Материалы к экзамену по функану (1135187), страница 3
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→ P (σ(A)) ´ÛC°g[*kieº¾d¯ YXZ]l±g3Ì jmkiYrnik³g»¼3e_°elje·ip¹gdÔmip¹jmi·Y\XZpm¹Ò±_Y\pi]le^ie_±m7[*kieº¾Y\Xr]l±_eË[*kiecre·¹jYrke_±kg σ(A)Ï;±XrÆÚf ²ije_]lk¿U± H m C(σ(A)) XZe_e]l±_Y\]_XZ]_±_Yrkkie ¯XZee_]l±Yr]_Xr]l±_Y\kikieH[*¿u²ekiYr²i^Yr^i¿R±kieXr]lm1²^ief³elj¾glY[ U kgor]lmF²i^eXZ]_^glkXr]l±g´ÛBXZ]»g_Yr]_XǼSXèf³YZj³g_]l®R²ieX½jYZf³kiminSÈdg_cf³e_p¹g_q\gl]_YZj®XZ]l±gi´ U A(P ck Ak h) = U (P ck Ak+1 h) = P ck Pk+1 =PPh#]º´ Y_´ U Ax = tU x mijmh²ie_j³glcZgl¼ y = U x ⇒ x = U −1y h U AU −1y(x) =t ck Pk = tU ( ck Ak h)Êhl]8eYZXl]®³frYnrXl]_±ml]YZj®kie¸eX ÊÎ YXZ]l±_ji¼iYr] kim]ºg_^iki¿RnËmqre[*e^ÀSmiq[ A X Λ ´ÊùÐkie±g¶²ie8kiYr²i^Y ¯ty(x)^i¿R±kieXr]lm1o\]leËUXZ²^gl±Y½fjm±_e8fji¼F±XZY\cre²i^ieXZ]l^g_kXZ]_±gi´µ¶gl²e[*kim[ÁÀSe_^[ jmi^e_±_p ]_Yre_^Y\[*¿ úÖmXrX\giÓO±XǼipimnjmikYrniki¿Rn¨À kipiÔime_kglju±ÌÍmp¹glpX½j³Y½fXr]l±_miYhÖ± C(K) hÐcÇf³Y K ¯ pÊe[*²³glpi]lke_Y3²ief[*keº¾dY\Xr]l±_e RÏH²^iY½fXr]l±g_m[ï±Â±_mÊC([a,fY F (fb])) =Rbh kYrpÊe_]le^g»¼VÀ kpiÔim¹¼Fe_c\^glkim·iYrkike_nñgl^im³glÔimim)´f (x)dΦ(x) Φ ¯(x, y)a])ûO$;)¥ t 痢3Ö?Mu§¥3 vt ¥3 v ¡;Ú£Oþ ¢C¢;«Ù¡; *t w Ö t ¡CÚ£O¥3v R¥3¥3u¡C t*w Cý^O¥3¢C)þz¢;;£Á¢ zt t C«XR¥3¢ wÛC°_e° Î m[²i^iYZf³¿Úf ÊÎ mn1°m¹jYr]ºh nÊf³¼Ve]]_^iYr°e_±g_kim¹¼Ãk³g»jmi·m¹¼VÔmip¹jmi·Y\XZpÊecreH±_Y\pi]le_^³gi´Õ Y\[+[+g´ ¬S XZ]l® H ¯ XZY\²gl^g_°_Yrj®_kieYRcrm¹j³®_°_Y\^i]le±_e¸²i^eXZ]_^glkXr]l±_e³h A : H → H ¯ Xrg_[*eXZe²i^¼¹¾dYrk ¯ki¿Rn´;ecÇfg¶± H [*eº¾kie¶±¿R°_^g_]l®¶pÊe_kiY\·iki¿RnHm¹jmÃXr·iYr]_ki¿RnËkg_°_e_^q\g[*pik ]l¿OÒ²ief³²^ieXr]l^glk³XZ]l± HnXZe¸Xr±_enXZ]l±g_[*mÓ Hn ⊥ Hm, ∀n 6= m h H = L Hn m AHn ⊆ Hn hl²i^mi·iY\[ A |H m[*Y\Yr]¶Ôimp¹jmi·iYXZpimn±_Y\pi]le^´î8e_p¹g_q\gl]_YZj®XZ]l±eÓ² Xr]l® h1 6= ´ ¬S XZ]_® H1 ¯ q\g[*¿RpÊg_kimiYFe_°elje·ipim h1, Ah1 , Ah2 , ...
´OÛC·iY ¯±_m¹f³kiehR·]le AH1 ⊆ H1 h h1 Ôimp¹jmi·iYXZpimnf³ji¼ A |H ?´ #g_[*Y\]lm[ðq\g_ef³kiehÚ·i]le A(H1⊥) ⊆ H ⊥ Óh*]leDYXZ]l® kg_X H = H1 ⊕ H ⊥ me_°g7²i^ie1 Xr]l^g_kXZ]l1±g(Ak, h) = (k, Ah) = 0 ∀k ∈ H1⊥ , h ∈ H11miki±gl^im³glki]lk¿e_]_kieXrmi]lYZj³®_kieH²i^im³[*YrkiY\kim¹¼ A ´Þ eº¾kieËXr·imi]»gl]l®³h·i]_e h = e ¯ ²iYr^i±¿Rn3°g_qrmXrki¿Rn3±_Y\pi]le_^3e^i]lekie_^[*m^ie_±glkikiecre°g_qrmX\g {e }± ´ Ü X½jmh)]leDkgl1nÊfYr]_1XǼ´ÑÚeqr®[*Y[m²i^ef³YZj³g_Y\[u] ¾dYËXrg[ nÆe_²HYr^g_ÔimiƲiHeXZ1]l^6=e_YrHkm¹¼ H2 h·i]le;mBf³jin¼ h: 1 ´ e¬ n e_∈/j H·im1 [ ¾dY H = hH21=⊕Hen2 ⊕(H1 ⊕H2)⊥ hXre±XrY\[*m¸Xr±_e_n ¯XZ]_±g_[*mVmiqBjY[+[*¿¶´ ¬ ^ief³e_ji¾gl¼Ë²i^ie_ÔY\XrXC²ieglk³g»jecrmim ° f³Y\[XZ]l^iemi]l®fe²e_°_YZf³kiecrepÊe_kiÔ³gÅÌÍ]º´ p´+°g_qrmX en XZ·iY\]lki¿Rnh*]le%²ieXZ]l^ieYrkimY1pÊe_^^iYrpi¯ ]_kieh*²ieXZ]l^ieYrkik¿RHn3, kHgl°4e_, ^ ... f³e±ljY\]l±_e^¼iYr] XZje_±m¹¼[Âj³Y\[+[*¿mVkYB°eljY\YS·iY\[XZ·iY\]lYrkϽ´¬ elj³®_q ¼XZ®8jY\[+[*enVmV]lY\e_^iY[*e_nF²i^ieȸjecre°_m¹jY\]»gde·iYrk®²^ieXr]ledf³e_p¹glqgl]l®Y\e_^iY[ Ór² XZ]l® A : H → H ¯ Xrg[*eXZe²i^¼¹¾dY\kiki¿RnhZ]_e_cÇfg A kmi]»gl^ikieRo\pi±_m±g»j³Yrki]lY\k¶e_²iY\^gl]_e_^ ±dkYrpÊe_]le^ie[ L 2(µn ) hcÇfY µn ¯ kiY\pÊe_]le^i¿RY;°e_^iYrjYr±XZpimiYB[*Yr^i¿kg σ(A) higA : (fn (t)) → (tfn (t)PXZp¹glji¼i^ikieYB²^ie_miq\±_YZf³YrkimYCqgºfg_Yr]_XǼVpÊg_p L (f,´g) = (fn , gn )µnnn1O$;)¥ t w Ö¥ w v t ¥3 v ¡;Ú£Oþ ¢C¢;L×Ö¡; tw ×ý v ¡ t t #ª x ¢;¥«³;ª x Ö w ýÖ¥ ¡C; vw t ¢ vw ýÖ§Ö¥3?Mu¢ ¡;) t*w A_O¥3¢C)þz¢;;£ð¢ t« t ¢;? ;¢C¢A` ý vw ýÖ¢;¢aM5R¢;?P;? ýï¡;; vw t ¢ vw ý L2 (R, ν) ÑÚkim[+g_kimiY_Ó#f³e_p¹glqgl]lYrj®Xr]l±_e1kiYr²eljkieÔiYrkkie_Y´)Ñ[*e_miÒDjY\piÔim¹¼¹ÒÌ ùÚ[*e_ji¼ikie_±³h)`laalbÏ;²ieX½jYZf³k¹¼¹¼jY\piÔim¹¼ ¯ ]_eRjmß_ß ¯ ¼hr]leRjmߺ` ¯ ¼Bm¶ekgOq\g_pÊg_ki·imi±glY\]_XǼBg_pip ^gl]R²^iY½f³¿Úf Î m[F^g_q½f³Yrje[O´rÑDjY\piÔim¹¼¹ÒÌ ø ecZgº·iYr±³hº`_a_a_íÏo\]»g;]lY\e_^iY[+gBY\XZ]_®hlkie³hºpHXreº¾gljYrkimÆ8hº±¶[*e_Y\[Dorpiq\Y\[*²¹j¼i^iYR±fe_p¹glqgl]lYrj®Xr]l±_YÚqrm¹¼iY\]°_e_j®_È ÊÎ g»¼7f³¿R^gi´ÑzãSm^im¹jije±_Y ¯ 7 ±_miÈdmglkimÌ÷äõ]lYre^iY\[*¿ÙmÂqgºfgº·imÂÀ kipiÔmie_kglj®_ke_creFglk³g»jmiqgºæõÏor]_e_]B^glqZf³YZj1ÌôXZ²iY\pi]l^glj®_k³g»¼8]lY\e_^imi¼8e_²iY\^gl]_e_^ie±Ï#^³glqr°mi^g_Yr]_XǼ¸²ief^ie_°kiem8fg»¾YÖ²ie_jkieXr]l®_ÆdhÒe_]¼mJ±_YXZ®[+gVXZ±e_Y\e_°_^³glqrke ¯ kie3m%]ºg[OhpJkiY\Xr·g_Xr]l®_Ædh#or]len=]_Yre_^Y\[*¿ïkiYr]º1´ glp·i]leF² XZ]l®F° fYr]VÒe]l®p¹glpÊe_n ¯ ]lekg_°_^ieXZe_p)´X\g_[*eXZe_²^¼¹¾Yrkik¿Rn!´ ecÇfg ]»glp¹g»¼°_e_^YZjY\±XZp¹gl¼8kiYre]l^imiÔ³gl]lYrj®Y\e_^iY[+giÓl² XZ]_®kgl¼7pekiYr·kg»¼7[*Yr^³g Aµ:kg HR→m7H]»glp¹¯ g»¼3e_c\^glkmi·iYrkkg»¼3°_e^iYZjY\±Xr∃pÊgl¼FÀ kipÔim¹¼ a(t) h·i]_e A kmi]»gl^ikie ¯orp±_mi±g»jY\ki]lYrkVe_²Yr^g_]le_^ Âx(t) = a(t)x(t) ± L2(µ) ´Lî8e_p¹g_q\gl]_YZj®XZ]l±eÓ#±_eXZ²ie_j®_q Y[+XǼ%^iY\q j®»]»gl]_e[²i^ieȸjecre7°m¹jYr]»g´´ Þ eº¾kie=XZ·im]»gl]l®³hÊ·i]le µn (σ(A)) = 2−n Óf³e[*keº¾Hm³[Âk ¾k ƨ[*Y\^ kgXZee_]l±Yr]_Xr]l± Æ HÎ mi=näè[+g_HXrÈ8n ]»gl°ki¿RnËpÊe_oÀ ¯¯ÀSmiÔimYrki]ºæ λ = 2−n(µn (σ(A))−1 ¯ ²ie_j ·im[ kimi]»g_^iki¿RnÂmiq\e[*e_^³ÀSmiq\[Oh]lelj®pÊe3±_mÊfÅe²iYr^g_]le_^³gF±miq[*YrkiY\kikie[²i^eXZ]_^glkXr]l±_Y° f³Yr] ¾dYkiYhg x(t) → λ−1 tx(t) ´#î8ee_²i^iYZf³YZj³m[u[*Y\^ → tx(t)kg [− kAk , kAk] ¯ kg_²i^im[*Y\^h³k j¼[*m=±kiY x(t)#h7m³^_gZXpmÊfglY[ÉkglÈdm7e_]_^iYrq\pimh#XZee_]l±Yr]_Xr]l± Æ Î miYσ(A)i²e#Ói²\Yi^_±R¿%nerXº]_g_±m[k1g*[YZXl]Yh_±_]_ei^eDn\q_gn*[rYF]*[Y\Xr]lem%]ºg_p7fg»jY\YXèf³±micZg_[*mHne_²^iY½f³YrjRm[kglÈ [*Yr^ kgH±XZY\[ R mijmFpekiYr·kie[Åe_]l^iY\qrpÊY ¯ ±[kAkq\g_±_m,Xr3mkAk][*eXr]lmVe_] Hn ´ù ·iY\]le[kg ¯È8Y\nD²i^Y½f³±gl^imi]_YZj®kie_n7ke_^[*mi^e_±_pmh³²elj ·iY\kikg»¼7±Vmi]lecrYH[*Yr^gð f³Y\]ÃpÊe_kiY\·ikienÅÌÝ[*Y\^g R ^gl±kgÏZ´¹ù ·Yr]le[XZeºÒ¹^g_kiYrkimi¼Xrp¹g»ji¼i^ike_cre8²i^ie_mqr±_YZf³Yrkm¹¼Ë²ieXr]l^ieYrkikgl¼ a ° f³Yr]¸ecr^g_kimi·iY\kikien˱dXrm¹j 2e_c\^glkmi·iYrkkieXr]lm1m³XÇÒef³kiecrede_²iY\^gl]le^gi´cÌ bºÏdegf Ö t § t*w #ª xÐvw ýÖz t ýÖ¢ vw ý t kAk2 = kA2k h w C¢;¢C;i;ÿSª;£vt ¥3 v ¡;Ú£Oþ ¢C¢;«Ù¡;) t*w t4jS¬ Xr]l® A ¯ X\g_[*eXZe_²^¼¹¾Yrkik¿Rnh]lecÇfg kAk2 = sup (Ax, Ax) = sup (A2 x, x) = A2 hXr[O´^glqZf³YZj7ßai´2k;e_^e_È8mn°m¹jY\]Ó ¯ Ͻ´Êµ¶gS±XǼipimnËX½j ·g_nkxk=1²^imi±_YZf jY\[+[ hqgºf³kxk=1Y\nXZ]l± Y[ Ʊ8°_m¹jY\]lY_ÓÕ Y\[+[+gVßÓi² Xr]l® A ¯ Xrg[*eXre_²i^¼¹¾dY\kiki¿Rnh Q (x) = (Ax, x) ¯ Y\creHpi±g»f³^g_]lmi·ikgl¼VÀSe^[+gi´)e_cÇfg´³î8Yrn³XZ]l±mi]lYZj³®_kie³he_·Yr±_m¹Af³kgeÔiYrkip¹g suphef³kglpÊeÃ[*eº¾ ¯supkxk=1 QA (x) = kAkkAkkxk=1 QA (x) ≤ákieV²ielj ·mi]l®Fm=e°_^g_]lk Æ8Ó²i^e_±_Y\^¼iYr]_XǼh·i]le QA(x + y) − QA(xhfgljYrYY\XZjm− y)= 4R (Ax, y)h]le 4R á (Ax, y) ≤ c kx + yk2 + kx − yk2 ≤ 2c kxk2 + kyk2 ²ieËkYr^g_±_Yrk ¯supkxk=1 |QA (x)| = cXZ]_± ²g_^g»jjYZjecr^g[+gi´ ¬ eljglcZg»¼ y = kxk Ax ²ielj ·m[ 4 kxk kAxk ≤ 4c kxk2 ¯ ²i^e_]lmi±e_²ie_jeº¾k ÆkAxk´e_ÔYrkip ´iÑmi]_e_crYsupkxk=1 QA (x) = kAk1Oü=¡;) w t ª x ¢ t £ w Ö;¥ t Mu¢×u;¥ t ¢ t dü=¡C w t ª x ¢ t £ w );¥ t v ¢;Öý t ¢;¢ t £ ¢ t v ¡;ª x §ý t ¢C; v ¡C w t ª x ¢; ¥3;Õ YrpÔimim%ùÚ[*e_ji¼ikie_±g [*Yrk¹¼De°_^i¿R±glÆC]_XǼF[*ke_cre1^glk®_È8Y8or]lecreË°m¹jYr]»g´ÑujY\piÔim¹¼¹Ò ø e_cZgº·iY\±g±X ¶e_²im³XrglkekY¸e_·iY\ki®²ief³^e_°_ke´iÑãSm^im¹jije±_Y ¯ 7 ±_miÈdmglkimFXZ²Yrpi]l^³g»j®ki¿RY¶]lYre^iY\[*¿¨YXZ]l®³hkie·i]_e]»g_[µSYrn[+g_kgih¹gH·i]_e ¯ Y Î YS·iY\cre ¯ kim° f³® ¯ kiYS^glq\°_Y\^iYrÈd®XǼ´ÛC²i^iYZf³YZj³YrkimiY´ ¬S Xr]l®1qgºfg_kie[*kieº¾Y\Xr]l±_e³h#kiYrpÊe_]_e_^gl¼ gljcrY\°_^g ²ief[*kieº¾Y\Xr]l± hXZef³Y\^¾Hg Î g»¼ X h)mÂq\g»fglkeVc\m¹j®°_Yr^X]le_±¯ eV²^ieXr]l^glk³XZ]l±e H ´ÛC]le°_σ^¯ gl¾dY\kimiY λB: B → L(H) ÌÍX±7²^ie ¯¯XZ]_^glkXr]l±_e3kYr²i^iY\^i¿R±_k¿OÒ=e_²Yr^g_]le_^ie±ÏBk³glqr¿R±glY\]_XǼD²i^ieYrpiÔime_kikienÂ[*Yr^ien%k³gXreqrk³gº·iYrkimi¼[*m± L(H) h+YX½jmɱ_¿R²eljkiY\ki¿ H →X½jeH±_m¹¼Ó λ(E) = λ(E)∗ ∀E ∈ B ÌÍ]º´ Y_´±XZYVmiq\(X,[*Y\^iB)m[*¿RY[*kieº¾Y\XZ]_±g²iYr^YZÒef³¼i]±ËX\g_[*eXZe_²^¼¹¾Yrkik¿RYCe²iYr^g_]le_^¿;Ͻh 1 ∩ E2) = λ(E1 )λ(E2 ) ∀E1 , E2 ∈ Bm λ(E1 t E2) = λ(E1 ) + λ(E2 ) fji¼VjÆ;°¿OÒ1kiY\²iYr^Y\XZY\p¹glÆ Î λ(Em¹ÒXǼ´µ¶glpÊe_kYrÔhʲieX½jYZf³kiYrY XZje_±miY+q\g_±_Y\^iÈdg_Yr]O²ieXr]l^ieYrkimiY λ ÓY\XZjm E ∈ B m ∃ lim E =EE1, EÌͱ2Yr^∈Ò¹Bkimin¸m¶kimi¾Hkmin8²i^Y½f³Yrj¿nnS T n→∞h²i^i²iekim[+glÆC]_XǼÂp¹glp limEn = T S Ek h limEY½fYZjX Î Y\Xr]l± Yr]ºhY\XZjm±_Y\^ÒÊkminÂmEkn =n k≥nn k≥nkim¹¾kimnV²^iY½f³Yrj¿^gl±ki¿;Ͻh¹]le s = lim λ(En ) X ÊÎ YXZ]l± Yr]mF^gl±Yrk λ(E) ´¬ ^im[*Y\^Ói² XZ]l® (X, B, µ) ¯ ²i^ieXZ]l^³glkXr]l±_eXSe_°¿R·ikien σ¯ gºfifmi]lmi±kie_n3[*Yr^e_n µ ´ ¬ elj³eº¾Hm³[ H =he²iYr^g_]le_^ [*kieº¾Yrkim¹¼3kgÒÊgl^g_pi]lY\^imXZ]_mi·iY\Xrp Æ À kipiÔimÆ E ´ Þ eº¾HkeL2 (X, µ) λ(E) = M (ξE ) ¯ °_YZf³mi]l®XǼh¹·i]leHor]le° f³Yr]H²i^ieYrpi]_e_^iq\kgº·ikgl¼1[*Yr^g´ã¶glp1mF±HXZj ·glYSe°_¿R·ikien1[*Y\^i¿¶hi²i^ieYrpiÔmie_kik ÆU[*Y\^ [*eº¾kieHmXr²ielj³®_qre±g_]l®¶fj¼Ve²i^iY½fYZjYkim¹¼=miki]lY\cr^glj³g Õ Y\°_Y\cZgi´ ¬S XZ]_® f ¯ B ¯ miq\[*Y\^im[+gl¼De_c\^glkim·iλYrkik³g»¼7·imXZje_±g»¼DÀ kipiÔm¹¼Dkg X ´î¸ji¼ ¯²i^ieXZ]lY\niÈ8miÒÀ kipÔiminh²i^imikim³[+glÆ Î m¹ÒHjmiÈd®8pÊe_kiY\·ikieYR·imX½j³e¸^glqrjmi·ik¿OÒHq\kgº·iY\kimin an miki]lY\cr^gljÕ Yr°YrcZg%Y\Xr]l® S (f, λ) = P a λ {x ∈ X | f (x) = a } hÖk gJ±Âe_° Î Y[zXZj ·glYFjÆC°gl¼miq[*Yr^im³[+g»¼À kipiÔimi¼%^g_±_kine[*Yr^ike1²i^mi°ljmin ¾glY\]_XǼ%Xr±_Y\^Ò mJXZknmiq ²i^ieXr]lYrnÈ8m[*mÌͱ7ãSm^im¹jije±_Yh#kgl²i^m[*Yr^h²ie_j³glcZg_Æ;] Sn (f, λ) = P k λ x ∈ X | k ≤ f (x) ≤ k+1 ÏBm%mki]lYrc\^g»j Õ Yr°YrcZgòie_kim³[+glYr]XǼhp¹glp2²i^iYZf³YZj7mki]lYrc\^g»j³e_±H²iek∈Z²i^ieXZ]lY\niÈ8m³[O\´ #g2[*Yr]lm[Oh³·i]leo\2]le_]mik]lYrc\^g»j3Y\Xr]l®jmkiYrnik³g»¼FkiYr²i^Yr^i¿R±kg»¼À kipiÔimi¼ H → H H¬S Xr]l®´ ¬ e_jeº¾m[hiqgºfg_±glY[*e_Y¶ÀSe_^³[ j³e_n´ ¬ ^ie_±Y ¯R η) &^¼iYr]_XǼh#·]lξ,e η(R∈fH(x)& λ(x)ξ, η) =λξηR :fB(x)→& λCξη (x) ´ÛC²i^iYZf³YZjm³[]lY\²iYr^i®Fλeξη²i(E)Yr^g_]l=e_^ (λ(E)ξ,A = f (x) λ(x)mVfji¼FkiY\e_cr^³glXkimi·Yrkikien B ¯ miq\[*Y\^imX[*e_n3À kipiÔmim f kg X ´ ¬S Xr]l® DA ¯ [*keº¾dY\Xr]l±_eH±YrpiX]le^ie_±H± H hfji¼pÊe_]le^i¿OÒ8XÇÒef³m]_XǼ R |f (x)|2 & λξξ (x) ´ Þ eº¾kieC²i^ie_±Yr^im]l®hº·i]leSo\]leBjmikiY\nikie_YÚ²ief³²^ieXr]l^glk³XZ]l±e± DA ´³î¸ji¼ ξ ∈ DA e²i^iXY½fYZjm[e_²iY\^gl]le^ A 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YrnF²ieX½jYf³e±gl]_YZj®kieXr]lm fˆn(∞) = 0 ´ q¸kiYr²^iYr^i¿R±kieXr]lm ¬ @m7^O(g_±_λki)e[*Yr^ikiecre²i^mi°ljmi¾dYrkm¹¼ f À ¯ ¼[*m fn¯±_¿R]_Yrp¹glY\]h¹·]lemF±²^iY½f³YrjY f(∞)´ˆ=0 Y8eË^DY8[g3ßY´ ô:êÔUÓ;æ¢ëÜ*×CÔâUÓסé¬ßFØXÞ%ï2öçÒë2ÕFÖ,á f ∈ L1 (R) ÞÀÙÖ,Ô î ì2ß x Ù¡Ý Û ÔÜ\اßFاÔ"ë2ÕàÜ\Ô¡Ù¡Þ\ß ÞXاަô:êë2ØXì2íXÞ;ïf (x) = (v.p.)Rg(t) =fˆ(y)eixy dy = limR→∞îïeHp3gq¶gH]YBj3®XS]H±e³´ ¬S Xr]l®R1IR (x) = √2π´ ¬ ^im³[*Yrk¹¼¹¼V]lYre^iY\[ °mikimh=12πÞXاÖ,ßwòcÓ\Þ¡Ó*ë2ßàæ×8ÙÀاßìÔxÖ,Ô«Ó*Ôxã5ÔxìÓ*ßwÕFÖ,ØXÔ¡ÕFÖ,ÞRRô Ô ßÙ!Þ§åØXÔ Ú Ô¡òcÓ)×!ØXÞ î ßا×öçfˆ(y)eixy dy fˆÚ îf (x+t)−f (x)tZRt = 0öç,èXÔ¡òåx×−Reixy1fˆ(y)dy =2π−RZReixy−RZe−iyz f (z)dzdyR+∞+∞+∞ZZZZR1eiy(x−z) R2 sin(y(R − z))1|−R dz =dz =f (z)f (z)f (z)eiy(x−z) dydz =2π−ix2πx−z−∞−∞−R1=π−∞+∞Zsin(Rt)f (t + x)dtt−∞Ñp ^³XZY[+gl]»g_kg»jmq\g¶[*¿É²ie_p¹g_qr¿R±gljm8·]le∀R limR→∞+RR−Rsin(Rt)dtt=π´ ¬S XZ]_®¶]lYr²iY\^i®¶q\g»fglke´ ¬ _e o\]le[ he_·iY\±_mÊfkieh)X ÊÎ Y\Xr]l± Yr] T1 > 0 h·i]_e f (x) RT sin(Rt) dt − f (x) <ε>0t π −TY¯ÝÌklg²e*[ikmO[#h·l]Ve*[ ¿^_g\XrX+[lg_]i^im±_g\Y[Êpeikipi^\Yl]ik_eYZÏ´UÑZX¹mjimikl]\Yrci^im^εx3 ∀T > T1 , ∀R > T1[*eXr]lm3kg_nÊf³Yr]XǼ T2 > T1 hi·i]le R f (x+t) < ε ´³µ¶g_pekiYrÔ)h¹miq XZje_±m¹¼Ëî8mkim g(t) = f (x+t)−f (x)t3tmiki]_Yrcr^mi^ Y[+g7kg [−T2, T2] Ìݱ7|x|>Te_p^iY\Xr]lkieXr]lmk ji¼ekgFmiki]_Yrcr^mi^ Y[+g7²ie X½j³e_±_mÆ8hg7±XZÆÚf ±_kYk ji¼ ¯ e_·Yr±_m¹f³kiehÐmiki]lY\cr^im^ Y[+gűXZm¹j ]lecre³hзi]_emik]lYrc\^imi^ Y\[+g f ϽhRXZ]»g»j³e°_¿R]l®³hÐk³glnÊf³Y\]_XǼ TRhi·l]dei²i^mÌͲie Õ Y\[+[*YºÏ½´Êù ·iY\]le[%o\]lm¹Ò]l^iYrÒHeÔiYrke_pf (x+t)−f (x) dt < 3εR1 > 0R > R1 sin(Rt)t−Tm[*Y\Y\[²i^iemiqr±elj®kie_Y222 ZT2ZT2ZT2 f (x) εsin(Rt)sin(Rt)sin(Rt)f(x)f(x)|IR − f (x)| ≤ IR −dt+dt − f (x) ≤ IR −dt+πttπt π 3−T2−T2−T2 ∞ZZT2sin Rtεsin Rt 1f (t + x)≤ + f (x)dt −dt ≤3 πtt−∞−T2 T ZZ2ZT2f (t + x) − f (x)sin Rt 1 sin Rt 1 εf (x)f (x + t)+sin(Rt)dt −dt + dt ≤ ε3 π ttt π −T2−T2|t|>T2·i]lemF]l^Yr°_e±gljeXr®¶f³epÊg_q\g_]l®´`_í.
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