Конспект лекций по математическому анализу. Шерстнев А.Н. - 426 стр.

oTS@DA VE SLEDUET (2) S
                               ZZ
                   C = kK k = [ jK (t; s)j2(ds)(dt)]1=2:
                               MM
dLQ POSTROENIQ POSLEDOWATELXNOSTI Tn RASSMOTRIM ORTONORMIROWANNYJ
BAZIS ffj (t)g W L2(M; ). w SILU P. 3 ffj (t)fk (s)g | ORTONORMIROWANNYJ
BAZIS W L2(M  M;   ). pO\TOMU
                                   X
                         K (t; s) = jk fj (t)fk(s);
                                        jk
GDE jk | KO\FFICIENTY fURXE FUNKCII K OTNOSITELXNO BAZISA
ffnj (t)fk (s)g, I RQD SHODITSQ PO NORME L2(M  M;   ). pUSTX Kn =
 P  f (t)f (s). tOGDA
        jk j   k
j;k=1
                                    Z
                       (Tnf )(t)  Kn (t; s)f (s)(ds)
                                   M

| KONE^NOMERNYJ OPERATOR (T. K. Tn = P jk h
; fk ifj ). pRI \TOM (SM.
                                                     n
                                     j;k=1
PUNKT (2) NASTOQ]EGO DOKAZATELXSTWA)
                                   X
                                   1                     X
                                                         n
        kT , Tnk2  kK , Kn k2 =             jjk j2 ,           jjk j2 ! 0 (n ! 1);
                                   j;k=1                 j;k=1
^TO I TREBOWALOSX. >
Z 5. u P R A V N E N I E. pOKAVITE, ^TO W USLOWIQH P. 4 (T f )(t) =
  K (s; t)f (s)(ds) (f 2 L2(M; )).
M




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