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2URAL MATHEMATICAL JOURNAL, Vol. 8, No. 2, 2022, pp. 27-45
DOI: 10.15826/umj.2022.2.003
ON ONE INEQUALITY OF DIFFERENT METRICS FOR TRIGONOMETRIC POLYNOMIALS
Vitalii V. Arestov
Krasovskii Institute of Mathematics and Mechanics,
Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russia
Ural Federal University, 51 Lenin ave., Ekaterinburg, 620000, Russia
vitalii.arestov@urfu.ru
Ural Federal University, 51 Lenin ave., Ekaterinburg, 620000, Russia
Krasovskii Institute of Mathematics and Mechanics,
Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russia
marina.deikalova@urfu.ru
Abstract: We study the sharp inequality between the uniform norm and Lp(0, n/2)-norm of polynomials in the system C = {cos(2fc + of cosines with odd harmonics. We investigate the limit behavior of the best
constant in this inequality with respect to the order n of polynomials as n ^ <x and provide a characterization of the extremal polynomial in the inequality for a fixed order of polynomials.
Keywords: Trigonometric cosine polynomial in odd harmonics, Nikol'skii different metrics inequality.
1. Problem statement, backgroung, and some preliminaries 1.1. Some notation
This paper considers classical spaces of complex-valued functions of one variable; in fact, we use not the spaces themselves but the norms of these spaces on some subspaces of polynomials. Let I = [a, b] be an interval on the real axis, and let u be a nonnegative, integrable function on I called a weight. For 0 < p < to, the space LU = LU(I) consists of complex-valued, Lebesgue measurable on I functions f such that the function u|f |p is integrable on I. The functional
is a norm in the space LU = LU (I) for 1 < p < to, but not for 0 < p < 1. Nevertheless, for all 0 < p < to, we will refer to (1.1) as a norm or, more precisely, as a p-norm. The space LU = LU(I) (here p = 2) is a Hilbert space with the inner product
Marina V. Deikalova
(1.1)
In the case of unit weight u(x) = 1, the weight symbol is omitted in the notation of spaces and their norms. By L= LTO(I) we mean the space C = C(I) of functions continuous (bounded) on the interval I with the uniform norm
11/iU = 11/iio(i) = max {|/(x)|: x € (I)}.
The space C = C(I) contains the subspace Co = C(I)o of functions / vanishing at the right end point of the interval: /(b) = 0. In what follows, the parameter a is equal to zero: a = 0, and b is 1, n/2, or n depending on the situation.
We define the spaces of 2n-periodic functions accordingly: the spaces , 0 < p < to, with p-norm
ii/iip = ii/ii^ = (^ fjmr<to) p
and the space C2n of continuous 2n-periodic functions with the uniform norm
ii/iU = ii/ilc2n = max{|/(x)|: x € R} = max{|/(x)|: x € [-n, n]}.
1.2. Nikol'skii C—LP inequality: the classical case
Let fn = fn(C), n > 1, be the set of trigonometric polynomials
n
fn(x) = — + ^(a-fc cos kx + bk sin kx) k=l
of order (at most) n with complex coefficients. In function theory and its applications, inequalities between two different norms of polynomials are of great importance. Such inequalities arose in Jackson's paper [19], but they were thoroughly investigated and applied by Nikol'skii [25]; [26, Ch. 3, Sect. 3.3], in this connection, they are called Nikol'skii inequalities or inequalities of different metrics. Large studies have been devoted to such inequalities, see [25]; [26, Ch. 3, Sect. 3.3]; [21, Ch. 3, Sects. 3.5-3.6]; [11, Ch. 8, Sect. 8.4]; [17, 24] and the bibliography therein. In this paper, the authors will need some information about the Nikol'skii inequalities
11 /n 11 TO < C(n)p h/nhp, in € fn, (1.2)
between the uniform norm and p-norm of polynomials (see, the same sources [21, Ch. 3, Sects. 3.5, 3.6]; [17, 24]). We assume that C(n)p is the best (the smallest possible) constant in this inequality. Employing harmonic analysis, it is easy to obtain (see, for example, [21, Ch. 3, Sect. 3.5, Theorem 3.5.1]) that if p = 2, then
C(n)2 = V2n + 1 (1.3)
and inequality (1.2) becomes an equality at the Dirichlet kernel
1 n
Dn(x) = - + cos kx> (L4)
k=i
i.e., the Dirichlet kernel is an extremal polynomial. The exact values of C(n)p for p = 2 are unknown. There are constructive estimates for C(n)p, 0 < p < to, mostly upper ones; see [11, Ch. 8, Sect. 8.4]; [1]; [12, Sect. 7.2]; [15-18] and the bibliography therein. Note for the future the upper estimate of Badkov [12, Sect. 7.2, Theorem 7.2]
C(n)p < 4n1/p, 0 < p < to. (1.5)
It is not the best at the moment, but sufficient for us in what follows.
Much research has been devoted to inequality (1.2) for p = 1; information about the history and results related to this inequality can be found in [10, 15-17, 29, 30]. The following estimates are simple and quite rough:
n + 1 < C(n) 1 < 2n + 1. (1.6)
The upper estimate follows from the representation of polynomials fn € Fn in the form of convolution
1 r
fn(x) = - / fn(t)Dn(x - t)dt, fn € Fn, n J-n
n
with the Dirichlet kernel (1.4). The Fejer kernel
n
^ T'k (t) =
Fn(t) = £ Dk (t) k=0
n + 1 IN , 1 /sin(n +1)1 \ 2
+ + 1 -k) cos kt = - — t 2 , t / 2z/vr, V e Z,
— 1 2 v sin2 /
provides the former inequality in (1.6). This kernel is nonnegative (see, for example, [27, Vol. 2, Part 6, Sect. 3, Problem 18]) and such that
(n + 1)2 im ii 1 f^ r- r u n +1
Halloo = Fn(0) = ||F„||i = - r Fn(t)dt
2 n J0
2 ' " ""* n J0 "w 2
This implies the former inequality in (1.6).
Taikov in [29] gives the result of S.B. Stechkin that (for p = 1) there exists a constant c > 0 such that
C(n)1 = cn + o(n), n —^ to. (1.7)
Stechkin's proof of this result is given in [30]. Estimates (1.6) imply that 1 < c < 2. Taikov [29] obtained substantially closer two-sided bounds for the constant c. Hormander and Bernhardsson have obtained [14] the best estimates currently:
1.081857643 < c < 1.081857645.
Let E(ct) be the space of entire functions of exponential type (at most) a > 0, and let E(ct)p for 0 < p < to be the space of functions f € E(<r) belonging on the real axis to the spaces Lp = Lp(R) with finite norms ^
llf llP = llf IIlp(r) = ( jr If (x)|pdx) P, 0 <p< to,
llf lU = llf IIC(r) = sup{|f (x)|: x € R}, p = to. For any 0 < p < to on E(<r)p, we have the inequality
< |f ||p, f € E(ct)p, (1.8)
ascending to Nikol'skii [25]; [26, Ch. 3, Sect. 3.3], in which Ap is a finite constant depending only on the parameter p; see details in [15, 16]. In what follows, we assume that Ap is the least possible, i.e., best constant in (1.8). The exact value of this quantity is currently known only for p = 2; namely, A2 = 1/aA; see, for example, [31, Ch. IV, Sect. 4.9, Subsect. 4.9.53, (28)].
Gorbachev [16] obtained a significantly more informative assertion in comparison with (1.7). Namely, he proved that the following limit relation is true:
lim = (1.9)
n^ro 2nn
Actuality, Gorbachev obtained [16] an even more precise result; namely, he proved the following two-sided inequality:
C (n)i n + 1 .
Ai < <-Au
2nn n
which entails, in particular, (1.9).
The following statement belongs to Levin and Lubinsky [22, Theorem 2.1, (2.1)]; it means that a relation similar to (1.9) holds for all 0 < p < to.
Theorem A. The following limit relation is valid for 0 < p < to:
Hm C{n) =
(2nn)1/P p v 7
This statement is part of a more general result of Ganzburg and Tikhonov [15, Theorem 1.5, (1.19)]. Nikol'skii inequalities and, more generally, Bernstein-Nikol'skii inequalities
||/nr) ||, < C M(n)p,q ||/n ||P) fn (1.11)
for trigonometric polynomials and similar inequalities for entire functions are a broad area of function theory. Ganzburg and Tikhonov, in the already cited paper [15], and Gorbachev and Mart'yanov in [18] studied the relationship between exact constants in Bernstein-Nikol'skii (C, Lp)-inequalities and, more generally, (Lq, Lp)-inequalities for polynomials and entire functions of exponential type. We presented here in Theorem A only some results on this topic, which we will need in what follows; for a rich overview of these studies, see [17].
Over the last century, extensive investigations have been carried out on sharp inequalities, i.e., the study of exact constants and extremal functions in inequalities (1.11) for trigonometric polynomials, as well as for algebraic polynomials and entire functions of exponential type; for specific results and further references, see [1, 2, 10, 12, 13, 15, 17, 21, 23, 24, 28, 31].
1.3. Nikol'skii inequality between the uniform norm and Lp-norm on the interval [0,n/2] for polynomials in the cosine system with odd harmonics
1.3.1. Nikol'skii inequality for ^-polynomials
Let Cn = Cn(C), n > 0, be the set of polynomials
n
0n(x) = ak cos(2k + 1)x (1.12) k=0
with complex coefficients in the cosine system with odd harmonics
C = {cos(2k + 1)x}£=0. (1.13)
The functions (1.12) will be called Cn-polynomials or C-polynomials of order n. The functions (1.12) are trigonometric polynomials; as trigonometric polynomials they have order 2n + 1. Note that the functions (1.12) vanish at the point x = n/2: ^>n(n/2) = 0, so none of them is the identical unity. The main goal of this paper is to study the sharp inequality
||0n|U < M(n)p||0n||p, 0n € Cn, (1.14)
between the uniform norm
ll^nlU = ll0n||c[0,n/2] = max{|0n(x)|: t € [0,n/2]}, and integral p-norm (0 < p < to)
/2 f ^/2 \1/P
№n\[p = \\(fin IIlp(0,7t/2) = I - J \(j)n{x)\Pdx I
of Cn-polynomials on the interval [0, n/2]. Inequality (1.14) appeared in the authors' paper [8] in connection with the study of a variant of the generalized translation in system (1.13) on an interval.
In connection with inequality (1.14), the question naturally arises of the sharp pointwise inequality
|<£n(t)| < M (n,t)p ll^nllLp (0,n/2), ^n € Cn, (1.15)
for points t € [0,n/2]. Especially important, as will be seen from what follows, is the inequality (1.15) at the end point t = 0:
|<£n(0)| < M(n, 0)p||^n||Lp(0,n/2), ^n € Cn. (1.16)
The study of inequalities (1.14), (1.15), and (1.16) includes, in particular, the study of the properties of extremal polynomials at which the inequalities turn into equalities. It is clear that if the polynomial 0n is extremal in one of these inequalities, then, for any constant c, the polynomial c^n is also extremal. If any extremal polynomial of this inequality has the form c^n with some constant c, then ^n is said to be the unique extremal polynomial.
The following statement is proved in the authors' paper [8, Theorem 4].
Theorem B. For 1 < p < to and n > 0, the following statements hold.
(1) The best constants in inequalities (1.14) and (1.16) coincide:
M (n)p = M (n, 0)p. (1.17)
(2) The polynomial extremal in inequality (1.16) attains its uniform norm at the point 0 and is also extremal in inequality (1.14).
The authors do not know whether equality (1.17) holds for 0 < p < 1; in this case, we can only state that M(n, 0)p < M(n)p.
1.3.2. Approximation interpretation of inequalities
The problems of studying inequalities (1.14), (1.15), and (1.16) can be reformulated as approximation problems; we will do this only for inequality (1.16). Consider the set
Cn[0] = {0n € Cn: 0n(0) = 1} (1.18)
of polynomials with fixed value at the point 0: 0n(0) = 1. On this set, we define the value
En[0]p = inf {||0n||Lp(0,n/2): ^n € Cn[0]} (1.19)
of the least deviation from zero of the class of polynomials (1.18) in the space Lp(0, n/2). It is clear that
En[0]p = 1/M(n, 0)p.
Moreover, extremal polynomials in problem (1.19) and inequality (1.16) coincide. More precisely, (every) extremal polynomial in (1.19) is also extremal in (1.16); conversely, if ^ is an extremal polynomial of inequality (1.16), then the polynomial ^>/^>(0) is extremal in (1.19). Thus, the problem of sharp inequality (1.16) is equivalent to problem (1.19) on the least deviation from zero of the class (1.18).
1.3.3. Christoffel—Darboux kernel for system (1.13)
Sometimes, we will use the following shorter notation for the functions of system (1.13):
nk(x) = cos(2k + 1)x, x € [0, n/2]. (1.20)
This system of functions is orthogonal with respect to the inner product
2 r!2 _
(.f,g} = - / f(t)g(t)dt. n Jo
More precisely, as is easy to see, for k, m > 1, the inner products
2 r/2
5k,m = {'Ik, f]m) = ~ cos(2k + l)i cos(2m + 1 )tdt
n J o
have the following values: ¿k>m = 0, k = m, and ¿k;k = 1/2.
Due to the orthogonality of the system {nk}k>1, the coefficients of the polynomial
n
0(x) = ak cos(2k + 1)x (1.21)
k=0
are expressed in terms of the polynomial itself by the formulas ak = 2(0, nk). Substituting these expressions into (1.21), we obtain
n n n
0(x) = flk nk (x) = 2^2 (0, nk )nk (x) = ^ 0, 2^ nk(x)nk), k=0 k=0 k=0
which can be written in the form
2 W2
<f>(x) = - <f>(t)Kn(x,t)dt, n Jo
where
n
Kn(x,t)=^ nk (x)nk (t) (1.22)
k=0
is the Christoffel-Darboux kernel for the system Cn. Convolute this kernel. Using the formula
2 cos a cos b = cos(a + b) + cos(a — b), (1.23)
we find
nn
Kn(x, t) = 2 ^ nk(x)nk(t) = 2 ^ cos ((2k + 1)x) cos ((2k + 1)t) =
k=0 k=0
n
= (cos ((2k + 1)(x +1)) + cos ((2k + 1)(x — t))). k=0
Using the formula (see, for example, [27, Part 6, Sect. 3, Problem 16])
En x „ sin2(n +1)0 „ .
cos (2k + 1)9 =----!-, O^vtt, 1/eZ,
v J 2 sin 6
k=0
we obtain the following representation for the Christoffel-Darboux kernel:
^ 1 /sin2(n + 1)(x +t) sin2(n + 1)(x - t)
K-n{x,t) = - -———--h
2 \ sin(x +t) sin(x — t)
The orthogonality of the system of functions (1.20) implies that, for a pair of polynomials
0(x) = ^ ak cos(2k + 1)x, ^(x) = ^ cos(2k + 1)x, k=0 k=0
a generalized version of Parceval's identity holds:
n
2 (</>,#) = y^'lj'l,-
k=1
In particular, the norm of the polynomial 0 € Cn is expressed in terms of its Fourier coefficients (a&} by Parceval's identity
n
2ihll2(0,n/2) =E laki'" k=0
Using this equality and Holder's inequality, we obtain the inequality
n / n \ 1/2
10(0)1 = < ATliEH2) = + 1) ||0||i3(o,7r/2),
k=0 ^ k=0 '
which at the kernel (1.22) turns into an equality. Thus, in the space L2(0,n/2), we have
\liu )■> = v/2(??, + l), n > 0. (1.24)
It is useful to compare this result with the corresponding result (1.2)-(1.3) for the classical case.
1.4. Main results
The authors consider the following statements to be the main ones in this paper.
1.4.1. Limit behavior of the best constants in inequalities (1.14) and (1.16)
For the best constant in inequality (1.14), we have an analog of the above Theorem A.
Theorem 1. The following limit relation holds for constants M(n)p in inequality (1.14) for 0 < p < to:
lim = Ap. (1.25)
n
n
1.4.2. Characterization of a polynomial extremal in inequality (1.16)
Denote by = € Cn a polynomial of order n > 1 with unit leading coefficient that deviates least from zero in the space LW (0, n/2) with the weight
w(x) = sin2 x (1.26)
on the interval (0,n/2). In other words, is a solution to the problem
min{ll0nhiW(0,n/2) : 0n € <} = y^n^LW(0,n/2) on the set C1 of polynomials (1.12) of order n with leading coefficient 1: an = 1.
Theorem 2. For all 1 < p < to and n > 1, the polynomial of order n with unit leading coefficient that deviates least from zero in the space LW (0,n/2) with weight (1.26) is the unique extremal polynomial in inequality (1.16).
There are statements similar to Theorem 2 in [5, Theorem 1; 6, Theorem 2; 7, Theorem 2; 9, Theorem 3; 4, Theorem 2; 3].
We will also give some estimates for the best constant M(n)p in inequality (1.14); see, in particular, Section 2.4.
2. Behavior with respect to n of the best constant in the Nikol'skii inequality
for C-polynomials
2.1. Case p =2
According to (1.3) and (1.24), for n > 0, we have
C(n)2 = V^TTl, M iuh = y/2(n + l).
Thus,
lim M^i = lim CO»*
n—^^o n
1/2 n—TO n
1/2
Both limits exist and coincide; this fact served as an argument for the authors that this property of the quantities M(n)p will hold for all 0 < p < to.
2.2. Expression of C-polynomials in terms of the classical trigonometric polynomials
Let Cn be the set of even (complex) trigonometric polynomials
n
fn(x) = ^ ak cos kx k=0
of order (at most) n > 1. On this set, we consider the uniform norm
H/Jto = ||/n|C[0,n] = max{ l/n(x)| : x € [0,n]}
and the integral p-norm (for 0 < p < to)
1 rn \ 1/p
Jo
Hp = II/«IIlp(o,7t) = {~ jo \fn(x)\pdx^J
It is well known (however, it is easy to show) that the best constants C(n)p in inequality (1.2) and the inequality
|/n(0)|< C(n)p ||/ra||p, /„ (2.1)
coincide.
Lemma 1. For fn € cn, the function
0n(x) = fn(2x) cos x (2.2)
is a Cn-polynomial. Conversely, every CCn-polynomial 0n can be represented in the form (2.2), where fn € cn. Thus, formula (2.2) establishes one-to-one correspondence between cn and Cn.
Proof. Let fn € cn be a trigonometric polynomial. Let
n
0(x) = fn(2x) cos x = ^ ak cos 2kx cos x.
Applying again formula (1.23), we have
2 cos 2kx cos x = cos(2k + 1)x + cos(2k — 1)x. Using this relation, we find
20(x) = 2/n(2x) cos x = 2 ^ ak cos 2kx cos x =
k=0
n n n n—1
y^ ak cos(2k + 1)x + ^ ak cos(2k — 1)x = ^ ak cos(2k + 1)x + ^ ak+1 cos(2k + 1)x. k=0 k=0 k=0 k=—1
As a result, we obtain the representation
nn
20(x) = ^ (ak + ak+1) cos(2k + 1)x = (2a0 + a1) cos x + ^(ak + ak+1) cos(2k + 1)x, (2.3)
where a—1 = 0 and an+1 = 0. The function (2.3) is a Cn-polynomial.
Let us prove the inverse statement, i.e., let us prove that an arbitrary Cn-polynomial
0(x) = ^ Ak cos(2k + 1)x k=0
can be represented in the form (2.2). Rewrite this polynomial in the form
n
0(x) = A'k cos(2k + 1)x, (2.4)
k=—1
where A—1 = A0 = A0/2 and A'k = Ak, 1 < k < n.
n
It suffices to represent the polynomial 20 in the form (2.3). The latter means that the coefficients {Afc}n=-1 of the polynomial (2.4) can be represented as
A'fc = ak + ak+1, —1 < k < n; a-1 = 0, an+1 = 0. (2.5)
It is easy to see that the formulas
n—k
afc = E(—!)1 Ak+i = ^k — ak+i, k = n, n — 1,..., 0,
1=0
give a solution to system (2.5). Lemma 1 is proved. □
Representation (2.2) implies that inequality (1.16) is equivalent to the inequality
|/n(0)| < C(n,a)p h/nllLg(0,n), /n € cn, (2.6)
on the set cn with weight a(t) = cosp(t/2). More exactly, the following assertion holds.
Lemma 2. For 0 < p < to and n > 1, inequality (1.16) on the set Cn and inequality (2.6) on cn are equivalent; specifically:
(1) the best constants in inequalities (2.6) and (1.16) are related by the equality
C(n,ff)p = M(n, 0)p; (2.7)
(2) extremal polynomials in these inequalities are related by (2.2).
Proof. Using relation (2.2), we find that, for an arbitrary polynomial 0n € Cn,
|fn(0)| = |0n(0)| < M (n, 0)p ||0n ilLp (0,n/2),
where
/ 2 rn/2 \ 1/P / 1 rn \ 1/P
11^11^(0,^/2) = ^-^ |/n(2a:)cosaf£teJ = ^ |/„(i) cos(i/2)|pdiJ = ||/„||lp(0^. Lemma 2 is proved. □
The following statement contains a quantitative relation between the constants in inequalities (1.14), (1.16), and (1.2).
Lemma 3. For 0 < p < to and n > 1, the best constants in inequalities (1.14), (1.16), and (2.1) (or, equivalently, (1.2)) are related as follows:
C(n)p < M(n, 0)p, (2.8)
M(n)p < C(n + 1)p. (2.9)
Proof. For polynomials /n € cn, we have ||/n||Lp(0,n) < ||/n||Lp(0,n). Therefore, the best constants in (2.1) and (2.6) are related by the inequality C(n)p < C(n, ct)p. This and (2.7) imply (2.8).
Let us prove inequality (2.9). A polynomial 0n € Cn has the form
0n(x) = E ak cos(2k + 1)x. k=0
Writing it in exponential form, we find
n n n
20n(x) = ^ ak (ei(2k+1)x + e—i(2k+1)x) = £ akei(2k+1)x + ^ ake—i(2k+1)x = k=0 k=0 k=0 n n /n+1 n—1 \
= e—^ akei2(k+1)x + e—^ ake—i2kx = e—i( ^ ak—1ei2kx + ^ ak+1e—i2k^ . k=0 k=0 k=1 k=0
The function
n+1 n—1
gn+1 (x) = ak—1eikx + ^ ak+1e—ikx k=1 k=0
is a trigonometric polynomial of order n + 1. The functions 20n and gn+1 are related as 20n(x) = e—gn+1(2x), x € R, or, equivalently, as
20n(x/2) = e—ix/2gn+1(x), x € R. (2.10)
The following inequality holds for the polynomial gn+1 (cf. (1.2)):
||gn+1||c2n < c(n + 1)p|gn+1|£pn. (2.11)
As a consequence of (2.10), we have ||gn+1 ||c2n = 2||0n||c[0,n/2] and
f i r2n \ 1/p f 1 /-n \ 1/p
\\9n+i\= \gn+i(x)\pdxj =[x/2 = t]=2^-j \Mt)\pdtj =
(2 fn/2 \1/p = 21-J \Mt)\pdt) = 2||0„||£P(o^/2).
Consequently, inequality (2.11) is equivalent to the inequality
||0n he[0,n/2] < C(n + 1)p ||0n|Lp(0,n/2).
Comparing this inequality with (1.14), we conclude that inequality (2.9) holds. Lemma 3 is proved completely. □
2.3. Proof of Theorem 1
By Lemma 3, for all 0 < p < to and n > 1, we have the inequalities
C (n)p < M (n)p < C (n + 1)p.
As a consequence, we have
C (n)„ M (n)„ C (n + 1)p f n \1/p
< -<
(2nn)VP " (2nn)1/^ _ (2n(n + 1))1/p \n + 1
Passing here to the limit as n ^ to and using the result (1.10) of Theorem A, we obtain (1.25). Theorem 1 is proved. □
2.4. Estimates
2.4.1. Monotonicity of a constant in (1.14) in p
Let 0 < p1 < p2 < to. For an arbitrary 0n € Cn, we have
(2 r n/2 \1/p2 /2 f n/2 \1/p2
u\\P2 = i - yo \mP2dt\ = i - Jo \<p(t)\pi\<p(t)\p2~pidtj < < (f £/2 mr/1" mc^-^ = mpi)pi/p2 mc^-^.
Consider inequality (1.14) with the parameter p2:
||0||c[0,n/2] < M(n)p2 ||0|p2 < M(n)p2 (n0hpi)pi/p2 (||0|| C[0,n/2]) (p2—pi)/p2 . Dividing by the latter factor, we obtain the inequality
(|0|c[0,n/2])pi/p2 < M(n)p2 (||0||pi)pi/p2 .
Raise it to the power p2/p1:
||0|C[0,n/2]< (M(n)p2)p2/pi ||0|pi. This inequality holds for any polynomial 0 € Cn; hence,
M(n)pi < (M(n)p2 )p2/pi ,
or
(M(n)pi)pi < (M(n)p2)p2 , 0 < p1 < p2 < to; (2.12)
this is the required property of monotonicity.
2.4.2. Estimates for a constant in (1.14)
As a special case of (2.12), we have the inequality
M(n)p > (M(n)1 )1/p , 1 < p < to. Using now inequalities (2.8) and (1.6), we obtain the lower estimate
M(n)p > (n + 1)1/p , 1 < p< to. Using (2.12) and (1.24), we obtain the estimates
M(n)p < (2(n + 1))1/p , 0 < p < 2,
p
a/p
M(n)p > (2(n + 1))1/p , p > 2. Finally, inequalities (2.9) and (1.5) imply the estimate
M(n)p < 4(n + 1)1/p, 0 <p< to.
3. Characterization of an extremal C-polynomial in the Nikol'skii inequality
The primary purpose of this section is to prove Theorem 2, which characterizes the extremal Cn-polynomial in the Nikol'skii inequality (1.16). As mentioned above, statements similar to Theorem 2 can be found in the authors' papers, personal and with co-authors: [5-7, 9]. The ideas contained in these statements have been summarized in [3]. The fact that we need [3] holds for inequalities similar to inequality (1.16) for algebraic polynomials in spaces Lp with weight. Because of this, we first rewrite inequality (1.16) in terms of algebraic polynomials, use Theorem 4 from [3], and then make a conclusion related to inequality (1.16).
3.1. Equivalent to (1.16) inequality for algebraic polynomials on an interval
We can associate inequalities (1.14) and (1.16) on the set Cn with equivalent inequalities on the set of algebraic polynomials. For this, we describe C-polynomials in terms of algebraic polynomials. Denote by pn the set of algebraic polynomials
n
Pn(i) = E Cl(3.1)
l=0
of degree (at most) n with complex coefficients. In further consideration, we will need some properties of the Chebyshev polynomials
Tv(t) = cos(vx), x = arccos t, t € [-1,1], (3.2)
of the first kind of degree v > 0; these properties can be found, for example, in [21, Ch. 2, Sect. 2.2]). A polynomial Tv, v > 1, can be represented as
V [V/2] (_1)k
TM) = Ô E ^rCk;-kC2t)v~k = v1? -v2v-3tv-2 + ■■■. (3.3)
2 v k fc=0
A polynomial Tv is even or odd in accordance with the evenness of the number v, and its leading coefficient (for v > 1) is 2V-1.
Lemma 4. For n > 1, the relation
0n(x) = tpn(t2), t = cosx, x € [0,n], t € [-1,1], (3.4)
establishes a bijection between the set of polynomials 0n € Cn of order n and the set pn of algebraic polynomials pn of degree n. Under this correspondence, the leading coefficient an(0n) of a C-polynomial 0n and the leading coefficient cn(pn) of the polynomial pn are related by the formula,
Cn(Pn) = 22nan(0n); (3.5)
in addition, the following equalities hold :
0n(O) = Pn(1), (3.6)
^p-D/2
7TV-L — U
Mlp(0,tt/2) = Hp™ 11^,(0,1), f(u) = v(u)p = —J==^. (3.7)
Proof. First, let us show that an arbitrary C-polynomial
n
0n (x) — ^ ak cos(2k + 1)x (3.8)
k=0
of order n > 0 can be presented in the form (3.4), where pn is an algebraic polynomial (3.1) of degree n. Replace cos(2k + 1)x with T2k+1(t), t — cosx, t € [—1,1], in ( ). A polynomial T2k+i is odd, and consequently, can be represented in the form T2k+1(t) — tRk(t2), where Rk is a real algebraic polynomial of degree k. The leading coefficient of the polynomial Rk is 22k and Rk(1) — 1. Therefore, polynomial (3.8) can be represented as (3.4), where pn is some algebraic polynomial (3.1) of degree n; the leading coefficients of the polynomials 0n and pn satisfy the relation (3.5).
Conversely, let pn € Pn. Consider the function f (x) — cos xpn(cos2 x). Based on the representation (3.1), we have
n
f (x) — cos x pn (cos2 x) — ^ ci cos2l+1 x. (3.9)
1=0
The function cos2l+1 x is an even trigonometric polynomial of order 21 + 1:
i
cos2l+1 x — ^ pi,k cos(2k + 1)x. (3.10)
k=0
This fact is, of course, known. However, it is easy to obtain (by induction on I) starting from the representations (3.2) and (3.3). In particular, it follows from (3.3) that the leading coefficient of the representation (3.10) is pl;l — 2-21. Thus, function (3.9) is a C-polynomial of the form (3.8).
Relation (3.4) establishes a bijection between Cn and Pn. Formula (3.6) is obvious. Let us verify (3.7):
( 2 rn/2 \ 1/P ( 2 rn/2 \ 1/P
Hz>(0,tt/2) = (-y \<j>n{x)\pdx\ ={ — J | cos xp„(cos2 x) \pdx ) = [cos x = t] = (2 f \ u^ ip dt \1/p .2 1 (1 f \ , MPu(p-1)/2du\1/p -1 \ 1/p y(P-1)/2
|pn(u)|P u(-u)du , u(u) — u(u)p —
n
\/l - U
Thus, we obtain equality (3.7). Lemma 4 is proved completely. □
0
Corollary 1. The representation (3.4) implies that for n > 1 the polynomial 0 € Cn, 0 ^ 0, can have on [0,n/2) at most n zeros, taking into account their multiplicities, i.e., Cn is the Chebyshev system on [0,n/2).
Corollary 2. Lemma 4 implies that inequalities (1.14) and (1.16) can be associated with equivalent sharp inequalities on the set of algebraic polynomials:
HpnlU <M(n)pl|pn 11Lu (0,1), pn €Pn (3.11)
with constant M(n)p — M (n)p and
|pn(1)| < M(n, 1)pllpnlLu(0,1), pn € Pn, (3.12)
with constant M(n, 1)p — M(n, 0)p. Moreover, a polynomial € Pn is extremal in inequalities (3.11) and (3.12) if and only if the polynomial ^>n € Cn related to by formula (3.4) is extremal in inequalities (1.14) and (1.16).
Let us reformulate the problem of studying inequality (3.12) as an approximation problem. Consider the set
pn[1] = K epn: Pn(1) = l} (3.13)
of polynomials with a fixed unit value at unity: pn(1) = 1. On this set, we define the value
e«[1]p = inf {||p„||lip(0,1): p„ € pra[1]} (3.14)
of the least deviation from zero in the space LU(0,1) of the class of polynomials (3.13). It is clear that en[1]p = 1/m(n, 1)p. Moreover, polynomials extremal in problem (3.14) and in inequality (3.12) coincide (up to a multiplicative constant). Thus, the problem on the sharp inequality (3.12) is equivalent to problem (3.14) on the least deviation from zero of the class (3.13).
Lemma 5. For 1 < p < to and n > 1, the following statements hold for a polynomial € pn extremal in inequality (3.12) and such that £n(1) = 1.
(1) The polynomial is characterized by the property
/ pn-i(u)(1 - u)u(u)|^„(u)|p-1sign £„(u)du = 0, pn-i epB-i. (3.15)
0
(2) The polynomial has degree n, all its n roots are real, simple, and lie on the interval (0,1); in this sense, the polynomial is real.
(3) The polynomial is unique.
Proof. Most of the results of this lemma are contained in [3]. We find it difficult in some cases to make precise references to this paper. Therefore, we have to repeat some arguments from [3] with clarifications and explanations.
In Theorem 4 of [3], in particular, the following properties of a polynomial extremal in the inequality (3.12) were proved.
(1') The polynomial is real; more exactly, it has real coefficients and, hence, takes real values on the real axis.
(2') The polynomial is characterized by the property (3.15).
It follows that the polynomial has n sign changes on the interval (0,1). Otherwise, the polynomial with simple zeros at the sign change points of the polynomial would have order at most n — 1 and the property (3.15) would not hold on it.
Finally, let us verify that an extremal polynomial is unique for all 1 < p < to. In fact, let £ra and Zn be two polynomials that solve problem (3.14). By the inequality ||gn + (n||lg <
||lU + ||(n||,Lg, their half-sum (gn + Zn)/2 has the same property; hence, we have the equality ||^n + Cn|lg = ||^n||L£ + ||(n||L£. For 1 < p < to, since the space LU(0,1) is strictly normalized, it immediately follows that Zn = If p = 1, then we can only assert so far that the signs of the polynomials Zn and coincide almost everywhere on [0,1]. The zeros of these polynomials are simple and lie on the interval (0,1), so the polynomials Zn and have the same set of zeros and the same value at the point u = 1: Zn(1) = £ra(1) = 1, hence, these polynomials coincide. Thus, the extremal polynomial is unique for p = 1 too.
Lemma 5 is proved completely. □
Based on the weight u defined in (3.7), let us define the following weight on the interval (0,1):
w(x) = (1 — u) v(u) = —- ^ (3.16)
n
Consider the problem on the least deviation from zero
u(Pn)L|L(0,1) — min{||pnHlP (0,1) : pn € pm (3.17)
in the space L^(0,1) of the set of algebraic polynomials of degree n whose leading coefficient
* _ *
is 1. Denote by ^ = a polynomial solving this problem:
(0,1) = IKIIlL(0,1);
it is called a polynomial of degree n with unit leading coefficient that deviates least from zero in the space L^(0,1).
Theorem 3. For all 1 < p < to and n > 1, the polynomial of degree n with unit leading coefficient that deviates least from zero in the space L^(0,1) with weight (3.16) is the unique extremal polynomial in inequality (3.12).
Proof. The polynomial ^ is characterized by the property that the function |^|p :sign ^ is orthogonal to the space pn-1 (see, for example, [20, Ch. 3, Sect. 3.3, Theorems 3.3.1 and 3.3.2]):
/ ro(x) pn-1(x)|^n(x)|p 1sign f?n(x) dx — 0, pn-1 €Pn-1.
0
This property is the same as property (3.15). Therefore, the polynomials £n and ^n can differ only by a multiplicative constant. Theorem 3 is proved. □
3.2. Characterization of a C-polynomial extremal in inequality (1.16)
Let us apply the results of the previous section to describing the characteristic properties of Cn-polynomials extremal in inequalities (1.14) and (1.16).
3.2.1. An analog of Lemma 5 in the set of C-polynomials
Let us reformulate Lemma 5 for the extremal Cn-polynomial of inequality (1.16).
Lemma 6. For 1 < p < to and n > 1, the following statements hold for polynomials ^>n € Cn extremal in inequality (1.16) and such that ^>n(0) — 1.
(1) The polynomial ^>n is characterized by the property
(■ n/2
/ 0n-1 (t)(sin2 t) |^n(t)|p-1sign Pn(t)dt — 0, 0n-1 € Cn-1. (3.18)
0
(2) The polynomial ^>n has order n. The polynomial ^>n has n simple roots on the interval (0, n/2).
(3) The polynomial ^>n is unique.
Proof. Let us employ the statements of Lemmas 4 and 5. Let € pn be an extremal (algebraic) polynomial in inequality (1.16) with the property £n(1) — 1. The polynomial £n and the polynomial ^>n are related by (3.4).
Let us check that the relation (3.18) coincides with (is equivalent to) (3.15). For this, we transform the functional on the left-hand side of (3.15). For a polynomial pn-1 € pn-1, we define a
polynomial € that is expressed in terms of pn_i by formula (3.4). Based on the left-hand side of (3.15), we find
/ Pn_i(u)(1 - u) u (u) | (u) |p_1 sign (u) du =
/0
1 Z"1 1
= - / p„_ 1 («)(1 - «) -I Vüßn(u)|p~1 sign £>„(«)du =
n 7o V1 - u
' ' ^ ■v/tt£»n(«)|p_1siffn Qn(u)d,u = \u - +2 n J 0
2
I tyn n ./0
2 r n/2
n ./o
/ p„_i(«)\/l — («) sign Qn(u)du = [u, = t ] =
0 [■1
/ ip„_i(i2) - i2|i£)„(i2)|p-1sign Qn(t2)dt = [t = cosx] = 0
0n_i(x) sin2 x (x)|p-1sign (x)dt.
Now you can see that the conditions (3.18) and (3.15) hold or do not hold simultaneously. Lemma 6 is proved. □
Consider a problem similar (3.17) on the value
U«)L|(o,n/2) = min{||0„||lp(o,n/2): 0n g <} (3.19)
of the least deviation from zero in the space (0, n/2) with weight a(x) = sin2 x of the set C1 of algebraic polynomials of degree n whose leading coefficient is 1. Denote by = the
polynomial of degree n with unit leading coefficient that deviates least from zero in the space (0, n/2), i.e., the polynomial that solves problem (3.19):
U(C1)LP(0,n/2) = y^n yLP(0,n/2) •
Lemma 7. For 1 < p < to and n > 1, the following statements hold for problems (3.17) and (3.19).
(1) The values of the problems are related by the equality
U(C1)LP(0,n/2) = 22n u(Pn(0,1).
(2) A polynomial ^n g pn extremal in problem (3.17) and a polynomial g C1 extremal in problem (3.19) are related by the equality
^ra(x) = 22nt^n(t2), t = cosx, x g [0, n/2], t g [0,1].
Proof. Let pn g pn, and let 0n g Cn be the polynomial expressed in terms of pn by formula (3.4). We have
1
||Pn||^(oa) = U{p~1)/2VT^Tl,\pn(u)\pdu = [u = t2} =
2 fi _ 2 />7r/2
= - J Vi ~ t2\tpn(t2)\pdt = [t = cos a;] = - J sin2 x\<f>n(x)\pdx = Un\\PLP(0 7T/2y
Consequently, the norms satisfy the equality
||pnIIl^(0,i) = y^n^L?(0,n/2).
According to (3.5), we have
€ Ci ^ 2_2np„ € Pi. From here, all the assertions of Lemma 7 follow. Lemma 7 is proven. □
3.2.2. The proof of Theorem 2
For an algebraic polynomial p € pn and a Cn-polynomial 0 € Cn related by (3.4), we will say that they are (3.4)-related.
(1) According to Corollary 2, the polynomials q € pn and 0 € Cn extremal in the Nikol'skii inequalities (1.14) and (1.16) and inequalities (3.11) and (3.12), respectively, are (3.4)-related. According to Lemmas 5 and 6, these polynomials are unique up to numerical factors.
(2) Let q* € pn and € C1 be extremal polynomials, i.e., polynomials that deviate least from zero in problems (3.17) and (3.19), respectively. According to Lemma 7, the polynomials pn and 22ntp* are also (3.4)-related.
(3) According to Theorem 3, the polynomial q* is extremal in inequalities (3.11) and (3.12). Consequently, is also extremal in inequalities (1.14) and (1.16).
Thus, Theorem 2 is proved. □
4. Conclusions
The system of functions (1.20) is in some sense a "quarter" of the classical trigonometric system. However, as it turned out (see Theorem 1 and Lemma 3), the best constants in inequalities (1.2) and (1.14) are very close. The reason for this is not clear to the authors. What will be the situation with other extremal problems in these systems, the authors also do not know.
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