The inequality in formula (6) becomes an equality only for Pk = 0 (k = n + 1, n + 2 .... ), i.e. .... X 14k' ((4s-- i) m+t6s a- i2S~+6s--t)--n ' (3m+iOs--i)l. (2s -- t) kn. 6 (m ...
BEST
APPROXIMATION
BY TRIGONOMETRIC N. I.
OF
PERIODIC
POLYNOMIALS
FUNCTIONS IN L 2
Chernykh
UDC 517.5
E s t i m a t e s a r e gotten f o r the b e s t a p p r o x i m a t i o n s in L2(0, 2v) of a periodic function by t r i g o n o m e t r i c p o l y n o m i a l s in t e r m s of its m - t h continuity modulus o r in t e r m s of the c o n tinuity modulus of its r - t h d e r i v a t i v e . The inequality En-1 (I)L, < (C~m)-'/"*'mfs'~/n;I)L~
Cl ,# const)
m -1 /2 is u n i m p r o v a b t e f o r the whole s p a c e L 2 (0, 27r). Two is proved, w h e r e the constant (C2m) titles a r e cited in the bibliography. In p a p e r [1] an investigation was made of the p r o b l e m of the b e s t constant in the J a c k s o n inequality f o r the b e s t a p p r o x i m a t i o n s En(f) in L 2 (0, 21r) of the functions f ( x ) ~ L 2 (0, 2~) by t r i g o n o m e t r i c p o l y n o m i a l s of d e g r e e n. A proof was given of the inequality
(-5;-;/)
E._,(/)
-0; let us a s s u m e a = 0, b = 1r/n, p(t) = sin nt, g(t) = t, q(u) = ~2(u; f ) . F r o m inequality (2) we t h e n get the e s t i m a t e
t and f o r any 8 E (0, 2~m/n (n -- t))
(8 < n/N)
let u s define an e v e n p e r i o d i c function ~ve (t) with p e r i o d 27r/N b y the equations ~ (t) =
( 8 - - t)/~, t ~ 10, 8l O, t ~ Is, ~/N].
It is e a s y to v e r i f y t h a t f o r the c h o s e n r e s t r i c t i o n s on ~ the inequality 2 v m / n _ 2 v / N - e and t h e r e f o r e , the function 1
~ ,
is s a t i s f i e d ,
7 tra
(t) = 2 ~,=, ( - t )'+'c~-' ~, (tt)
on the i n t e r v a l 8 ~ t ~ 2~[n
807
is identically equal to z e r o . Since ~, (tt) > ~, ((t + ~) t)
9
for 0 < t < ~ and C m - I ~_ /,ra-l-I
then F,(t)>o
(o
