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4URAL MATHEMATICAL JOURNAL, Vol. 4, No. 2, 2018, pp. 24-32
DOI: 10.15826/umj.2018.2.004
ORDER EQUALITIES IN DIFFERENT METRICS FOR MODULI OF SMOOTHNESS OF VARIOUS ORDERS
Niyazi A. Il'yasov
Baku State University, Baku, AZ 1148, Azerbaijan [email protected]
Abstract: In this paper, we obtain order equalities for the kth order Lq(T)-moduli of smoothness Wk(f ; S)q in terms of expressions that contain the 1th order Lp(T)-moduli of smoothness wi(f ; S)p on the class of periodic functions f € Lp (T) with monotonically decreasing Fourier coefficients, where 1 < p < q < tx, k, I € N, and T = (-n,n].
Keywords: Inequalities of different metrics for moduli of smoothness, Order equality, Trigonometric Fourier series with monotone coefficients.
Let Lp(T), 1 < p < to, be the space of all measurable 2n-periodic functions with finite Lp(T)-norm
/t
= (n-1 JT | f (x)|p dx)1/p,
where T = (—n,n]; let En(f)p be the best approximation of a function f in the metric Lp(T) by trigonometric polynomials of order at most n, n € Z+; and let wj(f; 5)p, where l € N and 5 € [0, +to), be the 1th order modulus of smoothness of a function f € Lp(T):
ut(f; 5)p = sup {Af (-)||p: h € R, |h| < 5},
where
ai/<s>=E<-I
The following statement contains known upper estimates for (f; 5)q in terms of wj(f; 5)p, where f € Lp(T), p < q, and l,k € N (see, for example, [3, Theorem 1]; the background and the corresponding references can also be found in [3]).
Theorem A. Let 1 < p < q < to, f € Lp(T), a = 1/p - 1/q, l,k € N, and let
i/q
x \ 1/q v=1 P
Then f € Lq (T) and the following estimates hold :
(1)
< C1(l,p,q){ ||f ||1 +Qi(f ; p; a; q)}; (2)
(x \ 1/q
£ J , 77, G N, l<k:
q
K VV=n+1 7
/ n X l/g\
+n~k[Y,v9(k+a)~luji{f]l)j }' nGN' l>k-
Hereinafter, Cj(k, l,p, q,...), where j € N, stand for positive values depending only on the parameters given in parentheses.
Remark 1. In addition to the background outlined in [3], certain facts the author has learned after the publication of [3] should be mentioned, which partly provide more detailed information about the situation in the matter under consideration.
1) The first part of Theorem A: Qi(f; p; a; q) < to ^ f € Lq(T) and estimate (2) for l = 1 were established by Ul'yanov [11, §3, Theorem 1, statement c, inequalities (3.6)].
2) The problem of establishing estimates of type (2) by methods different from those applied in [11] was also considered by Timan [9; 10, Theorem A]. In [10, first indention after the statement of Theorem A], it was noted that the first part of Theorem A above with estimate (2) (under the
assumption that / f (x) dx = 0, which ensures the absence of the term ||f ||1 on the right-hand
J T
side of (2); see [10, inequality (1.12)]) for 1 < p < q < 2 was obtained in [8]. Actually, [8] (see [8, Theorem 8]) does not contain estimate (2); instead, there were announced an assertion that leads to the implication Q^(f; p; a; d) < to ^ f € Lq(T), where d = min{2,p} = p < q.
3) Estimate (2) in various forms has also been obtained earlier by other authors (see, for example, [7, Introduction] and the references therein).
4) Estimate (3) for l = k = 1 was proved by Ul'yanov [12, §4, Theorem 4, inequality (4.4)] (its formulation was given earlier in [11, §3, second inequality in (3.6')], and the validity of this estimate for l = k > 1 was also mentioned there).
5) Estimate (3) follows immediately from inequality (2) (see, for example, [7, Sect. 4], where this fact was noted for the case l = k = 1). In the general case l < k, it is sufficient to apply (2) to the function Af (x), where h € R, |h| < n/n, and take into account the estimates (¡(Af; n/v)p < 2k(¡(f; n/n)p for v < n and ^(Af; n/v)p < 2k^¡(f; n/v)p for v > n + 1.
6) In [8, Theorem 8, inequality (40)], it was announced an inequality from which estimate (4) (with an additional term of order O(n-k) on the right-hand side) can be obtained with d = min{2,p} < q instead of q on the right-hand side of this estimate.
Estimate (3) can be strengthened in the case p > 1; more exactly, the following theorem holds.
Theorem B. Suppose that 1 < p < q < to, f € Lp(T), a = 1/p — 1/q, l,k € N, l < k, and condition (1) holds. Then, the following estimate is valid:
/ n X l/p , <x X 1/q
^ V=1 q ^ V=n+1
Estimate (5) was first obtained by Kolyada [7, Sect. 3, Theorem 2, inequality (3.8)] for the case l = k = 1; its validity for l = k > 1 was noted by Goldman [2, Sect. 4, proof of Lemma 6, inequality (11)]. Estimate (5) for k > l follows from the well-known order equality
V=1 v=1 q
where 1 < a < to and 0 < ft < min{k, l}.
Recall that an order equality pn x ^n means that there exist numbers 0 < C5 < C6 depending only on the parameters given (in this case, on k,l,,0, and a) such that C5^n < ^>n < C6^n.
For given p € [1, to), denote by MP(T) the class of all functions f € LP(T) whose Fourier coefficients satisfy the conditions ao(f) = 0 and an(f) ^ 0 and bn(f) ^ 0 as n t to. It is known (see, for example, [1, Ch. 1, Sect. 30]) that Fourier series of such functions converge everywhere expect maybe a countable set of points x = 0 (mod 2n); i.e., we have
Vn
n=1
f (x) = E(°n(f )cos nx + bn(f )sin nx)
almost everywhere in R.
In the present paper, which is a continuation of the author's research [5, 6], we consider the problem of optimality of inequalities (3), (4), and (5) in terms of order equalities on the whole class MP(T) for 1 < p < q < to.
Theorem 1. Let 1 < p < q < to, a = 1/p — 1/q, and l, k € N. A function f € MP(T) belongs to Lq(T) if and only if the condition ^¿(f; p; a; q) < to holds. Moreover, the following order equalities hold:
1/q
^ V=1 '
py 1/q
(6)
\ 1/q
E■ »€», (7)
7/-<tj -i- 1 '
"7
n/q V n/ P \ l V ^ Py
x v=n+1
^ \ 1/q
71 \ I I ST^ vqa-1. .9/ s.7r\
s x v=n+1 7
/ n N \ 1/q n
+n-k(YJvq[k+a)~luqi(f^) J j, neN, I > k]
(8)
(n \ 1/p / <x \ 1/q
) *( E ' nGN' (9)
v=1 v=n+1
/ n N 1/p
) > "6N, (10)
q p \ ^=1 q/
Remark 2. When evaluating from below in the order equality (7) the second term n°"wj(f; n/n)p cannot be omitted in the general case, because there exists a function g € Mp(T) such that n^w^g;n/n)p = O(wk(g;n/n)q). The function g e MP(T) is defined as follows (see [6, Sect. 3.1]):
g(x) = ^ an cos nx, where an = an(p; l) = n-(i+1-1/p), n € N, and the following order equalities
n
n=1
hold: En-1 (g)p x n l, n € N, and W^g; n/n)p x n 1 (ln(en))1/p, n € N. Since l > a, we have g € Mq(T); moreover,
En-1 (g)q x n -(i - n € N ^ Wi(g; n/n)q x n -(i - , n € N.
Thus, in view of these order equalities, we have
nCTWi(g; n/n)p x n-(i-CT)(ln(en))1/p x Wi(g; n/n)q(ln(en))1/p, n € N;
whence nCTwj(g;n/n)p = O(w£(g;n/n)q), n € N, and a fortiori nCTw^g;n/n)p = O(wk(g;n/n)q), n € N, in the case k > l.
q
However, it can be omitted if the sequence {(i(f; n/n)psatisfies Stechkin's (Si)-condition ({(i(f; n/n)p}^=1 € Si): there exists e € (0,1) such that the sequence {ni-£(i(f; n/n)p}^=1 almost
increases. This condition is equivalent to Bari's (B(a))-condition for every fixed a € [1, to)
({(if; n/n)pCi € B<a)):
(n X 1/a
V=1
(see [6, Sect. 3.2)].
Theorem 2. Suppose that 1 < p < q < to, f € Mp(T), a = 1/p - 1/q, l,k € N, l < k, and condition (1) holds. If {( (f; n/n)p}^c=1 € Si, then the following order equality holds:
1/q
(ro \ 1/q
v=n+1 7
Remark 3. The condition {(i(f;n/n)p}'TO=1 € Si guarantees the validity of the estimate na(i( f; n/n) < C7(k, l,p, q)(k (f; n/n) ,n € N, for all functions f € Mq (T), where 1 <p < q < to, l,k € N (see the proof of Theorem 2, inequality (22)). In the case l > k, this estimate holds for f € Mq(T) without any conditions on the sequence {(i (f; n/n)p}£°=1 (see the proof of Theorem 1, inequality (18)).
Remark 4. In connection with Theorem 2, note also the following fact, which is an obvious {orollary of the order equality (9) in Theorem 1: the order equality (11) is valid if and only if {(k(f;n/n)qe~=1 € B^. In addition, if {a(f;n/n)pe™=l € B(p) ^ {¿i(f;n/n)p^^=i € Si, then, in view of (11) and (9), we have {(k(f; n/n)q}^=1 € B^. On the other hand, in view of (11), the latter condition guarantees only that vqa-1aiq(f; n/v)p) 1/^^=1 € B— but does not
that {(i(f;n/n)pe~=1 € B(p).
Proof of Theorem 1. In the proof of Theorem 1, we will use neither estimates (2)-(5) from Theorems A and B nor the direct (see [6, inequality (0.3)]) and inverse (see [5, inequality (2)]) theorems of approximation theory for periodic functions in different metrics. We will use only certain known results that are characteristic of functions from the class Mp(T) (the corresponding statements are gathered in [6, Sect. 1]). Since the auxiliary inequalities needed for the proofs of Theorems 1 and 2 were established by the author in [5, 6], we will mostly refer to these papers instead of giving original references, which can be found in [5] and [6].
1) The first part of Theorem 1 and the order equality (6) were proved in [6, Sect. 2, the proof of statement (1) of Theorem 1]; more precisely, if f € Mp(T) and Qi(f;p; a; q) < to, then f € Lq(T) and ||f ||q < Cg(l,p,q)fii(f;p; a; q) (the sufficiency); if f € Mp(T) belongs to Lq(T), then f € Mq (T) and Qi(f; p; a; q) < C9(l,p, q)||f ||q (the necessity). Moreover, in the proof of statement (1) of [6, Theorem 1], we actually obtained the following estimates for a function f € Mp(T) under the condition Qi(f; p; a; q) < to (see also [5, Sect. 1]):
Cw(jp,q)E(f; p; a; q) < ||f ||q < Cn (p,q)E(f; p; a; q), (12)
where
/ \ 1/q / / n \ \ 1/q
v= 1 7 v= 1 V'
2) The upper estimate in (7): taking into account the inequality (see [6, Sect. 2, inequality (2.1)])
/ <x x 1/q
n'ui^l) <c12(l,p,q)[ E J > neN, I € N,
n/p ' " ' " \ v V/p ,
x v=n+1 7
(13)
in the estimation of En-1(f )q from above (see [6, Sect. 2, step The upper estimate in the proof of statement (2) of Theorem 1]), we obtain
( ^ x \ 1/q
£ J , neN, I € N.
7/-n_Ll P/
V/p
■ v=n+1
(14)
(15)
Further, using estimate (14) in the inequality (see [6, Sect. 1, Lemma 1, inequality (1.8)])
<Cu(k,p,q){En(f)q + naujk(yf]^)j J, ne N, k € N,
and taking into account (13), we obtain for l < k
T / <x \ 1/q ^
q ^ v=n+1
/x , \ \ 1/q
<Cl5(k,l,p,q)( >
^ v=n+1
where C15(k, l,p,q) = C14(k,p,q){C13(l,p, q) + 2k-iC12(l,p, q)}; whence,
(ro X 1/q
£ , neN.
v=n+1
The lower estimate in (7): applying to the inequality (see [6, Sect. 2, step The lower estimate in the proof of statement (2) of Theorem 1])
x / n \ \ 1/q
IZ^Mij^) <
v=n+1
( / n \ 1/p \ 1/q^l
< C16(l,p,q) n^PTvpi-1EP-1(f)J + ( £ vqCT-1E,q-1(f)p
^ v=1 ' v=n+1 ' '
Pi + \ L^ V Ev-1(f)P
v=n+1
the lower estimate in the order equality (1.7) from [6, Sect. 1, Proposition 5]:
n N 1/p
N / . \ 1/p
^ V=1
(16)
and inequality (10) from [5, Sect. 1]:
/ x \1/q / n \
f £ ^"^(/U (/;-), neN, k € N,
^ v=n+1 '
we obtain for arbitrary l, k € N
(ro \ 1/q
v=n+1 7 (17)
3) The upper estimate in (8): if f € Mp(T) and Qi(f;p; a; q) < to, then f € Lq(T) and, consequently, f € Mq(T). By the upper estimate in (16) and inequality (14), we have
, n \ / n \1/q
"*(/;-) < C2o(k,q)n-kl ^qk-lEqv-iU\) <
^ V =1 7
(n n
V=1 M=V+1 ^ p
n ro / n\ \1/q
+ V m-V J <
V=1 ^=n+1 r 7
/n ^ n ro / ^ \1/q
< c21(fc,z,p,<?)n-fc E(/;-J.}E+ EE -
^=1 ^ pV=1 V =1 ^=n+1 ^ V'
(n ro ^ 1/q
E^^-^fi/;-) E <
1 v p ,1 v p/
C / n \ \ 1/q / ro \1/qN|
<C21(k,l,p,q)\n~k E^(fc+<j)-V(/;^)j + E f> «eN.
The lower estimate in (8): let us first prove that the following estimate holds for l > k:
<C22(k,l,p,q)ujk(f]-) , ne N. (18)
n p n q
To this end, we will need the inequality (see [6, Sect. 2, inequality (2.6)]
< C23(k,p,q)uk(f-, -) , n € N, k € N. (19)
nq
Applying (19) in the upper estimate in (16) and taking into account the following known property of the modulus of smoothness: 5-k(k(f; S2)q < 2k5-k(k(f; ¿1)q for 0 < ¿1 < S2 ^ vk(k(f; n/v)q < 2knk(k(f; n/n)q for 1 < v < n, we obtain
/ n \ i n \1/p
n'^f;-) <C2o(l,p)n'-l( E^_1^-i(/)pJ <
V=1
" . . , n \ \ 1/p
< C2o(l,p)C23(k,p, q)na~l ( E I) )
V=1 v q
(n \1/p
E ^ ^(fc, l,p, q)Uk (/;
which implies the required estimate in (18).
Applying estimate (18) in inequality (17), we obtain the following upper estimate for the first term on the right-hand side of (8):
(ro \ 1/q
E + , ne N.
V=n+1 p q
The following estimate was obtained in [5, Sect. 1, the proof of the lower estimate for the second term on the right-hand side of the order equality (7)]:
/ n \ 1/q , \
n-M J2vq{k+a)~lEl-i(f)p) < c26(k,p,q)uk(f; ^) n € N.
v=1
Hence, in view of the known order equality (see, for example, [4, Sect. 2, Remark 7, order equality (15)])
n n
Y,vafi-lK-iU)v ~ E^'Vi/;^) > (20)
—: —: v V/p
v=1 v=1
where n € N U {+to}, 1 < a < to, and 0 < ^ < l, we obtain (l > k ^ l > k + a, a € (0,1))
/ n \ \ 1/q / n \ 1/q
n-fc(E^(fc+<J)"V(/;-), ) XB-1 E^tii/lp <
v= 1 V p v=1 '
< , ne N.
nq
The latter inequality implies the upper estimate for the second term on the right-hand side of (8). 4) The upper estimate in (9): the upper estimate in (12) implies the inequality
En-1(f)q < C27(p,q){nCTEn-1(f)p +( E vqCT-1 Eq_1(f)pY9\, n € N. (21)
^ v=n+1 ' '
Indeed, applying (12) to the function gn(f; x) = f(x) — Sn(f; x), where Sn(f; x) is the partial sum of order n € N of the Fourier series of the function f € Mp(T), and taking into account the estimate Ev-1(gn)p < ||gn||p < (1 + C28(p))En(f)p, v € N, where C28(p) is the constant in the well-known M. Riesz inequality ||Sn(f; -)||p < C28(p)||f ||p (1 < p < to, f € LP(T)), and the equality Ev
- 1(gn )P - Ev- 1(f)p, v > n + 1, n € Z+, we obtain for n € N
En(f )q < ||gn(f;') ||q < Cu(p,q)E(gn;p; a; q) < - n+1 \ 1/q / x \ 1/q^
_ vq--1E,q-1(gn)J + V vqa-1Eq-1(gn)p) :><
( /n+1 \ 1/q / x \ 1/q ^
< C11(p, q) E vqCT-1Evq-1(gn)p + E vqCT-1Eq-1(gn)p
^v=1 ' ^v=n+2 ' '
( /n+1 \ 1/q / x \1/qN|
Cn(p,q) (1 + C28(p))En(f)p E vqa-M + E vqCT-1Evq-1(f)p h
^ ^v=1 ' v=n+2 ' '
Cn(p,q){(1 + C28(p))C29(q,a)(n + 1)CTEn(f)p + ( E v^E^f^) ^X
^ V 7/--ni') / J
< Cn(o,q^(1 + C28(o))C2Q(q,a)(n + En(f)p + ( E v^^>p
v=n+2
and for n = 0
Eo(f )q < ||f ||q < C11(p,q)E(f; p; a; q) < Cn(p,q){ Eo(f )p + ( E V"^-^) ^ }.
Inequality (21) was used in [6, Sect. 2, step The upper estimate in the proof of statement (3) of Theorem 1] for obtaining the estimate
/ n \ 1/p / x \ 1/q
n°-<( J2^{l~a)~lEi-i(f)q) <Cao(l,P,q){ E uqa~luJi{f]l)j ' "
^v=1 ' v=n+1 P'
whence, by the order equality (20), we obtain the following upper estimate in (9):
/ n / \ \ 1/p / ro \ s 1/q
V=1 q' v=n+1
The lower estimate in (9): by inequality (17), we have for all l,k € N
(ro \ 1/q
v=n+1 7
The upper estimate for the first term (k(f; n/n)q on the right-hand side is obvious, because, in view of the fact that (k(f; n/n)q I (n t), we have
(n \ \ 1/p
E^'-'-M£
V=1 q/
(n \ 1/p
7/-1 '
nJAV=1 / v n/q
The following upper estimate for the second term n°"(i(f; n/n)p was established above (see the proof of the lower estimate in (8) at step 3):
/ n N 1/p
n p V=1 v q
Combining the obtained inequalities, we come to the required lower estimate in the order equality (9):
n\ \ 1/q / n , \ 1/p
p
/ ro \ 1/q / n \ 1/p
^V =n+1 p ^ V=1
5) The order equality (10) follows from (7) and (9). The proof of Theorem 1 is complete. □
Proof of Theorem 2. The upper estimate in (11) was obtained at step 2 of the proof of Theorem 1 (see inequality (15)):
/ ro x 1/q
"k(f;D <C15(k,l,p,q)( E J > ri&N.
q ^ V=n+1
To obtain the lower estimate in (11), we preliminarily prove that, if {(i(f; n/n)p}'TO=1 € Si, then the following estimate holds for all l,k € N:
nW/;-) <CM(k,l,p,q)uk(f;-) , n € N. (22)
n p n q
Indeed, since
(see [6, Sect. 3.2)]), in view of [6, Sect. 2, inequality (2.8)] and [6, Introduction, inequality (0.4)], we obtain
<Cs5(l,p,q)En(f)q<Cs5(l,p,q)Cs6(k)ujk(f]-) , n € N. n p n q
The required lower estimate in (11) follows from (17) and (22):
(x \ 1/q
£ <C19(k,l,p,q)(l + C-M(k,l,p,q))LOk(Kf]^) , ne N.
v=n+1 V' q
The proof of Theorem 2 is complete. □
Remark 5. By inequality (19), the upper estimate in the order equality (16) implies the inequality
(n \ 1/p
Y,uP{l~a)~luk{f^)q) ' hk,nen. (23)
V=1 '
Inequality (23) (the case l < k) and inequality (18) (the case l > k) for functions f € Mq(T) c MP(T) are inverse (in the sense of the upper estimate for ; 5)p in terms of wk(f; 5)q) to inequalities (3) and (4), respectively, which hold for all functions f € Lq(T) under the condition of convergence of series in (1). From inequalities (23) and (18), we can conclude that, in the passage from the class Mq(T) to the class MP(T), where p < q, the smoothness of a function f € Mq(T) increases by a value not larger than a in the case l < k (see [6, Sect. 3.3)], where the author considered the case k = l and ; 5)q x 5a, 0 < a < l, 5 € (0,n]) and increases by a value not smaller than a in the case l > k.
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