❧ ❙✐♠✉❧❛t✐♦♥✱ ✹✼✱ ✷✵✵✲✷✵✻✳
❬✸✾❪ ❨❛♥❣✱

❳✳✲❏✳✱

❚❡♥r❡✐r♦

▼❛❝❤❛❞♦✱

❏✳

❆✳✱

❇❛❧❡❛♥✉✱ ❉✳✱ ✫ ❈❛tt❛♥✐✱ ❈✳ ✭✷✵✶✻✮✳ ❖♥ ❡①❛❝t
tr❛✈❡❧✐♥❣✲✇❛✈❡ s♦❧✉t✐♦♥s ❢♦r ❧♦❝❛❧ ❢r❛❝t✐♦♥❛❧

❬✸✹❪ ❙❤❡♥✱ ❨✳✱ ✫ ▲✐♥✱ ❲✳ ✭✷✵✵✹✮

❚❤❡ ♥❛t✉r❛❧

✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ♦❢ ♣❧❛♥❡ ❡❧❛st✐❝✐t② ❛♥❞ ✐ts

❑♦rt❡✇❡❣✲❞❡ ❱r✐❡s ❡q✉❛t✐♦♥✳ ❈❤❛♦s✱ ✷✻✭✽✮✱
✶✸✾✲✶✺✹✳

✇❛✈❡❧❡t ♠❡t❤♦❞s✳ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛♥❞
❈♦♠♣✉t❛t✐♦♥✱ ✶✺✵ ✹✶✼✲✹✸✽✳
❬✸✺❪ ❱❛s✐❧✐❡✈✱

❖✳

❱✳✱

P❛♦❧✉❝❝✐✱

❙✳

✫

❙❡♥✱

▼✳ ✭✶✾✾✼✮✳ ❆ ▼✉❧t✐❧❡✈❡❧ ❲❛✈❡❧❡t ❈♦❧❧♦❝❛✲
t✐♦♥ ▼❡t❤♦❞ ❢♦r ❙♦❧✈✐♥❣ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧
❊q✉❛t✐♦♥s ✐♥ ❛ ❋✐♥✐t❡ ❉♦♠❛✐♥✳ ❏♦✉r♥❛❧ ♦❢
❈♦♠♣✉t❛t✐♦♥❛❧ P❤②s✐❝s✱ ✶✷✵✱ ✸✸✲✹✼✳

❆❜♦✉t ❆✉t❤♦rs

❈❛r❧♦ ❈❆❚❚❆◆■

✐s Pr♦❢❡ss♦r ♦❢ ▼❛t❤❡♠❛t✐✲

❝❛❧ P❤②s✐❝s ❛t t❤❡ ❊♥❣✐♥❡❡r✐♥❣ ❙❝❤♦♦❧ ♦❢ ❚✉s❝✐❛
❯♥✐✈❡rs✐t②✱ ■t❛❧②✱ ❆❞❥✉♥❝t Pr♦❢❡ss♦r ❛t t❤❡ ❚♦♥
❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❍❈▼❈✱ ❱✐❡t♥❛♠ ❛♥❞

❬✸✻❪ ❨❛♥❣✱ ❳✳ ❏✳ ✭✷✵✶✶✮✳ ❋r❛❝t✐♦♥❛❧ ❋✉♥❝t✐♦♥❛❧

❍♦♥♦r❛r②

Pr♦❢❡ss♦r

❛t

t❤❡

❇❙P

❯♥✐✈❡rs✐t②

❆♥❛❧②s✐s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❆s✐❛♥ ❆❝❛✲

✐♥ ❯❢❛✱ ❘✉ss✐❛ ✳

❞❡♠✐❝✱ ❍♦♥❣ ❑♦♥❣✳

♣❛♣❡rs✱ ❤✐s r❡s❡❛r❝❤ ✐♥t❡r❡sts ❢♦❝✉s ♦♥ ✇❛✈❡❧❡ts✱
❢r❛❝t❛❧s✱

❬✸✼❪ ❨❛♥❣✱ ❳✳ ❏✳ ✭✷✵✶✷✮ ▲♦❝❛❧ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉✲

❢r❛❝t✐♦♥❛❧

❆✉t❤♦r ♦❢ ♠♦r❡ t❤❛♥ ✷✵✵
❛♥❞

st♦❝❤❛st✐❝

❡q✉❛t✐♦♥s✱

♥♦♥❧✐♥❡❛r ✇❛✈❡s✱ ♥♦♥❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ s②st❡♠s✱

❧✉s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❲♦r❧❞ ❙❝✐❡♥❝❡ P✉❜✲

❝♦♠♣✉t❛t✐♦♥❛❧

❧✐s❤❡r✱ ◆❡✇ ❨♦r❦✱ ❯❙❆✳

♠✐♥✐♥❣✳

❛♥❞

♥✉♠❡r✐❝❛❧

♠❡t❤♦❞s✱

❞❛t❛

"This is an Open Access article distributed under the terms of the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited (CC BY 4.0)."

✷✸✽

❝ ✷✵✶✽ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮