Download On an Inequality of Sidon Type for Trigonometric Polynomials

On an Inequality of Sidon Type
for Trigonometric Polynomials
A. O. Radomskii
Moscow State University
Abstract
We establish a lower bound for the uniform norm of the
trigonometric polynomial of special form via the sum of the L1 -norm
of its summands. This result generalizes a theorem due to Kashin and
Temlyakov, which, in turn, generalizes the classical Sidon inequality.
We set, as usual, by
kf kp
the norm in
Lp (0, 2π), 1 ≤ p ≤
∞:
p1
Z 2π
1
kf kp =
|f (x)|p dx , 1 ≤ p < ∞,
2π 0
kf k∞ = ess sup |f (x)|.
[0,2π]
By the classical Sidon theorem (see [6]; pp. 393-395) if the
{nk }∞
k=1 , is lacunary with lacunarity
index λ (i.e., nk+1 /nk ≥ λ > 1 for any k), then there exists a
positive constant c(λ) depending only on the lacunarity index
sequence of natural numbers
such that, for any trigonometric polynomial of the form
T (x) =
N
X
ak ∈ R,
ak cos nk x,
k=1
the following inequality holds:
kT k∞ ≥ c(λ)
N
X
|ak |.
k=1
Kashin and Temlyakov obtained a more general result in the
special case. Namely, they proved (see [1] and also [2])
Theorem A (B. S. Kashin, V. N. Temlyakov).
trigonometric polynomial of the form
f (x) =
2l
X
k=l+1
pk (x) cos 4k x,
1
For any
2
where the pk are real trigonometric polynomials of deg(pk ) ≤
2l , where k = l + 1, . . . , 2l and l = 1, 2, . . . , the following
inequality holds
kf k∞ ≥ c
2l
X
kpk k1 ,
k=l+1
where c > 0 is an absolute constant.
Kashin and Temlyakov also introduced
QC -norm
(see [1]
and [2]) and proved that for this norm the previous result is
true in a more general case:
for a function
f ∈ L1 (0, 2π) with Fourier series f ∼
P∞
s=0 δs (f, x),
where
1
δ0 =
2π
2π
Z
QC -norm
f (x) dx,
0
of
X
δs =
2s−1 ≤|k|<2s
f
is dened as
Z 1
∞
X
rs (ω)δs (f, ·)
≡
0 kf kQC
s=0
where
fˆ(k)eikx , s = 1, 2, . . . ,
{rk (ω)}∞
k=0
dω,
L∞
is the system of Rademacher.
For any real
function f ∈ L (0, 2π) the following inequality holds
Theorem B (B. S. Kashin, V. N. Temlyakov).
1
∞
kf kQC
1 X
≥
kδs (f )k1 .
16 s=0
A similar result to the theorem A is proved by the author
in the case of an arbitrary lacunarity index and under weaker
constraints on the degrees of the polynomials
pk
(see [3] and
[5]).
Suppose that a sequence of natural numbers
satises the condition nk+1 /nk ≥ λ > 1, k =
1, 2, . . . . There exist the constants c = c(λ) > 0 and γ =
γ(λ) ∈ (0, 1) depending only on λ such that, for any trigonometric
Theorem 1.
{nk }∞
k=1
3
polynomial of the form
f (x) =
N
X
pk (x) cos nk x,
k=l
where pk are real trigonometric polynomials of deg(pk ) ≤
[γnl ], k = l, . . . , N, N ≥ l, l = 1, 2, . . . , the following
inequality holds
kf k∞ ≥ c
N
X
kpk k1 .
k=l
In connection with research of
QC -norm
Grigor'ev proved
the following result (see [4]).
There exists the sequence of trigonometric
such that
Theorem (Grigor'ev).
polynomials
σk (x) =
{σk (x)}∞
k=1
X
cj eijx , kσk k1 ≥
2k ≤|j|<2k+1
and
n
X
σk k=1
∞
π
, kσk k∞ ≤ 6, k = 1, 2, . . . ,
4
√
≤ A n, n = 1, 2, . . . ,
where A > 0 is an absolute constant.
Using some ideas of Grigor`ev`s work we have proved (see
[5]) the following result which shows that in theorem 1 (in
nk = 2k ) the condition deg(pk ) ≤ [γ2l ] can`t
k−k ε
substituted for deg(pk ) ≤ [2
] neither any ε ∈ (0, 1).
the case
be
Suppose that ε, εe are real numbers such that
< ε < εe < 1. For any W ∈ N there exist the real
trigonometric
pk (x), k = 1, . . . , W , such that
polynomials
Theorem 2.
1
2
ε
deg pk ≤ 2k−k , kpk k1 ≥ 13 , kpk k∞ ≤ 70, k = 1, . . . , W,
and
n
X
k max pk (x) cos 2 x ≤ CW εe,
1≤n≤W k=1
∞
where C = C(ε, εe) > 0 is a constant depending only on ε
and εe.
4
REFERENCES
[1] B. S. Kashin, V. N. Temlyakov, On a norm and approximation
characteristics of classes of functions of several variables, in
Metric Theory of Functions and Related Problems in Analysis,
Collection of articles dedicated to P. L. Ul`yanov on his seventieth
birthday (AFTs, Moscow, 1999), pp. 6999 [in Russian].
[2] B. S. Kashin, V. N. Temlyakov, On a certain norm and
related applications, Math. Notes, Vol. 64, No. 4 (1998), pp.
551554.
[3] A. O. Radomskii, On an Inequality of Sidon Type for
Trigonometric Polynomials, Math. Notes, 2011, Vol. 89, No.
4, pp. 555-561.
[4] P. G. Grigor'ev, On a sequence of trigonometric polynomials,
Math. Notes, 1997, Vol. 61, No. 6, pp. 780-783.
[5] A. O. Radomskii, On the possibility of an extension of
inequalities of Sidon type, Mat. Zametki, (in press).
[6] A. Zygmund Trigonometric Series. V. 1. Ì.: Mir, 1965
[in Russian].