Kan Fashe-fashe na Sassakan Firam Biyu a Sararin Hilbert

Teburin Abubuwan Ciki

• 1. Gabatarwa da Abubuwan Farko
• 2. Firam da Firam da aka Saka
• 3. Sakamakon Fashe-fashe don Sassakan Biyu
• 4. Siffanta ta Ma'aikaci
• 5. Bayyani na Ƙididdiga
• 6. Muhimman Fahimta
• 7. Ƙarshe

1. Gabatarwa da Abubuwan Farko

An gabatar da iyalan firam da aka saka ta Bemrose et al. a cikin 2015, wanda aka ƙarfafa ta aikace-aikacen sarrafa siginar da aka rarraba a cikin hanyoyin sadarwar na'ura mai auna firikwensin. Babban ra'ayi ya ƙunshi shirya siginar ta amfani da iyalan firam masu dacewa da nodes na firikwensin, tabbatar da ingantaccen sake gina sigar ba tare da la'akari da wane ɓangaren ma'aunin da aka samu ba. A ilimin lissafi, iyalin firam {f_ij}_{i∈I, j∈I_n} don sararin Hilbert mai rabuwa H an saka su ne idan, ga kowane rabo {σ_j}_{j∈I_n} na saitin ma'anar I, saitin {f_ij}_{i∈σ_j, j∈I_n} ya zama firam don H tare da iyakoki iri ɗaya. Wannan bayanin yana mai da hankali kan sassakan firam biyu (F, G), inda F = {f_i}_{i∈I} da G = {g_i}_{i∈I}, yana bincika ƙananan fashe-fashe waɗanda ke kiyaye kaddarorin da aka saka. Muna amfani da ma'aikatan haɗawa don sauƙaƙe hujjoji da bincika siffofi waɗanda suka haɗa da aikin tsinkaya mai karkata da kusurwoyin sararin mara komai.

2. Firam da Firam da aka Saka

Bari H ya zama sararin Hilbert mai rabuwa, kuma B(H) ya nuna aljabar ma'aikatan layi masu iyaka akan H. Ga ma'aikaci T ∈ B(H), R(T) da N(T) suna wakiltar kewonsa da sararin mara komai, bi da bi. Firam F = {f_i}_{i∈I} don H yana gamsar da A∥x∥² ≤ ∑_{i∈I} |⟨x, f_i⟩|² ≤ B∥x∥² ga duk x ∈ H, tare da mafi kyawun iyakoki A_F da B_F. Ma'aikacin haɗawa T_F : H → H an ayyana shi ta hanyar tushe na al'ada B = {e_i}_{i∈I} kamar T_F e_i = f_i, tare da ma'aikacin bincike T_F* da ma'aikacin firam S_F = T_F T_F*. Muhimman kaddarorin sun haɗa da: F firam ne idan T_F yana da yawa, kuma S_F yana da inganci kuma yana jujjuyawa. Firam na al'ada na biyu S_F^{-1}(F) yana ba da damar sake ginawa: x = ∑_{i∈I} ⟨x, f_i⟩ S_F^{-1} f_i.

Firam da aka saka, kamar yadda ayyana 2, suna buƙatar cewa, ga kowane rabo {σ_j}_{j∈I_n} na I, sakan {f_ij}_{i∈σ_j, j∈I_n} ya zama firam tare da iyakoki iri ɗaya A da B. Raunin sakan yana jujjuya buƙatun daidaitawa. Theorem 1 (daga [2]) ya kafa cewa sassakan biyu masu rauni an saka su, yana sauƙaƙe bincike. Wannan bayanin yana mai da hankali kan biyu (F, G), yana amfani da ka'idar ma'aikaci don samar da yanayin fashe-fashe.

3. Sakamakon Fashe-fashe don Sassakan Biyu

Sakamakonmu sun dace da wallafe-wallafen da suka wanzu ta hanyar bincika ƙananan fashe-fashe δF = {f_i + δ_i}_{i∈I} na firam F. Ƙarƙashin wasu sharuɗɗa akan ∥δ_i∥, biyun (F, δF) ya kasance an saka shi. Musamman, idan ∥δ_i∥ < ε ga duk i kuma ε yana da isasshe ƙanana dangane da iyakokin firam, to fashe-fashe yana kiyaye kaddarorin da aka saka. Hujjoji suna amfani da ma'aikatan haɗawa: Bari T_F da T_δF su zama ma'aikatan haɗawa na F da δF. Idan ∥T_F - T_δF∥ < A_F / 2, to T_δF ya kasance yana da yawa, yana tabbatar da δF firam ne. Don sakan, la'akari da kowane rabo σ; Ma'aikacin haɗawa T_σ don sakan {f_i}_{i∈σ} ∪ {δ_i}_{i∈σ^c} dole ne ya gamsar da iyakoki iri ɗaya. Muna nuna cewa ∥T_σ - T_F∥ za a iya sarrafa shi, yana kiyaye jujjuyawar S_σ.

Muhimmin lemma: Idan F da G an saka su tare da iyakoki A, B, kuma ∥T_F - T_G∥ < A/2, to ƙananan fashe-fashe na kowane firam suna riƙe da kaddarorin da aka saka. Wannan ya shimfiɗa zuwa jerin inda fashe-fashe suke da adadi, yana haɗa sakamakon da ya gabata.

4. Siffanta ta Ma'aikaci

Muna siffanta sassakan biyu ta hanyar kusurwa tsakanin sararin mara komai na ma'aikacin haɗawa gauraye da kewayon aikin tsinkaya mai karkata. Ayyana ma'aikacin T_{F,G} : H → H × H ta T_{F,G} x = (T_F x, T_G x). Biyun (F, G) an saka shi ne idan ga kowane rabo σ, ma'aikacin da aka iyakance T_σ = (T_F|_σ, T_G|_{σ^c}) yana da yawa. Wannan yawan yawa yana daidai da sharadin cewa kusurwa tsakanin N(T_{F,G}) da R(P_σ) an iyakance shi a ƙasa, inda P_σ aikin tsinkaya mai karkata ne akan sararin da ya dace da σ.

Musamman, bari θ_σ ya zama mafi ƙarancin kusurwa tsakanin N(T_{F,G}) da R(P_σ). Sa'an nan, (F, G) an saka shi ne idan inf_σ θ_σ > 0. Wannan yana kama da siffofi na firam Riesz, inda daidaitawa a cikin rabowa yake da mahimmanci. Aikace-aikacen sun haɗa da tabbatar da sakan don firam masu alaƙa ta hanyar fashe-fashe masu ƙanƙanta ko bambance-bambancen matsayi na iyaka.

5. Bayyani na Ƙididdiga

6. Muhimman Fahimta

• Sassakan biyu masu rauni an saka su, yana sauƙaƙe bincike zuwa wanzuwar iyakokin firam kowane rabo.
• Ma'aikatan haɗawa suna ba da hanya ɗaya don hujjojin fashe-fashe, suna guje wa gardama masu dogaro da daidaitawa.
• Yanayin kusurwa ya shimfiɗa zuwa sararin Banach da firam na haɗawa, kamar yadda aka nuna a cikin ayyukan da suka gabata.
• Ƙananan fashe-fashe a cikin ƙa'idar ma'aikaci suna kiyaye kaddarorin da aka saka, tare da bayyannun iyakoki da aka samo daga ƙa'idodin firam.
• Aikace-aikacen a cikin hanyoyin sadarwar na'ura mai auna firikwensin suna buƙatar ƙarfi ga gazawar node, wanda aka ƙirƙira ta hanyar rabowa.

7. Ƙarshe

Wannan bayanin yana ci gaba da ka'idar firam da aka saka ta hanyar kafa sakamakon fashe-fashe ta hanyoyin ka'idar ma'aikaci. Muna nuna cewa ƙananan fashe-fashe a cikin ƙa'idar ma'aikacin haɗawa suna kiyaye kaddarorin da aka saka, tare da iyakoki da aka bayyana cikin sharuɗɗan ƙa'idodin firam. Siffanta ta hanyar kusurwoyi tsakanin sararin mara komai da kewayon aikin tsinkaya yana ba da sabon hangen nesa, yana haɗa sakan da kaddarorin geometric a cikin sararin Hilbert. Aikin nan gaba zai iya ƙaddamar da waɗannan sakamakon zuwa K-firam da firam masu ci gaba, ƙara haɓaka aikace-aikacen a cikin sarrafa rarrabawa.