A review on harmonic wavelets and their fractional extension

In this paper a review on harmonic wavelets and their fractional generalization, within the local fractional calculus, will be discussed. The main properties of harmonic wavelets and fractional harmonic wavelets will be given, by taking into account of their characteristic features in the Fourier domain.

❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r ❆ ❘❡✈✐❡✇ ♦♥ ❍❛r♠♦♥✐❝ ❲❛✈❡❧❡ts ❛♥❞ ❚❤❡✐r ❋r❛❝t✐♦♥❛❧ ❊①t❡♥s✐♦♥ ❈❛r❧♦ ❈❆❚❚❆◆■ 1,2,∗ ❊♥❣✐♥❡❡r✐♥❣ ❙❝❤♦♦❧✱ ❉❊■▼✱ ❚✉s❝✐❛ ❯♥✐✈❡rs✐t②✱ ❱✐t❡r❜♦✱ ■t❛❧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 − 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) − cos(2πx) +i 2πx 2πx ✷✷✺ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r Re(φ ) 0.6 Im (φ ) π π - 0.25 - 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 ❞❡❢ − 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 − = 2πix ∞ k=0 (2πi)k k x (k + 1)! ✭✻✮ ▲❡t ✉s s❧✐❣❤t❧② ♠♦❞✐❢② t❤❡ ❤❛r♠♦♥✐❝ s❝❛❧✐♥❣ lim ϕr (x) = lim ϕi (x) = x→∞ − 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) = ❞❡❢ x→0 ϕα (x) = ▼♦r❡♦✈❡r✱ s✐♥❝❡ eπin =   1,  −1, Eα (2απix) − , 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) = ✭✹✮ 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 − = 2παix ✭✶✸✮ ❢r❛❝t✐♦♥❛❧ ❛♥❞ ✐ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s ψ(ω) = ϕα (x) = | ■❙❙❯❊✿ ✹ | ✷✵✶✽ | ❉❡❝❡♠❜❡r 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 χ(2π + ω)χ(2π + ) 2π ω = χ(2π + ω)ϕ( ˆ ) ϕ (ω) = ✷✳✸✳ ❙❝❛❧✐♥❣ ❢✉♥❝t✐♦♥ ✐♥ ❋♦✉r✐❡r ❞♦♠❛✐♥ s♦ t❤❛t✱ ϕ (ω) = H ❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥ ✭✶✮ ✐s ❞❡✜♥❡❞ ❛s ϕ(ω) = ϕ(x) = 2π ❞❡❢ ✇✐t❤ ∞ −∞ ❙♦ t❤❛t✱ ✐♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥✱ ✐✳❡✳ ✇✐t❤ r❡✲ s♣❡❝t t♦ t❤❡ ✈❛r✐❛❜❧❡ ω ω 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 ψ (ω) = H 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) ✇❡ ❤❛✈❡ ω ω − 2π ϕˆ 2 ω χ 2π + = χ (ω) 2π = 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 = χ(2π + ω) ❜♦✉♥❞❡❞ ❢r❡q✉❡♥❝②✮ ϕ(ω) = ✭✶✺✮ H ω + 2π ✇❡ ♦❜t❛✐♥ ω χ (4π + ω) χ(2π + ) = 2π ∀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 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) = 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 ±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) = 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 (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) = , ✇❤✐❧❡ ❢♦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, = ✭✷✺✮ ▼♦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 = ✷✷✾ ❱❖▲❯▼❊✿ ✷ Pr♦♦❢✿ ❆❝❝♦r❞✐♥❣ t♦ ✭✶✵✮✲✭✷✷✮ ♦♥❡ ❤❛s t❤✉s ❜❡✐♥❣ ∞   αk                    αk∗             ϕ(x)❞x −∞ = 1, ϕ(x) = 2π 1, ϕ(ω) ∞ δ(ω) = 2π −∞ 2π 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 = = −∞ = 2π f (x), ϕ0k (x) 2π = = f (ω)e−iωk ❞ω 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) =           ∗    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 = ❲❛✈❡❧❡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) + + ∞ n βn k ψ k (x) + ❡r❢ (π √ σ) ϕ00 (x) + ϕ00 (x) ❡r❢ (2π √ σ) − ❡r❢ (π √ σ) × ψ00 (x) + ψ (x) , n=0 k=−∞ k=−∞ e−u du t❤❡ ●❛✉ss✐❛♥ ∗ n β n k ψk (x) + x = √ 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) ψ00 (x) + ψ (x) = n=0 k=−M N M M 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 ∼ = √ sin 2πx x √ ❡r❢ (2π σ) − ❡r❢ (π σ) 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 = e−(16x) ✐s ✭s❡❡ ❋✐❣✳ ❊①❛♠♣❧❡s ♦❢ ❍❛r♠♦♥✐❝ ✇❛✈❡❧❡t r❡❝♦♥str✉❝t✐♦♥ sin 2πx e−(16x) ∼ = 2πx π π − ❡r❢ 16 π π − cos 2πx ❡r❢ − ❡r❢ 16 π π − cos 6πx ❡r❢ − ❡r❢ − (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 /σ × ❚❤❡ tr✉♥❝❛t❡❞ ,M = ✇❛✈❡❧❡t s❡r✐❡s ✇✐t❤ N = ✐s s♦ t❤❛t ✐❢ ✇❡ ❝♦♠♣✉t❡ t❤❡ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ❜② ✉s✐♥❣ t❤❡ ❊qs✳ ✭✷✻✮ ✭♦r ✭✷✼✮✮ N ❡r❢ π − ❡r❢ π ✇❡ ✇✐❧❧ ❣❡t ❛ ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥✳ ✷✮ ✇❡ ❣❡t √ ❡r❢ (π σ), √ β ∗00 = 12 [ ❡r❢ (2π σ) α0 = α0∗ = β00 = ❡r❢ ❆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 ✐✳❡✳ f (ω) ∼ = 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 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 χ(2π + ω) f (ω) ∼ = 2π 0.2 ❋✐❣✳ ✸✿ ❍❛r♠♦♥✐❝ ✇❛✈❡❧❡t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ N f (x) = e−(16x) ❛♥❞ t❤❡ ✵✲s❝❛❧❡ N = 0, M = ❛♥❞ ✷✲s❝❛❧❡ N = 2, M = ❛♣♣r♦①✐♠❛t✐♦♥✳ M αk e−iωk + eiωk k=0 −n/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✱ 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❡ 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 + 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 ∞ e2πiα k k=−∞ × ψkn (x) + ψ nk (x)     αk            αk∗            n   βk                 β ∗nk            2π ϕα (ω)eiωk ❞ω = = ❇② t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t ❊qs✳✭✶✮✱ ✭✺✮✱ t❤❡ ❜❛s✐❝ ❢✉♥❝t✐♦♥s ♦♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞ 2π ϕα (ω)e−iωk ❞ω = = ✭✸✹✮ 2π αΓ(1 + α) n+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 = = 2π αk∗ = 2π αΓ(1 + α) ×                  β ∗n  k           = 2−n/2 2π e2πiα αΓ(1 + α) 2n+2 π f (ω)eiωk/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 ✐♥ 2π = e2πiα k αΓ(1 + α) 2n+2 π e2πiα ✭✸✻✮ 2n+1 π = 2−n/2 ∞ ψα (x) 2π e2πiα k αΓ(1 + α) f (ω)e−iωk ❞ω = 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✐♦♥ 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 = ✱ 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 = 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) = t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✇❛✈❡❧❡t ❝♦❡✣❝✐❡♥ts ✭✷✼✮✱ ✐♥ ❣❡♥❡r❛❧✱ ✈❛♥✐s❤ ✇❤❡♥ ❡✐t❤❡r f (ω) = 0, ∀k or f (ω) = Cnst., k = ❛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 ✐♥ ✭✸✽✮✳ −∞ = = 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 ❨❛♥❣ ❬✸✻✕✸✾❪✿ αk = ϕ (x) + ϕ0k (x) k x=2πh ✺✳✶✳ ❆❝❝♦r❞✐♥❣ t♦ ✭✷✶✮ ✐t ✐s −i2πhk e χ(2π + 2πh) (3) = χ(2π + 2πh) 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