Конспект лекций по математическому анализу. Шерстнев А.Н. - 447 стр.

   (ii) eSLI 1 | SOBSTWENNOE ZNA^ENIE T , TO URAWNENIE () RAZRE[IMO,
ESLI FUNKCIQ g ORTOGONALXNA WSEM SOBSTWENNYM FUNKCIQM, PRINADLE-
VA]IM SOBSTWENNOMU ZNA^ENI@ 1.
   3. pOLU^IM RE[ENIE URAWNENIQ (), ISPOLXZUQ SPEKTRALXNU@ TEOREMU
DLQ KOMPAKTNOGO SAMOSOPRQVENNOGO OPERATORA. pUSTX (fn ) | ORTONORMI-
ROWANNYJ BAZIS IZ SOBSTWENNYH WEKTOROW OPERATORA T I
                     X
                 T = n h
; fnifn ; n 2 R; n ! 0;
                        n
| EGO PREDSTAWLENIE PO SPEKTRALXNOJ TEOREME 253.4. pUSTX
              0 = fn 2 N j n = 0g; 1 = fn 2 N j n = 1g;
                = Nn(0 [ 1):
rE[ENIE URAWNENIQ () I]EM W WIDE f = Pn nfn , GDE n = hf; fn i | NEIZ-
WESTNYE KO\FFICIENTY fURXE WEKTORA f . tOGDA RAWENSTWO (I ,T ) Pn n fn =
Phg; f if PEREPI[ETSQ W WIDE
      n n
n
    (I , T ) Pn n fn = nP
                         20
                              n (I , T )fn +
                                               P  (I , T )f
                                               21 n
                                              nP
                                                             n
                         P                               P
                      + n2 n (I , T )fn = n2 n fn + n2 n (1 , n )fn
                      = P hg; f if + P hg; f if + P hg; f if :
                                                 0
                                 n n                n n         n n
                        n20                n21          n2
s U^ETOM EDINSTWENNOSTI PREDSTAWLENIQ \LEMENTA RQDOM fURXE POLU^A-
EM W SLU^AE 2(i) (TOGDA 1 = ;):
                           8 hg; f i; ESLI n 2  ,
                           < n                   0
                      n = : hg; fni ; ESLI n 2 .
                              1 , n
iSKOMOE RE[ENIE IMEET WID
                     X              X
                 f = hg; fn ifn + (1 , n ),1hg; fn ifn:
                     n20             n2
w SLU^AE 2(ii) (TOGDA 1 6= ; I NEOBHODIMO hg; fn i = 0 (n 2 1)) ISKOMOE
RE[ENIE IMEET WID
                X             X         X
          f = hg; fn ifn + n fn + (1 , n ),1hg; fn ifn ;
              n20             n21           n2

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