Download On a certain norm and related applications

Mathematical Notes, Vol. 64, No. 4, 1998
BRIEF COMMUNICATIONS
On a C e r t a i n N o r m and R e l a t e d A p p l i c a t i o n s
B. S. K a s h i n and V. N. Temlyakov
KEY WORDS: Lebesgue normed measure, quasicontinuous function, Rademacher system, trigonometric polynomial, entropy number, Kolmogorov width.
w
Introduction
Let p be the normed Lebesgue measure on the unit circle. For a function
series
f 6 Ll(dp) with Fourier
oo
60 =
6,(S,~),
s~ ~
~0 2~"/ d ~ ,
6~ =
~
~
.
/(~)e'(""),
k = 1,2,...,
2*-t_<l.l<2 k
k=O
w e consider the value
II/llqc -
d~,
r k ( ~ ) ~ ( / , x)
k=0
(1)
L~
where {rk(w)}~~
is the Rademacher system. By the space of qu~iconti~uoua functio~ (hence the
notation [[ 9 [[QC) we mean the closure of the set of trigonometric polynomials with respect to the norm (1).
Spaces of quasicontinuous functions can also be introduced in the multidimensional case. This may be
done in a number of ways. In what follows, we shall consider one of the variants: the closure of the set of
trigonometric polynomials of d variables ( d = 2, 3, ... ) with respect to the norm
lifllQc --
IIIIf(-, =')llQclloo,
(2)
where, by definition, for x = ( X l , . . . , x d ) E T d we take x I = ( x 2 , . . . , x d ) E T d-1. In other words,
identity (2) involves the QC-norm with respect to the variable xl and the sup-norm with respect to the
other variables.
In the paper inequalities for real-valued trigonometric polynomials in the norms (1), (2) axe established.
Note that our interest in these norms is related, first of all, to their possible applications to the study of
the approximation properties of classes of functions of several variables (in particular, see w In addition
to the study of the norms (1), (2), a comparison is carried out in this paper (see w between the norm
II*ll~ and the discrete norm
II*lloo,n -- max
I*(x)l
(3)
zEf/
(f~ c T d is a finite set of points) on the subspaces
T(Q~) of trigonometric polynomials with spectrum in
the staircase hyperbolic crosses:
Q~ =
U
pc,),
p(~) _ {n = ( n , , . . . ,
n~) c z a : [2 s~-l] _ I-jl < 2 sj , J = 1 , . . . , d}.
II~lh<k
Translated from Matematicheskie Zametki, Vol. 64, No. 4, pp. 637-640, October, 1998.
Original article submitted June 22, 1998.
0001--4346/98/6434-0551520.00
C)1999 Kluwer Academic/Plenum Publishers
551
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Inequalities
T h e o r e m 1. For any f E Ll(d#), the following inequality is valid:
1 co
[[fllQc >- ~ ~ [I,~2s(/,z)llLt(d~,).
(4)
s=O
R e m a r k 1. It is readily seen that in the proof of Theorem 1 we can confine ourselves to the case in
which f is a trigonometric polynomial (we assume that (4) necessarily holds if Ilfllqc = oo).
R e m a r k 2. It follows from Theorem 1 and Grigoriev's results [1] that
sup
tET(2 h)
Iltl q._.___s> cv~.
Iltlloo
On the other hand, it is readily seen from results on gap series that
sup
Iltlloo > ClV/~
tET(2k) I l t l l q c
here T ( m ) is the space of real trigonometric polynomials of degree _< rn.
R e m a r k 3. In the two-dimensional case the following inequality is valid (see [2]):
(5)
where, by definition, for even k we have
r / = {s = ( 2 k l , . . . ,2k. O, kx + k2 + " " + kd = k / 2 } ,
d= 1,2,...,
~,(f) =
y ~ f"(n)e i("'z).
-ca(s)
In [1] (see also Remark 2 above) an example showing that there is no analog o f inequality (5) in the
one-dimensional case was constructed. The problem of the validity of the corresponding analogs of the
estimate (5) for d > 3 remains open (see the discussion of this problem in [3]). For the d-dimensional
case, let us cite an inequality similar to (5) but with norm II 9 ][qc instead of II 9 libT h e o r e m 2. Suppose that for an even k the following polynomial in d variables ( d = 2, 3 , . . . ) is
~ven"
s 1 = ( s ~ , . . . , ~d),
/= stEEZk [ I s t lEl t = k - s t
t/2
Zt = {21},=0
with the property: /or some G C Zt
1) I[~s(f)[14 < 1 if sl e G;
2) the following estimate is valid:
E
~
116s(/)ll2 ~ bk~-l,
st EG [Ist Ih = k - s t
where b > 0 is an absolute constant.
Then
I[fllqc -> ckd/2,
c = c(b) > 0.
The following result shows that also in the one-dimensional case under additional constraints on the
polynomial f its uniform norm admits a lower bound similar to (4).
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T h e o r e m 3. For any polynomial of the form
21
f=
E Pk(X)Cos4kx'
k=l+l
where Pk E T(2t), k = l + 1 , . . . , 2I, the following inequality is valid:
21
Ilfll~ -> c ~ Ilpkll~,
k..~l+l
w
c > O.
Estimates of entropy numbers and Kolmogorov widths
In what follows, we preserve the notation and definitions used in our joint paper [4], in which the
approximation characteristics of classes of functions of d variables were studied.
T h e o r e m 4. For r > m a x ( l / q , 1/2), the following relations are valid:
em(H;, QC) • m - r ( l o g m ) r(d-1)+d/2,
era(W~, QC) • m - r ( l o g m ) r(d-D+l/2,
1 < q <_ cr
T h e o r e m 5. For r > 1/2 and 2 < q < cr the following relations are valid:
draCHm, QC) • m-"Clogm) "(d-')+d/2 ,
draCW~, QC) • rn-"Clogm) "('t-1)+'/2.
Inequality (5) was used in [2] to obtain lower bounds for the entropy numbers of function classes. With
the help of Theorem 2, similar arguments yield the lower bounds in Theorem 4. The lower bounds in
Theorem 5 follow from Theorem 4 and well-known inequalities connecting the entropy numbers era and
the Kolmogomv widths dra (e.g., see [5]). The upper bounds in Theorems 4, 5 can be established similarly
to the corresponding upper bounds in the metric L o~ from [6].
Theorem 3 allows us to obtain the correct order of the entropy numbers and Kolmogorov widths of the
classes L G r of functions of a single variable with smoothness of logarithmic type. Let us define the classes
LG r , r > O, by the following condition on the binary blocks of the Fourier series of their members:
LG r = ( f E L ~ 1 7 6IIg,(f)ll~o ___(1 + s ) -~, s = 0, 1 , . . . }.
T h e o r e m 6. Let r > 1. The following relations are valid ( m --. c~ ):
(log lift) - r + l
f o r p = (X),
era(La~' LP) • dra(Lar' LP) •
(logrn) -~+1/2 for 1 <_ p < oo.
In particular, Theorem 6 shows that the order of era(. L p) and dra(. L p) changes by a j u m p under
the transition from p < co to p = oo. A similar phenomenon in the two-dimensional case occurs for the
classes H ~ (see [2, 31).
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The discrete L~~
for p o l y n o m i a l s in d v a r i a b l e s f r o m T(Qk)
It is well known that for the space T(II) of trigonometric polynomials in d variables with spectrum in
a parallelepiped II there exists a finite set ~ such that the number [fl[ of elements in fl has the same
order as the dimension of T(II) and the following equivalence holds:
Iltlloo,~ • IIt[Ioo,
t ~ T(n)
(see also (3)).
Theorem 7 (stated below) shows that the situation is different for the spaces T(Qk): the equivalence
of the norms IItHoo,a and IIt11r162
for polynomials from T(Qk) can occur only if the number of points in fl
is much larger than the dimension d i m T ( Q k ) • 2~kd-l: Jill > 20+~)k, 7 > 0.
Theorem
t e T(Q~,),
7. Suppose that the set fl C T d possesses the following property: for any polynomial
Iltll~ <_ bk"lltlloo,,,
1
0 <_ o, <_ -i
Then
I~1 > c, IQkl e x p ( c k ' - ~ ) ,
c=c(b)>O,
c, = c,(b) > 0.
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References
1. P. G. Grigor'ev, Mat. Zametki [Math. Notes], 61, No. 6, 935-938 (1997).
2. V. N. Temlyakov, J. Complexity, 11, 293-307 (1995).
3. V. N. Temlyakov, East J. Approx., 2, No. 2, 253-262 (1996).
4. B. S. Kashin and V. N. Temlyakov, Mat. Zametki [Math. Notes], 56, No. 5, 57-86 (1994).
5. G. G. Lorentz, B u l l Amer. Math. Soc., T2, 903-937 (1966).
6. E. S. Belinskii, in: Studies in the Theory of Functions of Several Real Variables .[in Russian], Yaroslavl State Univ.,
Yaroslav! (1990), pp. 22-37.
(B. S. KASHIN) V. A. STEKLOV MATHEMATICS INSTITUTE, RUSSIAN A C A D E M Y OF SCIENCES
E-mail address: kashinQmi.ras.ru
(V. N. TEMLYAKOV) UNIVERSITY OF SOUTH CAROLINA
E-mail address: [email protected]
Translated by N. K. Kulman
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