SCIENTIFIC ABSTRACT STECHKIN, S.B. - STECKIEWICZ, I.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001653020014-9
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: August 25, 2000
• Sequence Number: 14
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

STEC1H1K1':, Sz. 11.
USSR/Mathematics - Approximations, NaYIJUU~51
Optimum
"The Order of the Best Approximations of Con-
tinuous Functions," S. B. Stechkin
"Iz Ak Nauk SSSR, Ser Matemat" Vol XV, No 3,
pp 219-242
Investigates best approximation of continuous
periodic functions by means of trig polynomials.
Gives necessary and sufficient conditions so the
best approximation may possess given (deg) order
of decrease. Cf. G. Jack*on, "The Theory of
Approximation," 1930, NY, and A. Zygmund, "Smooth
Functions," "Duke Math Jour-" 12, 1945, pp 47-76.
Submitted by Acad S. N. Bernshteyn 11 Hk~ SO -
. 186T5o

0
-
Ste6kin, S. B. On absolute
convergence of orthogo
-
-
-
Mat. Sbornik 1r.S.
series.
77
2V(717, 225-232 (1951).
=Russian)
Let'f(x) be a. continuous function of period 27, E.(f) its
best approximation by trigonometric polynomials of order
f) its Modulus of continuity. S. Bernstein proved
that the Fourier series of f(.0 converges absolutely if either
Ci* En-IL.(f) or Zn-ico(n-', f) converges CC, R. Acad. Sci.
P;iris 199, 397-400 (1934)]. The author shows first that CA
FJ these'two conditions are equivalent. He then extends the
First theorem (and its generalizations) to more general
orth6gonal series. Let Iriki be -in increasing sequence of
into-ers, F(u) an increasing con cave function vanishingatO.
2 1 Theor[cm 2. If feLl and c. are its Fourier coefficients with
em 4, = (0.(x) 1" 1, then
respect to the ortlionormal syst
E.,
-wbere- Elie expression in braces is the mean-square difference
between f(x) and the surri of the first 7t- I terms of its
expa.tnsion in terms of Theorm 3.
C., 4"'3)
D be the nican-square modultis of continuity of
and a., b. its ordinary Fourier co-,ffit:ients Theorem 4.
E~ jla.k 1 + :_5 (na-', Several theorems on
absolute convergence of Fourier ~!ovie, follow as corollaries.
,
The
propfs of Theorems 3 ind .1 6 on the following
W inequali 0% if 0, 0,

i r vanston, BL).

U.,

TE-C H K i r)
Nt The. best appro rWion of functions.repre-
z k ~~fns. ~B.
-177 - A
sented by lacunary tri99IIDP!e-rI - Serl -s.
p 77r~. kad,
Naul- SSSR (N.S.)IV,' =tMIM71Russian),
Let f(x)-E(ak cos trkx+b~ s:it irk.0 ho a I'miction Avidi
a lacunary Foarier scries, i-k~.
a-a,2+b*I, Let s~(x) lie the lmrtial ~mllq
ph
.4 E" >.Pk tile bL5L appril'allm i toll to r k
nometric pulytionlials of order Bernsleill provol Olaf
E,,=R, for a spevial choice of it, [Extrumal propaties of'
polynomials ONTI, pp.
31-36]Alvri- theamhor I)ro%
and, if qi-,~c, R. 1'. litvs, Jr,
12 7
Smirce i Mathematical Reviews, V61 No.

S TF C 14

Ste&ln, S. B. On de I& VxWe Poussin'ffums. Dakfady
-Akad. Nauk SSSR (N.S.) -4 S4-',-MTP3nr (Russiqn)
s,4qis of a Fourier series with
The de la Vall6e Poussin
.
partial sums s (x) are
"'.(X) On+ E S4(4

The author lnveatikaleq In inott detall tban has previoual,
Y
been' done the Lebesgue constants N41 . for these ~ sums,
N.,. = sup Jjr.,.(x, f) 11 for f(x) continuous with Jjfjj;9 4 the
norm being the maxiniurn. Arnong the results are the follow-
ing. If n,,-.mo-6 (even integers) and

then
N, --)2w-l Isinri-iginill-VI.
As a function of (n +1)1(tn + 1), N.. . is increasing and con.
cave. There is the following asymptotic formula generalizing
that for the ordinary Lebesgue constants (m=O):

4 loge n +1 1ogP
-+2r
-+3 log 2+-jj 71

+O'(n
_
r' nt+l -'
4y' n+
,-,
R. P. Boqs, Jr. (Evanston* 111.).

SOurcel Mathematical Reviews,
0-'
V03 N

U88R/~%tb4ImatIC6 ApproKlmatioss, Optl- 1-1 Apr 52
ME
"The Best Approximations of Periodic Functions by
Trigonometric Polynomials," S. B. Stechkin, Math
Inst imeni Steklov, Acad Sci USSR
"Dok Ak Nauk SSSR" Vol LWIII, No 5, pp 651-654
Considers continuous periodic function f(X); its
best approximations En(f) (n-l,..) by means of
trigonometric olynomials of order n-1; and its
ModLullus wk (d,fT of continuity of k-th order. Pre-
vious investigations indicate that En(f) 'w n`e` if vk,
da (d is a small pos fraction). A natural problem
218T56
USSR/Mathematics - Approximations, Opti- 11 Apr 52
TMIM (Contd)
arises here. Whether it is impossible to demonstrate
the equivalence of the condition E (f)-n-a in classi-.
cal terms of the moduli of continuHY of f (X) and
its derivs. Subject problem is studied here. Sub-
mitted by Acad S. N. Bernahteyn 11 Feb 52.
21OT56
E_
1-0

7Ste-Odn. S. B. absol convergeme of Foudet
e
Izvesti Nauk SSSPL Ser. Mat..17, 87-98- (1953).
ya A
(Russian) - , I..: .. I - I 1 1, ~ . ;
jkgatoin C: R. Acad.,Sci.'
A well-AnOW11 theorem of
Matbematical Revievs ,
Paris 199, (1934)] states that if the period;c
*

10 function f(x) has modulus of continuity w(8,f),
if (1)
VOL 14 No- and (2),E"(1/n) converges, then the
NOT. 1953 Fourier, series of f(x) conveiges absolutely. The author
AmilysIS gives ~a complete solupon of the problem sbggested by Bern-
stein's theorem: he-constructs a certain majorant WO(S) of
L..o, w (6). and shows that (1) implies the. absolute convergence
of the Fourier series if and only if converges.-,
The more, difficult part of the theorem is, of course, the
construction of a counterexample when the series diverges.
To construct the majorant w*, put
W(G)(O)=O, inf (G'%') the wimber of representations 6f m i~.
t e funil
h
jank,j: - - -
A
o" -~d As (s=
:does n
t mc 1 2 A sequence NY c is he
:Uliion of a finite mimber of scq6ences of the Class'D is said
tto be of Ow class 11?,. T1w mitutrestilt-of.the paper is that
is
(a it, rcls ?qx b". Sine n'kx)
is the Follriec I 'of a fuliction bounded. above
'below), thell is fillite. This geller, JU5 a
1previous result of sirloa ["Icta Ulliv. Szeged. Sect., SCI.,
r'blath. 10 (1943), 206-25-3; 8, 1,501 who r1lade a
I~itroii-cr assamption ijimiL (nLj" namely that it is the,.
a finite iiumber I lactinary (in the sens
union of
f
Had a rnard) sequ encez, Anathei- result of the present p,~per
is vj
tliat if tnAj 'k the union of fiH, te nurnber of lacunar,

:sequences, and is the Foutier series of a co, Wous
n th
function /, then the ratio of
any two numbt~i
-
R,,(I), A.(J) is contained between two positive nunibars*
independent of u, where E,,U) is the best approximation:
trigonometric, polynomials (jf*order ;.,, and,
of by
.

niwx
R, (a,, cos nk-v+ b,, sin nkz)
nk>n

nA;>tL
A. Zygmund (Chicago, IIL).

j-

STE-Cw 1 , so that

Chebyshev Sets in Banach Spaces SOV/20-121-4-3,,"34
S(Xpy)-