and let En(f)p be the best approximation of f by trigonometric polynomials of order n with respect to the norm in Lp. For a positive integer m let A?f = At(A~-if)
for
m > i
and
A~f(. ) = A t f (.) = :(. + t/2) - f(. - t/2). We put L ; = { f ( - ) : T + C [ 3 f(r) E L , } , where for fractional values of r we understand the derivative f(r) in the sense of Weyl [1]. T h e following Jackson-Stechkin inequality is well known in the theory of approximations: VmEN,
r,s>0,
l_
3A>0
VnEN, fELp
r
En_i(f)v < An_~wm(f(r) 2 : e )
(i) P
where Win(g, t)p = supM
Vp, q, 1 < p < q < oo, ~C > 0 V f E Lp: 1
where r . . . . P
1
Ilf[lqS
c]lf(r)llp,
, and the inequality
q
En(f)p<_-~En(f(")), (see [4, p. 237]) imply the following generalization of (1) to the case of distinct metrics: VmEN,~>0, 3A>0
VnEN, fEL~
l < p <_oe, l <_q < ~ , r >_ (1/p-1/q)+ E,_l(f)q<_An-~+(1/P-1/q)+wm(f (r),
2~E /
*This research was supported by the ISF Grant No. MC 5000.
Moscow State University. Translated from Matematicheskie Zametki, Vol. 57, No. 4, pp. 551-579, April, 1995. Original article submitted August 5, 1994. 0001-4346/95/5734-0375512.50~
Plenum Publishing Corporation
381
We denote by [e I and {a} the integral part and fractional part of a number a, respectively, by II~ll = min({a}, 1 - {a}) the distance to the nearest integer, and we put [a] = min{d e Z [ d _ a}. For d E N let I d = { ( z l , . . . , X d ) e R d I Vi 0 ~ xi ~ 1} be the d-dimensional unit cube, and let r - - {(1,y) 1 0 < y ~ l } L 3 { ( x , 0 ) 1 0 < x < l } . For x e S d (d > 2) we put Ilxlt = m ~ < L < d I1~11. Instead of using the continuity moduli, Yudin [5-6] proposed to estimate the value of the best approximation in the case p = q = 2 by quantities of the form mjaxliA~,~os/,~f(')l[2, where a E I d is a finite collection of particularly selected numbers. The following result has been proved in [7]. T h e o r e m . Suppose that the numbers r, m, d E N satisfy the condition d - 1 < r / m < d, let ( a 0 , . . . , Old) E I a+l, and suppose that infy>l ylldilayi[ > O. Then there exists Y > 0 such that V n e N, f E L~:
En-l(f)2 < Y n -r O~_j~_d" max -
-
A~.<~...f(')II . ~
J/
112
On the other hand, it has been proved that for every finite collection of numbers a 0 , - - . , ad,
sup{E:_~(f)~(~
E L~.) = oofor everyn E N.
D e f i n i t i o n . We call the following problem the Yudin problem. Determine the set of all collections of the parameters (p, q,r, rn, d) and vectors el E I d such that the following inequality holds: 3Y>OVnEN,
E:_,(:), <_ g. ,:+('/,-'/,~+
V f E L pr :
9 ~11:,7':<,,/::~'>11,,.
(2)
The present paper is devoted to the study of this problem. Assume that 7 > 0 and r, m, d E N; assume also that d > 1/7. I n w 5 w e p r o v e t h a t if r ~ r n T , l < p <_ ~o, and l <_ q < oo, or if r > m-/ +'T and l <_ p,q < oo, then inequality (2) holds provided that the collection ( d o , . . . , ad) has the property l i m i n f k r max Itk~z/~oll > O. kEN
l
(3)
On the other hand, we prove that if O < r < rn 7 and 1 < p,q <_ 0% or if r = roT, d = 1, and (l/p, l/q) e r, then there exists a vector ( d o , . . . , ad) satisfying (3) such that (2) fails. We also find certain conditions on p and q under which inequality (2) holds for fractional r. The question of the validity of the Jackson-Stechkin inequalities is intimately connected with the problem of estimating the norms in one class of multipliers in spaces of periodic functions. In w1 we prove two results concerning the properties of multipliers in the spaces Lp. w1. T w o t h e o r e m s o n m u l t i p l i e r s D e f i n i t i o n . We denote by L0 the set of equivalence classes of measurable functions on T coinciding a.e. It is endowed with the metric 2,r if(x ) _ g(z)l dx P(f'g) ---1 -f-If(x) - g(x)i 2"-'~"
L
Note that Lo is a complete linear metric space. 382
Definition. Let A = {Ak}k=-oo +oo be a complex number sequence, and suppose that 1 < p _< c~ and 0 < q < oo. We denote a linear operator M(A) on the set of trigonometric polynomials by the formula M(A)e ik= = ~ke ik=. If M(A) can be extended to a continuous linear operator from Lp into Lq, which we shall also denote by M(A), then we call M(A) a multiplier of type (p, q). R e m a r k 1. We mention one useful property of multipliers. If Pl > p, qa < q, and M(A) is a multiplier of type (p, q), then M(A) is also a multiplier of type (pc, ql). L e m m a 1. Let f , g E L and let {nk} be a sequence of positive integers such that no = 1 < n2 < n3 < " ~ ,
A0={0},
nk+x/nk > a > 1,
Ay={keZInj_l
1 < p < 2,
2 <_ q < oc;
(j=1,2,3,...).
We also set f j ( x ) = ~-]keai ]'(k) eik= and gj(x) = ~'].kS,xj "g(k)eik= (J E Z+) and suppose that [[fJllq < AI[gjl[ p uniformly in j. Then
Ilfllq < B 9 A- Ilgh, where B = B(p, q, a) > O. P r o o f . By a theorem of Littlewood and Paley [3, p. 346],
Ilfllq -< Cq,~[{ 11{fj(x)}~%ollt=llq, Ilgllp > cp,o I[[I{gi(x)}j%o il,~ lipBy the Minkowski inequality [8, p. 38] we have ~ , II I1{fJ(~))j_-oll,~ll~ -< II {11f sllqL.=011,= tl {llg~llpL-%ll,~ -< II II {gi(x))j%oll,~llp.
The conditions of the lemma imply the inequality
II{llY~ll~b%oll,= -< A [l{llaJll~}7=o[I,=. Hence Ilfllq < ACq,,~cp,,~llg]12,, which was to be proved. Suppose that r > 0. We set P r = {(x,y) e R2 : O<_x < l ; O < y < _ l ; x - y < _ r }
for r >
1/2 and
Pr-- {(x,y) ER2 : O
0o=
r+l x,y) e R 2 : O < x < l ; O < y < l ; z < y ; z < - - ~ ; y > ~ - - } ,
D ~ = {(x,y) D2~=
1-r
ER 2 : 1 / 2 < z < l ; 0 < y _ < l / 2 ; x - y < r } ,
{
l+r (Z,y) e ~ : x > y ; x <
03,= co{Dl~ U Dr }, 2
l-r}
2 ; y >-y-
'
where co{X} is the convex hull of the set X,
Dr = D Ou D~.
Note that Dr = I2 \ F = Pr for r >_ l. 383
be a complez number sequence such T h e o r e m 1. Suppose that r > 0 and n 6 N, and let A = {Ak}k=_o~ +~ that Ak = 0 for ikl < n/2.
For each j 6 Z+ Ietaj: {1,...,2i} -+ {2i,...,2 j+l - 1} be a one-to-one mapping and let
G{ =
Z: n a j ( s ) -
{k 6
n12 <
Ikl <
n~j(s) +
n/2}.
Suppose that the sequence A has the following properties: J~I1 =
sups" E IAx~I <
(4)
o~,
(~)
M2 = sup sup s"l~/~ I < oo. J,~ kea':
Then M(A) is a multiplier of type (p,q) for every p and q such that (l/p, 1/q) 6 P,. If, in addition, we have (l/p, 1/q) 6 D,, then the estimate
IIM(A)lb,q _
9
n (I/p-I/q)+
is valid with C = C(p, q, M i , M2) > 0. P r o o f . Let the sign << in an inequality of the form A << B mean that A < cB, where c is a positive constant which can depend only on p, q, M1, and M2. For each k e G~ we put Ikl = n a~(s)+I, - n / 2 _ l < n/2, and we set (j(k) = sign k. ( n ( s - 1 ) + I + [n/2]). We define an orthonormal system of functions ~,n (1 < [m I < 25 9 n) by the equality ~ j ( k ) = eikx For an arbitrary function f E L we set
B~(f)=
~
If(k)l~lG(k)l 1-2/'
)1/2
k6O:
k6GJ
By a corollary of the Hardy-Littlewood-Pa/ey theorem on rearrangements of the Fourier coefficients with respect to the system {~m} [3, p. 190],
IIM(A)fjllq << B~(M(A)fj) (2 <_ q < oo),
B~(fj) << IIfr
(1 < p S 2).
Assume that l < p < 2 < q < o o a n d l / p - 1 / q < r . By property (5) of the coefficients Ak,
2J --
<<
) 1/2 kEGJs
~ l/(kll21(,(kl/nl-~/'+Wql(j(k)l '-2/' kEG~
=nX/p-~/,.
If(k)l'l~j(k)ll-~/') kEGJ
384
-'-rtl/p-1/qn~(yj),
(6) (7)
Consequently, in view of inequalities (6) and (7),
IIM(A)f~II,<< ,~/'-~/qllfjll~. By Lemma 1 (in which we set nj equal to n 2 j - [n/2]),
IIM(A)fllq << ,F/P-~/qlIIIIp,
(8)
if the point (l/p, 1/q) is in the domain D). By the properties of multipliers (see Remark 1), this implies that if (1/p, 1/q) E P,., then M(A) is a multiplier of type (p, q). In what follows we always assume that ~," = ~ 2J - [~/2] (j ~ Z+). oo ~+~ where A~ = Ak if For each s E N we set Gs = Uj=[log2s ] G j and denote by As the sequence ts Mk~k=-~, k e Gs, otherwise A~ = 0. Conditions (4) and (5) on the coefficients Ak imply that nj+z --1 max
E
j~z+ Ikl=n~ Consequently, if 1 < p < co, then Maxcinkiewicz's theorem [3, p. 346] yields that M(As) is a multiplier of type (p,p) and IIM(As)fllp < s-~Ilfllp. (9) Assume again that 1 < p < 2 < q < oo. We put fj,~ = )'-]~keG{ ~k'f(k) ei~ and observe that fj,s can be represented as fj,s (x) = eiM~p1 (x) + e-iM~p2 (x), where P1 and P2 are trigonometric polynomials of order In/2] and M > n. Consequently, by the Nikol'skii inequality [9] we have IIP~,211~ << nl/~'-X/allPa,~ll,,, while by the Riesz theorem IIP~,211p << Ilfi,sll~. Hence Ilfi,sllq -< IIPlllq + IIP21lq << nl/~-l/qllfJ,sllp. Therefore, by inequality (9), the obvious identity M(As)fj,s = (M(As)f)j, and L e m m a 1 we obtain IIM(As)fllq << ~-~ n ~/p-~/q
(10)
II$11p.
To prove an analogous inequality for the indices satisfying the relation 1 < p < q < 2 or 2 <_ p < q < oc, it is sufficient to apply the Riesz-Thorin interpolation theorem [3, p. 144] to the operator M(As) and to use inequalities (9) and (10). We consider the bilinear operator R: n • ~o~ --+ L acting by the formula R(f, u) = ~']~s~176us. s ~. M(As)f. By the Parseval identity and inequality (4),
lie(s, u)ll ~, : ~
lull ~ 9 s~rllM(As)/ll~ << sup I~sl ~ ~
s=l
~
IA Is = II~,ll~llYll~.
(11)
s = l kEG,
s
Thus, R is a bounded operator from L2 • ~oo into L2. Assume that 1 < p < q < cx~ and that the sequence u is absolutely converging. By inequality (10) proved above and by the Minkowski inequality, oo
IIR(f, u)llq = ~
luslsrllM(A~)fllq
<< ~l/P-1/qllullallfllp.
(12)
s---~l
The estimates (11) and (12) of the norms of R that we have just proved and the multilinear version of the Riesz-Thorin theorem [3, p. 160] yield the estimate IIn(f,~)llql <<~
11Pl 11~1 U
-
II Iltllfll,1,
(13) 385
where Z/q 1 O/q ~t_ (1 - - 0)/2, 1/pl O/p -~ (1 - - 0)/2, and 1/t = 0, 0 _< 0 < 1. It is easy to see that R(f,u) = M ( A ) f ifu~ = s - " and I[u[[, < oe if0 < r. We also observe that the d o m a i n D ~ is the image of the domain {(p,q, 0) E R a : 1 < p <_ q < c~, 0 _ 0 < r} under the map =
=
(p, q, o) ~+ (O/p + (1 - o)12, Olq + (1 - o)12). Therefore, [IM(A)fHq << nZ/P-1/qllfllp ,
(14)
if (Vp, Z/q) e D~. Using the Riesz-Thorin theorem, from (8) and (14) we obtain a similar inequality in the case where the parameters 1/p and 1/q are in the domain D~. Finally, the properties of multipliers (see Remark 1) and inequality (14) yield the estimate
IIM(A)fltq << I[fllp in the domain D o . The theorem is completely proved. T h e o r e m 2. Suppose that a sequence A = tI)kkl+~176 Jk=-oo C C has the property
limsup IXkl > 0.
(la)
Ikl-*oo
Then in order that the operator M(A) belong to the class of multipliers of type (1, 0) it is necessary and sufficient that ?e > 0 lim #{k I lkl e [gJ,2J+1), I~kl > ~} = o~. (16) j-*+oo P r o o f . By condition (14) there is an infinite sequence of
indices
{kj}y=l C N such that
in.f ma,~(I.X-k, !, I% I) > O. 3
Without loss of generality we assume that kj+a/kj >_ 4 (j E N). Then 4 4k, k s + k~ + . . . + kj-1 <_ ~(1 - 4-J)kj < 3 J
kj - kl . . . . .
kj-1 >_
Consequently,
Mj - {+k s + ~ , , k , l ~ , e { - 1 ; 0 ; 1 }
2 + 4 -1+1 2 kj > kj.
}
c (-2'+~,-2']u
[2',2'+b,
(17)
where t = [log2 kj]. We shall prove the theorem using the argument by contradiction. Assume that condition (16) fails. Then for each e > 0,
~(~) - sup # {k I lkl e [2J, 2J+'), lxkl > ,} < oo. jEz+
We fix a positive integer n > 3 which is sufficiently large for the inequality ~(3 -'~) > 0 to hold, and we put L = [3"~(3-")],
,Sxj : {]g(./_l)Lq.l,... ,kjL} 386
(j = Z , . . . , n - -
1),
L~n -~- {k(n_l)L-kl } 9
P r o p e r t y (17) a n d the definition of the function qo imply t h a t for a fixed collection of indices m2 E A 2 , . . . , m , E A , , the number of indices n h E A1 such t h a t at least one of the inequalities
IA•177
___3-"
(18)
(/= 2,...,n)
fails is not greater t h a n
1=2
<- ~ - 2 # { k E M, I [~/r > 3 - " } <- 2 ( n - 1)qo(3-") < L. I--=2
Hence there is an index m ~ = & ( m 2 , . . . ,rn~), m ~ E A1, such t h a t (18) holds for ma = m ~ In the same way, for a fixed collection of indices rn3 E A n , . . . , rnn E A , , the number of indices m2 E A2 such t h a t at least one of the inequalities
(19)
[ <- 3-~
[~•
(I = 3 , . . . , n ; ~1 E {--1;0; 1}; m ~ = ~ ( m 2 , . . . , m n ) ) fails is not greater t h a n
~3. #{k~MtllAkl>3-n} <-6(n-2)~(3-")
Consequently, there is an index m2~ = & ( m 3 , . . . , rn~), rn ~ E A2, such t h a t the system of inequalities (19) holds for rn2 = m ~ Assume t h a t 3 <- j <- n - 1 and t h a t we have already constructed the maps {~ : A~+I x . . . x A , --+ A~ (s = 1 , . . . , j - 1) such t h a t
IA•177
<- 3--
where m ~ = r ,rnn), , m. ~ =. &(too,. j--i . . . . . . Note t h a t for fixed values of j and I the sets
'
(l = j,...,n),
mj_l,ms,O
"'''
rn ~ Mj,, {mmz+Dm+~imO+.-.+es_am Os_~la,
m,).
D=+l;e~e{-1;0;1)}
(m E AS) are pairwise disjoint and, therefore, the n u m b e r of indices rn s E A s such that at least one of the inequalities
l~+m,~+,imO+...+,~_~o_,
I<-- 3-"
(20)
(here l = j + 1 , . . . , n , the coefficients e l , - . . , e j - x E { - 1 ; 0 ; 1}, a n d ~,TzO
j-1
= ~s-l(,~s,
. .
9 m,),...
,m 0
0
= &(m ~ 9 9 9 ms_l,
ms,.
.., m,))
fails is not greater t h a n 2 x 3 J - l ( n - j ) c p ( 3 -n) < L. Hence we can choose the index rn I = m j0 E A i satisfying all inequalities (20) a n d we can set { / ( m / + l , . . . , m , ) = m j0. Finally, after the (n - 1)th step of the iteration procedure we set mn -- k(,_l)L+l, m,_~ = {,_~(rn,), ...
, m~ = &(m~,. .. , m , ) . 387
The inequalities (18)-(20) imply that ]A~,+...+~,~,,,,, [ _< 3 -n for e x , . . . , e ~ E
{-1;0; 1} and
E l e s [ > 1.
(21)
8 71 We now set Rn(x) = I-Ij=x(1 + cosmjx) and observe that R,~ is a trigonometric polynomial with the following properties:
(a) ILR.llx = 1,
(b) R,,(• = 1/9. (j = 1 , . . . , ~ ) , (c) o _< k,,(~) _< 1, (d) k.(~) # o .,. ,,. S = e l r n x + . . . + e = r n n ,
e, E {--1; 0;1}.
We put
Tn(3$) "-~E ~skn($)eisx' TI(z)= E Tn(~q)eixs' sEZ sEM
T : = T . - T~,
where M = { 0 , ~ m l , . . . , + m n } . Properties (c) and (d) and inequality (21) imply the estimate
IG(~)I _< 3-"
(2 r M).
(22)
Below assume that the sign << (>>) in inequalities of the form A << B (A >> B) means that A < cB
(A > cB) with the constant c > 0 independent of n, Since T~ is a trigonometric polynomial with gap coefficients in the sense of Hadmnard, it follows that [tT~][1 >> I]T~[[2, while by condition (15) and property (b) we have [[Tnl[12 >> V~. Inequality (22) and properties (c) and (d) imply the inequality
IIT~ll, _ ~
@.(41 << 1
(1 _< p_< oc).
s~M Consequently, (23) for sufficiently large n. On the other hand, property (c) implies that liT.x~I1~ << v% hence
IIT~II~ ~ IIT~II~+ IIT~II~ << v ~ .
(24)
Since ]lR~llx = 1 by property (a), it follows that [IR. * fill -< 1 for all n E IN and f E L such that = 1. Therefore the assumption that the (Lx, L0)-norm of M(A) is bounded implies that there exists a constant C > 0 such that
tlflll
sup sup p(T, 9 f, 0) _< C.
]l/lll_
It was shown by Pichugov in [10] and [11] that this property implies that, uniformly in n, we have
liT, Ill ~ A + Sln+ 11%112
(A, B > 0 ) .
However, this inequality contradicts (23) and (24). The theorem is proved. R e m a r k 2. Theorem 1 indicates conditions on the domain formed by pairs (p, q) such that M ( A ) is a multiplier of type (p, q). These conditions are exact in the following sense. If (1/p, i/q) ~ Pr, then there exists a sequence A satisfying the assumptions of Theorem 1 such that M(A) is not a multiplier of type (p, q). 388
In the case p = 1 or q = ~ this result follows from Theorem 2. oo
We set h ( x )
-,, E coskx k=2 k l - I / p I n k "
The function h
is of order x - 1 / ' I n - l ( 1 / z )
near the point x = 0
[8, p. 300]. We set Ak = Ikl - ' , k e g \ {0}, and Ao = O. Then oo
gP(~) ~
oo
XkCk(A)e~k~ =
E
k=--oo
E
k=2
Thus, fp E Lp and gp ~ Lq if 1/q < l i p - r. Let now
C O S kx k ~ + ~ - ~ / P l-n-k
oo
~ ~-~/P+~ln-'~(1/z)"
COS/r
S(~) ~ ~ ~kl/~l~k k----2
a n d let Uk ---- -t-1. It is known that one can select the sequence {Vk} so t h a t S E Lp for all p > 2. For r < 1/2 we set Ak = Ikl-'~,kl (k # 0) and ~o = o. If 1/q < 1/2 - r, t h e n oo
oo
k=--vo
k=2
COS k X
[see 3, p. 191]. w2. N u m b e r - t h e o r e t i c
statements
For "7 > 0 and n a t u r a l d we construct the sets B~ =
> O} =a B,=
{a E Id ] li~fCllaqll
U
d= r~l-d
Note t h a t by a theorem of Dirichlet [12, p. 24] we have B~ = O for "7 < 1/d. Jaxnik [13] has proved t h a t if 7 > 1/d, then there exists a vector c~ E I d such t h a t
li~iNnfq'rllaql I >
0
liminfq'rln-l(lqEN + q)llaqH < oo.
and
(25)
On the other h a n d if 3' = 1/d, then the inequalities
0 < li~f
q'tllaql I < cr
hold if the c o m p o n e n t s of the vector a E I d axe algebraic n u m b e r s of degree d + 1 such t h a t 1 , a l , . . . , a d are linearly independent over Q [12, p. 99]. L e m m a 2. Suppose that "7 > O, and let a = ( a l , . . . ,ad) E B.r and Ak = (kTilakll) -1 (k E N). For each j E Z+, let A~ 'J > A~ 'J > ... > X *J be a nonincreasing rearrangement of the collection {A2j,... , AV+I_I}. Then sup sup s~A *J < oo. __
__
__
-,2,T
jEZ+ l
P r o o f . We define a j ( s ) by the formula A~,J -- A~j(8). Below we shall prove the inequality d s,./
vml
389
(This inequality clearly implies the assertion of the lemma.) For s = 1 the inclusion a E B-~ implies d
in.f~(s) ~
I l a ~ i ( s ) l I > c > O,
3
where c depends only on a. Consequently,
II
in.f27j ~ 1
j(2)ll > c2-
v
f o r 2 = 1. Assume now that s > 2. Clearly, there exist 21 and 22 such that 1 < sx < s2 <_ a and 2J/(2 - 1). The inclusion a e B.y implies the estimate
]O'j(22)'-- O'j(Sl) ] <
inf2~J(s - 1) -~ ~ l l a ~ ( c r j ( s 2 ) - ~j(sl))ll >_ c. 3
Is
Since, by the definition of o'j, 2 7(j+l, ~[[O~O'j(2)l I > O'~(S)~
]lot~O'j(s,)l[d
I[O~uO'j(8)[[ > m K x o ' ~ ( 8 i ) ~ --
v
i=1,2
v
v
>-- --/,_-1,= .---.2"J max
II .oj(2,)ll ->
27J-1
-
12
,r
v
2 "tj c it follows that infj ~-- )-], IIot~crj(s)l[ ~ d27+-----Y. The lemma is proved. L e m m a 3. Suppose that T , m > 0 and r > ma.x(1,mqt), and let ~k = k-rllv~k]l - m
(k e N) and a e B~.
Then ~-,k=l .k~, < oo.
P r o o f . For each j e Z+ let )~'J (d = 1 , . . . , 2 j) be the nonincreasing rearrangement of the collection (d = 2 J , . . . , 2 j+I - 1).
d-7]ladll - I
By L e m m a 2 we have d%~*d~ < C, where C is independent of d and j. Consequently, oc
co
2j
9
i d=l
j=0
oo
Cm
d=l
2j
i j=0
1 d=l
2J 1 We first assume that 7m >_ 1. Then Y~ d~-'--~<< j ' and by the condition r > 7m, d=l ~
1
2-/
1
2(r--Trn)j E ~ j=0
~
d-----1
On the other hand, if 7rn < 1, then ~ d--~
j
<<~
2(r
m)j < OCt.
/=0
<< 2 j ( l - ~ ) ,
and by the condition r > 1 we obtain
2-----1 ~ j=0
i
2./
2(r_~rn) j ~
i
d=l
d---~ << ~
j=0
Thus, we have in b o t h cases oo
2=1
which was to be proved. 390
cQ 2 ( l _ ~ m ) j
2(r_,m)j < cr
L e m m a 4. Let n C Z, u,k E R \ {O}, and ( a 0 , . . . , a d ) C I d+l, ao # O. If ( n -
l / 2 ) U a o I _~ k <
(n + 1/2)~.o ~, then max IIn~,~o~ll < (1 + m a x . , ~ o
l_~s_~d
--
1) 0
P r o o f . The following inequalities axe obvious:
-<
( oe) 11 + ll , ll_<
The hypothesis of the lemma implies the equality
In -
aok/Ul =
9 II~oklull,
IIn':'~/~oll < (~,O'O' + l) m~ --
l_~s_~d+i
therefore
II'~,k/ull.
This yields the required assertion w3. A u x i l i a r y s t a t e m e n t s L e m m a 5. Let p and r be positive numbers, let a E I d, m e N, ~ E {0; 1}, and s(x) = G id= , Isinm .~i~l. Suppose that l < l < d , and let At(x) = x-~s(x)-l(sinzratx) ~ sign(sin m 7ralx). Suppose also that s(x) > 0
everywhere on R \ {0}. Then one can divide each interval Ia = (a, a + p), a > 0 into no more than d / + 1) 19 9 d 9 2 d c 2rn+d+l(P
subintervals such that At(x) is either convex or concave on every subinterval. P r o o f . It is easy to verify that the points of the form naT1 (j = 1 , . . . , d, n E Z) divide the interval I~ into no more than d + 2 + p(al + . . ~ + ad) intervals Ia,k such that the functions sinzrcUx do not vanish on these intervals. Since a > 0 and s(x) > 0, the zeros of the derivative A~'(x) on Ia,k coincide with those of the function
g(=)=='+=s(=Y~7(=). We introduce the sets
T~=
x " ( a v c o s z q 3 x + b , sinzrflx) l a ~ , b v e R
(n = 0 , 1 , . . . )
and T~ 1 = {0} and the multi-index s = ( s , , . . . ,sd) e Z d, Isl = Isll + . . . + [sal. Using the transformation formulas for products of trigonometric functions we can show that g(x) can be represented as
g(=)=
~
t(,,o)(=),
I,,l_<2m+l where t(s,~) E T(s,~), (s, a) = s i a l + " " + sda4. We shall consider the linear differential operator D f l f = f " + (Tr/J)2f. This operator has the following properties: (a) V t e T~ D~t e T~, (b) V t e T~ D~t e T~ -1, (c) if f E C2(/~,k) and f has N zeros, then the function D ~ f has no fewer than N - 2 zeros. 391
We shall prove the last inequality. On the interval I~,k we have the identity
D # f - sin~rflx sin 2 7r/3x Rolle's theorem implies that the function fl =
fix
has no fewer than N - 1 zeros. Hence the
function f2 = (sin 2 7r/3Xfl)l has no fewer than N - 2 zeros9 We fix the number k of the interval and select an index so such that t(s0,a) ~ 0 on Ia,~. We put D =
II Isl < 2 m + l
(D(s,~)) 3. Then by properties (a) and (b), ,s:~s 0
(D(so,co)2Dg(x) =aocosTr(so,a)x +bosinTr(so,a)x,
a 2 + b 2 > O.
It is easy to show that the number of zeros of the last function on the interval Ia,k is bounded by IIa,kl 9 [(So, a)l + 1. Consequently, by virtue of (c) the number of zeros of g on I~,k is not greater than
I/~,~1 I(s0,~)l + 1 + 4 + 6 9
"
# { s I lsl < 2m + 1} < Izo,~l (2m + 1)(~1 + . - - + ~d) + 5 + 6 . 2 d e ~ __
--
2 r n + d + l "
Summing over all k we obtain that the number of convexity segments and concavity segments of A~(x) on I~ is not greater than
~lzo,,I
(2m + 1)(~1 + -
+ ~) + 6 + 6
2 d~d ~+~+~
<_ 19 "
2~Cd2m+d+ld(p +
I).
k
The lemma is proved. D e f i n i t i o n . Let A = {ak}k~ C R and L E N. We say that the number of sign changes in the sequence A is not greater than L if there exists no more than L iudices kl < -.- < kL such t h a t akj 9 akj+l < O. In L e m m a 6 we denote by AAk and A2~k the quantities Ak -- ,kk+l and ~k -- 2)~k+l + Ak+2, respectively. Lemma6. Lets, L 9149 1. L e t M = {k 9 {)%}~0=1 C R , and 1-1kl/a, 0<_ [k I < a , H ( x ) - - E 5k-s=Akcos(kx+~). 6k = 0, Ik[ >_ a, keM
Suppose that the number of sign changes in the sequence {A2,~k} on M is not greater than L. Then = O I)~kI+ a max ]AAkl)/)"
IIHl[~
X-eM(L (km-ax-
kEM
P r o o f . We put b = [(s - 1)a], c = [(s + 1)a], and I = c - b. We represent H(x) as
where ~
I
l
k=O
k=O
= )~+~6~_~+~.
The Telyakovskii inequalities for the norms of trigonometric polynomials [14] imply the estimates
k=O
1
=
k=l
392
"
=
The definitions of ~k and ~ imply
,_1 Z
l -k
k=l
l--I
< -
-Z-;
kEM
l--1
<
(~k--saq-.b
--
k----1
S, kEM
1 I-1 I - k + 1
< ma~lAkl ~
a
--
k q- sa :(;:-Z)
b
k=l
<4
(2s)
I~1-
In the same way,
[~kl < max I~kl
t-~
k
k=l
- keM
k=l
< maxl)~k -
-
keM
< max
k
- keM
~<4 k
k=l
k=l
ak
(29)
tkl.
--
The formula &2(Ak_lBk_l) = Ak_l&ZBk_x + 2AAk-IABk
+ A2Ak-lBk+l
implies the inequality IA2~k-1 IX .marlI)~kltA2~k_as+b_ll + 2 ~
IA~kllAgk_~s+bl+ ]A2Ak+b-al.
kEM
Therefore l--1
l--1
,-1 k(t 7- k)i/x2Nk_l I <_ ~ ~l/x=Nk_l I = O(maXkEM l~kl) + k=l
O(a max IANkl))kEM + ~
k=l
klA2~k+b-~l"
(30)
k-----1
The condition of the lemma concerning the number of sign changes in {A 2)~k} and the formula S
~_, kA2Ak-~ = As+l + (s + 1)AA~ - As+x - (S + 1)AAs k=s-F1
imply the estimate l-1
(31)
klA2Ak+b-ll = O(L(maxl&kl + ama~ laAkl)). k
k=l
"kEM
kEM
Consequently,
IIHIla <
y~keos k=0
+
=0 L
~ksinkx k=l
1
[Akl+amaxl/XAkl kEM
by (26)-(31). The lemma is proved. 393
w4. E s t i m a t e s of t h e b e s t a p p r o x i m a t i o n of a p e r i o d i c f u n c t i o n by c o n s t a n t s L e m m a 7. Let n e N, and let A t = {A/k}kez (l = 1 , . . . , d ) be given sequences o] complex numbers. holdingfor Suppose that the functions ft G Lp and f E Lq are related by the formula d Ikl _>_ ,~, and let M(A') be multipliers of type (p, q). d Then E,-a(f)q <_E,=I IIM(A~)IIp,q maxa
d
d
E._~(f)q < llgiIq <- E [IM(AZ)flIiq <- E [[M(AZ)lIP,qm~xlIfzlIq' 1=1
1=1
which was to be proved. We set A t = { Ak}ikleT., / where A~ =
sign(sin" 7talk)
for
k r 0 and
Al0 = 0.
(ik)"(2i)" E~=I ]sin" rrajk I Note that if ft = A2%~ f(~), then ~(k)
=
(2i) m sinm(Trka,)(ik)"f(k),
and for k r 0 we have d
EA/kfz(k) = t = l
d
d
Esin'~(Trkat)sign(sin"Tra'k)(j~-l -
~-1
j
f(k)
=y(k) The set P~" that we use in the statement of Theorem 3 is defined in w 1. T h e o r e m 3. Let rn, d E N, and suppose that ~" > 0 and that the conditions in one of the two following
groups are satisfied: (a) r > max(1,mT) and (l/p, 1/q)E 12;
(b), >
and (1/p, 1/q) e P,". Suppose also that a E I d and liminfk~Hka[[ > O. Then ken
Eo(f)q
sup
f~Z~,f#coa~t max ]lAr~aj f(~)lip
3
P r o o f . We assume that the conditions in the group (a) are satisfied. Then by Lemma 3 and the inequality [sinTrx[ >_ 21[xH we h a v e E ? = I [A/k[ < 0(3. The norm of the multiplier M ( A t) regarded as an operator from La into Loo is not larger than 2)"]k~176[A/k[. Consequently, the properties of multipliers (see Remark 1) imply that M(A l) is a bounded operator from Lp into Lq for all 1 <_ p, q _< c~. This and Lemma 7 (for n = 1) imply the required assertion. Assume now that the conditions in group (b) are satisfied. 394
Let 1 < l < d. Note that
[Alkl~-4-rnlkl-r\j~-lllaJkllm]_
~dm-la-m[kl-r+m'~
[kl'r j = l
[[~162
"
Therdbre, by Lemma 2 there exists a sequence of one-to-one maps at(s): { 1 , . . . , 2/} --+ { 2 J , . . . , 2 j+x - 1} (j E Z+) such that sup s u p s3'm,~lj(s) < cx3. jEZ+ l
By Theorem 1 this implies that M ( A l) is a multiplier of type (p, q) for (l/p, l/q) E P'w~. L e m m a 7 (for n = 1) yields the required assertion. L e m m a 8. Let m , d E N and a E I d, and suppose that 7 > 0 and 0 < r <_ rn 7 and that either the conditions (a) liminfk'qlka]l = 0 , (1/p, 1/q) E 12, k---+oo
or the conditions (b) d = 1, liminfkTl[kall < c~, (1/p, 1/q) E F are satisfied. Then
k ---~oo
E0(f)~
sup IEL;, .f~const max 11A~r~ f(~)lip
=00.
3
P r o o f . (a) Let the sequence {k~}~~176 1 satisfy the condition lira kZIl~k~ll = 0.
v ---).~
We put g~(x) = eik~z. It is easy to observe that Eo(gv)q = 1 for each q, 1 <_ q <_ oo, and that
m?xlla=m ojg(r) II, = 2"k~ m.axl m Jsin
~GI
_
< (27r)m(k~llaGII)mkS-~m ~ 0
(~ -+ ~ )
for all p, 1 < p < oo. Consequently, Eo(g.)q(maxjll/xr~jg(d)llp) -~ ~ o~ (~ -+ ~). (b) The inequality liminfk~[lkal[ < oc and the fact that IAkk[ = IA~[ imply that k--~oo
lim sup I)~l > 0. L e m m a 2 implies that there exists a constant K > 0 such that sup sup s7mA *'i < K, jEZ+ 1
[A~J+~-x]}"
sup # {kl Ikl E [2J,2J+x), lax[ > e} = sup # {s E [1,2J+x] [ A*'i > e} /EZ+ /EZ+ < sup # {s e [ i , 2 i + ' l I K s - ~ m > ~} < ~ .
jEz+ Consequently, by Theorem 2 the operator M(A 1) is not a multiplier of type (1,0) and, therefore, it is not a multiplier of type (1, 1). Hence there exists a sequence of trigonometric polynomials g,~ such that 02~g.(z)dz
= 0 and I]M(A~)y.[l~/llg.ll~
--~ ~ (n -+ ~ ) .
We set f,~ = M(A1)gn. Then
IlY.lll /llA';',~j(f lll = IIM(A~)g.ll~/llg.lll. To finish the proof we observe that
Eo(f)p >>
I[f - ff(O)llp and use Remark 1 and the duality principle.
395
C o r o l l a r y . If r < m i d and I < p, q <<_co or if r = m, d = 1, and ( I / p , I / q ) E P, then for" each vector a E I a we have sup feL;,f#const m a x I
Eo(f),
IIA2%o f( )lip
= oc.
P r o o f . By a theorem of Dirichlet [12, p. 24],
l i m i n f k l / a H k a l l < co k --+ oo
for each a E I d. Consequently, we can set "f = r / m if r < m / d and ~ = 1 if d = 1 a n d r = m a n d use the assertion of L e m m a 8. T h e o r e m 4. Let m, d E N, and suppose that ",/>_ 1/d and that the conditions in one of the two following groups are satisfied:
(a) 0 < r <
(1/;, i/q) e
(1/p, 1/q) E r . Then there ezists a vector a E I a such that liminfkXllkall > 0 and (b) r = m y ,
d=l,
k~N
sup fEL;,f:#const m a x 1
Eo(f)
II/xm.oj f(~)II,
--OO.
P r o o f . If conditions in group (a) are satisfied, then we take a vector satisfying inequalities (25) as a. Assume t h a t a sequence {k,}~~ 1 C N satisfies the inequality sup k 7 ln-1(1 + yEN
k.)ll k ll < oo.
As in L e m m a S, we set g~(x) = e ik'*. T h e n Eo(gv)q = 1 (1 _< q _< co) and rn
max]) A m J'--'2~ooVv~(r)llllp -< (27r)m (k~ ln-l( 1 + for a l l p , l < _ p < o o . Consequently, E o ( g ~ ) q ( m a x j l ] A ~ s g ( ~ ) l } p ) - I
14 ~l~ r - ' l r n
• ln~( 1 + .~v,,.~v
~ o
(v -+ oo)
-+ co (u ~ oo). The theorem is p r o v e d in case (a).
Assume now t h a t the conditions in group (b) are satisfied. It is known [15, p. 76] t h a t for each value of ~/> 1 there exists a n u m b e r a , 0 _< o~ < 1, such t h a t
0 < liminfk~llka]] < co. kEN
T h i s a n d L e m m a 8 imply the required assertion.
w 5. E s t i m a t e s o f t h e best a p p r o x i m a t i o n 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 T h e set D~m t h a t we use in the s t a t e m e n t of Theorem 5 is defined in w1.
T h e o r e m 5. Let m, d E N, and suppose that ~ > 0 and that the conditions in o n e of the two following groups are satisfied: (a) r > m a x ( 1 , m ~ / + ~ ) , (1/p, 1/q) e I2; (b) r >__m % ( l / p , l / q ) 6 D~,~. 396
Suppose also that 0 < ao < ... < ad < 1 and sup sup nEN /6L[,,f#const
l i m i n f k 7 max
IJkm/~oJl >
0.
Then
n r-(1/p-1/q)+ 9 E~-l(f)q < co.
mp 11aZ.,/~ f(')II,
P r o o f . We shall prove the theorem in the case where conditions (a) are satisfied. r We flx a n u m b e r n 6 N and a function f E Lp, f 7~ const, a n d set u = n/no, where no = [2/a0]. We introduce the sequences A t = { ) J } k e z (I = 0 , . . . , d), where
Atk = sinrroqk/u .
7talk/u)
sign(sin m+l
(ik)~(gs)m Ei=01 sin''+' ~'aik/ul We put
S(z)
= Ej=o I sin~+~
a n d t h a t for k
(kr
~d
~rajxl and ft = A ~ , / ~ f (~1. Note that
~o=o.
~(k)=
(2i) m sin'~(Trka,/u)(ik)~'f(k)
# 0, d
d
.X[~(k) = ~ sin"(r&adu)sign(sin m+' , ~ , k l u ) 1=0
.
sinrrcqk/u
/--0 d
x (E I sinr~+' ~'o,jk/ui)-'f"(k):
(32)
T(k).
j:O
We set A ) ~ = A~ -- I~+ 1. We shall estimate the value of iA,~I from above. We have
1 [sinrratk/u I r IzX;~l < I,~,krlfk -r + k-rl/x(,X~k~)l ___ 2 m
1 (u'~" (risinrra,k/ul <-_2~_lu-----~\~] \ ~
S(k/u)
k~+ ~ + k-rla(~k~)l
)
(33)
+lA(l~k~2m)[ ,
u
for k > ggg~o We set Yk = { x : k/u < x < (k + 1)/u}; then
]Asinrratk/u " signm+l sinrralk/u[ <- ul mvkaXl(sinrratx'signm+l S(x) sinrra'X ~]
1
< -- m a x
/'Tra,I cos ~azxl
- ~ v~ \
s(~)
< --Trm a x S -1 (x)
+
lsinr:azxlIS'(x)i"~ s~(~)
( + (m+l)lsin~'xl
- u v~
]
E~=o Icos ,~,~ll sin,~jxl~.) S(x)
Using the relations
j=l
we obtain the inequality 1
IAsinTrmk/u
1 s(k/~) sin rratk/u I << -~ m~v~s(~)-i"
sign m+l
i Here a n d in t h e t h e o r e m below t h e e x p r e s s i o n A ( ( B m e a n s t h a t t h e i n e q u a l i t y c a n d e p e n d only o n d , a , r , m , p , a n d q.
A <_ C B
h o l d s w i t h a c o n s t a n t C > 0 which
397
This and inequality (33) imply
la2~l << ~-'-'
(34)
max Vk s(=)-'.
1 For each x > - - - we determine the integer dz 6 N from the relations - 2ao
-r1.
d=
-l_< x<
(d=+2
-')oo'
"
We set Dk = {d~ } x E Vk}. This definition of Dk implies that Dk # ~ for k >__u/(2Oeo) and #Dk_<
#r ~tEN]
aok u
1 ao(k + 1) 2
+~.}
<_i+ --<4. ~~ u
Lemma 4 implies the inequality (for k > u/(2o~o))
,
)_1
Hence
(35)
(~)" n~ xS(z)-I << tEDkmax At, where ,kt = t - ' ( E j = l d I sin~raj/aotl"+')-' Since d= E Dk for x = k/u, by the definition of ~ we have
(36)
I~1 << ~-~ max ~t, tED~ while inequalities (34) and (35) imply
IAA~I << I t - - r - 1
max tEDk
~t.
(37)
For s E N we denote
M , = {k e N I u ( s - 1)% 1 < k < u ( s + 1 ) a o l } , ..= ~' e ikx IkleM, 1 -I=1~o/~,, ~= =
0,
o < Ix) < u/~o, I=l-> u/~o.
It is easy to see that for k E N,
)~ = e2~iAl-k,
where
~ = ( r - m)~r/2
ei=Cr+m)/2,k~ E N.
and
Consequently,
YA' ,-'~" kEMs
398
Ai~~k') a - l ) "
" kEMs
.,o.(.+..),/2
~os(k~ +
().
L e m m a 5 implies that the number of segments of constant sign of the sequence {A2(,k~err(r+m)ip-)} on the interval Ms is bounded from above by a constant which depends only on ra, ~', and d. Consequently, by L e m m a 6,
IIH~lla
max IA~I + u kEM. max IAA~I .
<<
kEM~
Hence by inequalities (36) and (37),
II//~llx <<
u -~
max max )~t ~ u-~max)~t,
kEM. tEDk
whereA~={teNls-l
tEA.
In the same way,
kEM.
, -.I)~l[~k-usaffx << u 9 u - r m a x / \ t . tEA.
We set
H%) = ~ H~(~) (l = 0,...,a). S~---2
This definition of H t implies that, first, if k >_ 2uaff 1 , then by (32), d
d
Z ~'(k)it(k)= ~ xs 1=0
=/(k),
/=0
and, second, Ht(k) = 0 for k < U/ao. L e m m a 3 implies that ~ 1 ~t < e~. Consequently,
8=2
s--.~2
This and the HSlder inequality imply the following estimate of the norms of the functions HI:
IIn;ll~ << u 1 - ~ u -~ Consequently, by the Young inequality for convolutions we have (for all g E Lp; here 0 < I < d and q > p)
IIH z * allq <-- IlH*llx/(1-x/p+l/q)llgllp << ul/P-x/%-~llg[lp
9
Hence by L e m m a 7, E[2a;,ul_l(f)q
~<~~z-r-t'(1/p-1/q)+
9 max
o
Ilfzllp.
Since n = u[2/ao], it follows that En-i(f)q
< Ef2~ZX,l_l(f)q << u -~+O/p-x/q)+ 9 o_
,
-
<< n -~+O/p-i/q)+ . max I I A ~ ,nf(~)llp, o
""
where the last inequality holds by virtue of the following well-known property of finite differences:
IIA~ongflp S no IIAh gIIpm
~
m
399
The theorem is proved i n t h e case where conditions (a) are satisfied. We now assume that the conditions in group (b) are satisfied. We introduce the sequences A t = {%k}k=-o~ l by the formula
)~
sign(sin~Tralk/n) for Ikl _> ", (ik)~(2i)~ ~ j =d0 ] sinm 7rc~jk/nl
=
for Ikl < n.
0 As in the above case, we have d
Z ~/~,(k) = f(k)
(Ikt_>~),
/=0
wherefl=A~' /,f (~). Lemma 5 implies that the number of monotonicity segments of the sequence %~ 9 ir+m on every interval Gt = {k e Z: ( t - 1/2)nao I < k < (t,+ 1/2)nao 1} (t E N) is not larger than some constant C1 = Cl(m, d, a) > O. Consequently, IAA~I << max ])~tk]. (38)
E
kEGt
kEGt
By the inequality I sinzrxl >
21[xll
we have
It Hence
d
max ken, ]~[
(39)
~rrt
<
Therefore, by Lemma 4, kEGt
(40)
kl_
We put ~, = t-~ (maxl_<,<_dI1~,~o111) -1. By Lemma 2, sup
sup s~;,~ < o~,
(41)
j E Z + 1 < s <2-~
where Xt 'j is a nonincreasing rearrangement of the collection {X2J,...,X2J+I-1). We determine the map aj(s) : {1,... ,2J} -+ {2J,... ,2 r - 1} from the equation A],J = X~j(s). The inequalities (38)-(41) imply
supff E j,s
kEG~j(s)
I~ 1
<~'
sup
sup
j,s
kEG~j(.)
srlnr~l<
o'3.
Consequently, by Theorem 1 the operator M(A l) is a multiplier of type (p, q) provided that D-~m, and we have the inequality ]lM(Al)llp,q << n-~+(1/p-1/q)+. Hence, by Lemma 7, E , , _ l ( f ) q << n -~+(1/p-1/q)+ n)~, IIA'~.,~j/,,f(~)llp. u~_3~a
The theorem is completely proved. Theorem 6 stated below is a trivial consequence of Theorem 4. 400
(1/p, 1/q)
T h e o r e m 6. Let m , d E N, and suppose that 7 >-- 1/d and the conditions in one of the two following groups are satisfied: (a) O < r < m% (1/p, 1/q) E I2; (b) r = m e , d = i, ( 1 / p , l / q ) e r . Then there exists a vector a e I e+l, ao = 1, such that liminf max k~]lkazll > 0 and ken ~
sup
hEN fEL~,f:~const
n r-cl/p-llq)+ 9 E n - l ( f ) q
mff II
f( )lip
The author is grateful to S.V. Konyagin and S.B. Stechkin for their attention to the work and to A.M. Minkin a~d S. A. Telyakovskii for useful discussions. REFERENCES
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