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_ 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). 552

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 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 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~