Конспект лекций по математическому анализу. Шерстнев А.Н. - 417 стр.

                                                                  Z
kAK IZWESTNO (SM. 171.7), OTOBRAVENIE f ! f ] (t) = (2),1=2 f (x)e,ixt dx
(t 2 R) | BIEKCIQ S NA S . pOKAVEM, ^TO ]) | IZOMETRIQ. mY IMEEM
               Z                 Z                  Z
        ] 2        ]    ]          ]
      kf k = f (t)f (t) dt = f (t)((2)     , 1 = 2   f (x)e,ixt dx) dt
               Z               Z                      Z
            = f (x)((2),1=2 f ](t)eixt dt) dx = f (x)f ][(x) dx
               Z
            = f (x)f (x) dx = kf k2:
(W TRETXEM RAWENSTWE MY ISPOLXZOWALI LEMMU x165).
   tAKIM OBRAZOM, ]) PRODOLVAETSQ DO OGRANI^ENNOGO LINEJNOGO OTOBRA-
VENIQ U : L2(R) ! L2(R). iZ WYKLADKI
                  Z             Z             Z
         ]           ]
       hf ; gi = f (t)g(t) dt = (2)  , 1 = 2   f (x)e,ixt dx 
 g(t)dt
                   Z                  Z                 Z
              = f (x)((2),1=2 g(t)eixt dt) = f (x)g[(x) dx
              = hf; g[i (f; g 2 S )
I NEPRERYWNOSTI SKALQRNOGO PROIZWEDENIQ, POLU^IM
               hUf; gi = hf; g[i (f 2 L2(R); g 2 S );
OTKUDA U g = g[ (g 2 S ). oPERATOR U QWLQETSQ UNITARNYM OPERATOROM W
L2(R) I NAZYWAETSQ OPERATOROM fURXE-pLAN[ERELQ. f dOSTATO^NO UBE-
DITXSQ, ^TO U UDOWLETWORQET USLOWI@ 2(B). pUSTX f 2 L2(R) PROIZWOLEN
I POSLEDOWATELXNOSTX fn 2 S TAKOWA, ^TO fn ! f W L2(R). tOGDA
          UU f = lim UU fn = lim U (fn[) = lim fn[] = lim fn = f:
                   n              n                 n       n
tAKIM OBRAZOM,   UU    = I . aNALOGI^NO, U U = I .g
   4. z A M E ^ A N I E. pRIWEDEM FORMULU DLQ \WY^ISLENIQ      " OPERATORA U
                                        ]              ZN
NA FUNKCIQH IZ L2(R). oBOZNA^AQ fN (t) = (2),1=2 ,N f (x)e,ixt dx (t 2 R)
(W PRAWOJ ^ASTI STOIT INTEGRAL lEBEGA), IMEEM
    kUf , fN] k2 = kUf , U[,N;N ] f k2 =Z kU (f , [,N;N ] f )k2
                   = kf , [,N;N ] f k2 =    jf (t)j2dt ! 0 (N ! +1):
                                            jtjN

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