The Olovyanishnikov inequality for multivariate functions

APPROXIMATION THEORY: A volume dedicated to Borislav Bojanov (D. K. Dimitrov, G. Nikolov, and R. Uluchev, Eds.) additional information (to be provided by the publisher) The Olovyanishnikov Inequality for Multivariate Functions Vladislav Babenko ∗, Yuliya Babenko 1. Introduction. Let G be the real line R, space Rm , the negative half-line R− or the octant m : x1 ≤ 0, · · · , xm ≤ 0}. Let Lp = Lp (G), 1 ≤ Rm − := {(x1 , · · · , xm ) ∈ R p ≤ ∞, be the space of functions x : G → R, integrable in the power p on G (essentially bounded when p = ∞), with usual norm. In the case when G = R or G = R− by Lrp = Lrp (G), r ∈ N, we will denote the space of functions x : G → R, that have locally absolutely continuous derivative x(r−1) such that x(r) ∈ Lp (G). For 1 ≤ p, s ≤ ∞ set Lrp,s = Lrp,s (G) = Lrs (G) ∩ Lp (G). Great amount of work has been done on finding inequalities of the form ° ° ° °β ° (k) ° ° α° (1.1) °x ° ≤ K · kxkp °x(r) ° . q s for functions x ∈ Lrp,s (G). Inequalities with the best possible constants (sharp inequalities) are especially interesting, and research of a lot of mathematicians was devoted to obtain such inequalities. The first sharp result was obtained by Landau [9] in 1913 for the case x ∈ L2∞,∞ (R− ), k = 1. One of the first complete results in this area was obtained by Kolmogorov [7] in 1939: for any k ∈ N, and k < r and any function x ∈ Lr∞,∞ (R) ° ° ° (k) ° °x ° ∞ ≤ kϕr−k k∞ 1−k/r kϕr k∞ 1−k/r kxk∞ ° ° ° (r) °k/r °x ° , ∞ (1.2) where ϕr is the rth periodic integral with zero mean value on the period of the function ϕ0 (t) = sgn sin x. Such functions are called Euler splines. The Kolmogorov inequality (1.2) becomes an equality for any function of the type ϕr (λt) for a positive real λ. Since this result inequalities of type (1.1) are often called Kolmogorov type inequalities. ∗ Supported in part by FFIU under project No. 01.07/00241 2 Multivariate Olovyanishnikov Inequality So far there are only few cases where for some p, q, s, exact constants in inequalities of the type (1.1) are known for all pairs k, r ∈ N, k < r. For review of known general and particular results see, for example, [1]. In the case when the domain is the half-line R− and q = p = s = ∞ inequality (1.1) with a sharp constant was obtained by Landau [9] (case r = 2), Matorin [10] in 1955 (case r = 3) and Shoenberg and Cavaretta [13, 14] in 1970 (case r ≥ 4). For the full picture of the problem development let us introduce the result of Shoenberg and Cavaretta in detail. Let σn,r : [0, 1] → R be the perfect spline of degree r with n knots, having the minimal L∞ -norm on the segment [0, 1] among all perfect splines of the following form σ(t) = n r−1 X X 1 r t +2 aν tν , 0 ≤ t ≤ 1. (−1)i (t + ξi )r+ + r! ν=0 i=1 For any n ∈ N we shall choose λn such that λ−r n |σn,r (0)| = |σ0,r (0)| and set sn,r (t) = λ−r n σn,r (λn t). Shoenberg and Cavaretta showed that there exists a limit lim sn,r (t) =: sr (t), n→∞ t ∈ R− . and sr (t) ∈ Lr∞,∞ (R− ). Moreover, sr (t) is the perfect spline of degree r with an infinite number of knots. Shoenberg and Cavaretta called it ”one-sided Euler spline”. They proved that the best possible constant in Kolmogorov type inequality in the case when G = R− and q = p = s = ∞ is equal to k (0)| |σn,r |skn,r (0)| |sr−k (0)| = lim = . 1−k/r n→∞ |σn,r (0)|1−k/r n→∞ |sn,r (0)| |sr (0)|1−k/r lim Hence, their inequality of Kolmogorov type looks like the following ° ° ° (k) ° °x ° ∞ ≤ °k/r ° |sr−k (0)| 1−k/r ° (r) ° x kxk ° . ° ∞ ∞ |sr (0)|1−k/r From this one can see that the constant in the Kolmogorov type inequality for the class Lr∞,∞ (R− ) in the case r ≥ 4 is not explicit. Before the result of Shoenberg and Cavaretta, Olovyanishnikov [11] in 1951 considered the more narrow class of (r−2)- monotone functions from Lr∞,∞ (R− ) (functions that have non-negative derivatives up to order r − 1). For this class he obtained a Kolmogorov type inequality for the case p = q = s = ∞. 3 Vladislav Babenko, Yuliya Babenko To introduce his result we need the following definition. For any positive parameters a and l, set ( 0, −∞ < t ≤ −l; (1.3) φr (a, l; t) = a(l+t)r , −l ≤ t ≤ 0. r! Set also φr (t) := φr (1, 1; t). Theorem 1. (Olovyanishnikov, [11]) For any k, r ∈ N with k < r, and for any function x ∈ Lr∞,∞ (R− ) that is non-negative with all its derivatives up to and including order r − 1, the following inequality holds: ° ° ° °k/r ° (k) ° 1−k/r ° (r) ° (1.4) °x ° ≤ Crk kxk∞ °x ° , ∞ where Crk = ∞ ° ° ° (k) ° °φr ° ∞ 1−k/r kφr k∞ = r!1−k/r . (r − k)! This inequality becomes an equality for any function of type (1.3). In [3] we generalized this result to the case of arbitrary q and p. m Let us discuss now the multivariate case. Let G = Rm − or G = R . For m r r = (r1 , ..., rm ) ∈ Z+ denote by L∞ (G) the space of functions such that x(r) = ∂ r1 +...+rm x ∂ r1 t1 ...∂ rm tm (derivative is understood in Sobolev sense) is in L∞ (G). General results on sharp inequalities for intermediate derivatives in multivariate case are known only for the cases connected with L2 -norm (Dinh Dung and Tikhomirov [5], Subbotin [15], Buslaev and Tikhomirov [4] ). The only known sharp results for the uniform norm are: (3,0) (0,3) 1. V. Konovalov [8]: if x ∈ L∞ (R2 ) ∩ L∞ (R2 ) ∩ L∞ (R2 ), kx(1,1) k∞ ≤ (3kxk∞ kx(3,0) k∞ kx(0,3) k∞ )1/3 . (3,0) (0,2) 2. O. Timoshin [12]: if x ∈ L∞ (R2 ) ∩ L∞ (R2 ) ∩ L∞ (R2 ), (3,0) 1/3 kx(1,1) k∞ ≤ 32/3 21/2 kxk1/6 k∞ kx(0,2) k1/2 ∞ kx ∞ . (2,1) (1,2) 3. V. Babenko [2]: if x ∈ L∞ (R2 ) ∩ L∞ (R2 ) ∩ L∞ (R2 ), kx(1,1) k∞ ≤ 3/22/3 (kxk∞ kx(2,1) k∞ kx(1,2) k∞ )1/3 . These results all are for the case of two variables. We do not know any 2 sharp result of this type when the domain is the octant Rm − (or even R− ). The main purpose of this paper is to obtain multivariate Olovyanishnikov inequality. In the second section of this paper we shall present a different from original proof of inequality (1.4) with somewhat weaker assumptions. Then in Section 3 we shall show how this argument can be generalized to the case of many variables. 4 Multivariate Olovyanishnikov Inequality 2. Alternative proof of the Olovyanishnikov inequality For t ∈ R set t+ = 0 if t ≤ 0 and t+ = t if t > 0. We will need the following lemma. Lemma 1. Let k ∈ N, x ∈ Lk∞,∞ (R− ) and function x is non-negative on R− together with all its derivatives up to order 1 + (k − 3)+ . If parameters a > 0, l > 0 of φk (a, l; t) in (1.3) are chosen in such a way that φk (a, l; 0) ≤ x(0) = kxk∞ and i.e. set a ≥ kx(k) k∞ (k) kφk (a, l; ·)k∞ ≥ kx(k) k∞ , ´1/k ³ . Then for any t ≤ 0 and l ≤ k! kx|x(0)| (k) k ∞ x(t) ≥ φk (a, l; t). (2.1) (2.2) (2.3) Proof. For k = 1 the lemma is obvious. Assume k ≥ 2. Since x(t) ≥ 0 for t ∈ R− , (2.3) is true for t ≤ −l. Let us prove now that it is also true for t ∈ [−l, 0]. Assume to the contrary that there exists a point ξ ∈ [−l, 0] such that x(ξ) < φk (a, l; ξ). This implies that for the difference δ(t) := x(t) − φk (a, l; t) the following is true δ(−l) ≥ 0, δ(ξ) < 0, δ(0) ≥ 0. From here it follows that for the derivative δ ′ (t) there exist points ξ1′ ∈ (−l, ξ) and ξ1′′ ∈ (ξ, 0) such that δ ′ (ξ1′ ) < 0, δ ′ (ξ1′′ ) > 0 and also δ ′ (−l) ≥ 0. Hence, there exist points ξ2′ ∈ (−l, ξ1′ ) and ξ2′′ ∈ (ξ1′ , ξ1′′ ) such that δ ′′ (ξ2′ ) < 0, δ ′′ (ξ2′′ ) > 0 and also δ ′′ (−l) ≥ 0. Repeating the same argument k − 2 times we arrive to the conclusion that ′′ ′ ′′ ′ ′ ) such that , ξk−3 ∈ (−l, ξk−3 ) and ξk−2 ∈ (ξk−3 there exist points ξk−2 ′ δ (k−2) (ξk−2 ) < 0, ′′ δ (k−2) (ξk−2 )>0 ′ ′ ) and also δ (k−2) (−l) ≥ 0. This means that there exist points ξk−1 ∈ ([−l, ξk−2 ′′ ′ ′′ and ξk−1 ∈ (ξk−2 , ξk−2 ) such that ′ δ (k−1) (ξk−1 ) < 0, ′′ δ (k−1) (ξk−1 ) > 0. Therefore, function δ (k−1) (t) on (−l, 0) changes sign from negative to positive, which contradicts the condition (2.2). ¤ 5 Vladislav Babenko, Yuliya Babenko Theorem 2. Let k, r ∈ N, k < r. Assume that function x ∈ Lr∞,∞ (R− ) satisfies the following conditions : 1) lim x(t) = 0, t→−∞ 2) function x(k) is non-negative with all its derivatives up to and including order 1 + (r − k − 3)+ . Then ° ° °k/r ° ° (k) ° 1−k/r ° (r) ° (2.3) °x ° ≤ Crk kxk∞ °x ° , ∞ where Crk = ∞ ° ° ° (k) ° °φr ° ∞ 1−k/r kφr k∞ = r!1−k/r . (r − k)! This inequality becomes an equality for any function of type (1.3). Proof. Observe that (k) φ(k) k∞ = x(k) (0). r (a, l; t) = φr−k (a, l; t) and kx Let parameters a, l of φr (a, l; t) be chosen in such a way that (k) φ(k) (0) r (a, l; 0) = x (2.5) and (r) kφ(r) k∞ , r (a, l; ·)k∞ = kx ³ ´ 1/r−k (k) k∞ i.e. set a = kx(r) k∞ and l = (r − k)! kx . kx(r) k∞ Applying previous lemma we obtain that for any t ≤ 0 x(k) (t) ≥ φ(k) r (a, l; ·). (2.6) (2.7) For t ≤ 0 set ∆1h x(t) := x(t) − x(t − h) and ∆kh x(t) := ∆1h (∆k−1 h x(t)). Define also a function ( 1, s ∈ [−h, 0]; Θ1,h (s) = 0, s ∈ R \ [−h, 0]. Z s+h Θk−1 (τ )dτ . Note that Θk (s) is positive for any For k ≥ 2 set Θk (s) = s k ∈ N and s ∈ R− . Moreover, supp Θk = [−kh, 0]. It is easy to check that for any t ≤ 0 Z x(k) (s)Θk (s − t)ds = ∆kh x(t). R− Since Θk (s) is positive, from (2.7) and (2.8) it follows that for any h > 0 Z x(k) (s)Θk (s − t)ds ∆kh x(t) = R− (2.8) 6 Multivariate Olovyanishnikov Inequality ≥ Z k φ(k) r (a, l; s)Θk (s − t)ds = ∆h φr (a, l; t). R− As we let h → ∞ because of assumptions on x we obtain ∀t ∈ R− x(t) ≥ φr (a, l; t), (2.9) and, hence, kxk∞ ≥ kφr (a, l; ·)k∞ . Therefore, (k) kx(k) k∞ = kφr−k (a, l; ·)k∞ = kφr (a, l; ·)k∞ 1− k kφr (a, l; ·)k∞ r 1− k r kφr (a, l; ·)k∞ r−k l a (r−k)! (r!)1−k/r 1− k ≤ ¡ r ¢1−k/r kxk∞ r = kxk1−k/r ak/r ∞ l (r − k)! a r! (r!)1−k/r kφr−k k∞ 1−k/r 1−k/r kxk∞ kx(r) kk/r (2.10) kxk∞ kx(r) kk/r ∞ . ∞ = 1− k (r − k)! r kφr k∞ Inequality (2.3) is proved now. It is clear that (2.3) becomes an equality for functions x(t) = φr (a, l; t) The theorem is proved. ¤ Remark. We stated Theorem 2 this way to see the direct generalization of it in multivariate case by Theorem 3. Now let us compare assumptions in Theorem 2 with assumptions in the original theorem of Olovyanishnikov. If about the function x in Theorem 2 we assume that it is nonnegative with all its derivatives up to order 1 + (r − 3)+ including, then the assumption (1) in Theorem 2 can be removed. Indeed, in this case function x is non-decreasing and function x(t) − lim x(τ ) satisfies all assumptions of Theorem 2. Hence, = τ →−∞ we obtain kx(k) k∞ ≤ kφr−k k∞ 1− k kφr k∞ r ≤ 1−k/r kx(t) − lim x(τ )k∞ kx(r) kk/r ∞ kφr−k k∞ 1− k kφr k∞ r τ →−∞ kxk1−k/r kx(r) kk/r ∞ ∞ . Therefore, we proved Olovyanishnikov inequality under assumptions that for r > 2 are weaker than conditions of Theorem 1. On the other hand, observe that while proving inequality (2.10) for a fixed k we did not need all derivatives up to order 1+(r−k−3)+ to be non-negative, we used the fact that only derivatives x(k) , ..., x(r−2) (up to x(r−1) when k = r − 1 ) are non-negative and the fact that x(t) → 0 as t → −∞. From this and (2.9) it follows that x(t) is non-negative. Moreover, the condition lim x(t) = 0 and the fact that x (r) is bounded imply that for any k < r, ′ (k−1) Hence, in a similar way, we can show that x (t), ..., x t→−∞ lim x(k) (t) t→−∞ = 0. (t) are non-negative. 7 Vladislav Babenko, Yuliya Babenko 3. Multivariate case For t = (t1 , ..., tm ) ∈ Rm denote by t̂j vector from Rm−1 obtained from t by removing the jth coordinate. In this case we shall write t as t = (t̂j , tj ). Let m m m {ej }m j=0 be a standard basis in R . We will say that the function x : R− → R rej belongs to the space L∞ (Rm − ), r ∈ N, if the following conditions are satisfied: 1) For every fixed t̂j ∈ Rm−1 there exist continuous on R− derivatives − k x(kej ) (t̂j , tj ) := ∂∂tkx (t̂j , tj ), k = 1, ..., r − 1; j 2) for every fixed t̂j ∈ Rm−1 the derivative x((r−1)ej ) (t̂j , tj ) is locally abso− lutely continuous on R− ; 3) kx(rej ) k := sup kx(rej ) (t̂j , ·)kL∞ (R− ) < ∞. m−1 t̂j ∈R− Observe that the last condition means that there exists a constant M such that for any fixed t̂j ∈ Rm−1 the function x((r−1)ej ) (t̂j , tj ) satisfies the Lipschitz − condition with the constant M with respect to the variable tj , i.e. ∀ t′j , t′′j ∈ R− |x((r−1)ej ) (t̂j , t′j ) − x((r−1)ej ) t̂j , t′′j )| ≤ M |t′j − t′′j | Recall that the function φr (a, l; t), t ∈ R, was defined by (1.3) and that φr (t) := φr (1, 1; t). For t ∈ Rm − and a collection b = (b0 , b1 , ..., bm ) of nonnegative numbers define Φr (b, t) := b0 φr (b1 t1 + ... + bm tm ). (3.1) Note that when m = 1, Φr (b, t) = φr (a, l; t) with a = b0 br1 and l = 1/b1 . We will need the following generalization of Lemma 1. Lemma 2. Let l ∈ N and let function x : Rm − → R satisfies the following conditions:   m \ m  j  ) ∩ 1) x ∈ L∞ (Rm Lle − ∞ (R− ) ; j=1 (kej ) 2) x is nonnegative on Rm , j = 1, ..., m up − along with its derivatives x to and including order 1 + (l − 3)+ . Assume that the parameters b = (b0 , b1 , ..., bm ) of the function Φl (b; t) are chosen so that kΦl (b; ·)k∞ ≤ x(0) = kxk∞ (3.2) and (lej ) kΦl (b; ·)k ≥ kx(lej ) k, j = 1, ..., m. (3.3) Then for any t ∈ Rm − x(t) ≥ Φl (b; t). (3.4) Proof of Lemma 2. Ee shall proceed by induction on the dimension m of the domain. Lemma 1 provides the basis of induction. Assume that the statement of Lemma 2 holds true for m = d − 1 ≥ 1 Let us prove now that it is true for m = d. 8 Multivariate Olovyanishnikov Inequality First of all we need to make sure that on any coordinate hyperplane (to be more precise, on its intersection with Rd− ) the conditions of Lemma 2 are satisfied in the case when m = d − 1. From the definition of Φl (b, ·) it follows that its norm is achieved at t = 0. Hence, condition (3.2) is satisfied. Further, ° ° ° (lej ) ° (le ) °Φl (b; ·)° = Φl j (b; 0), and since the norm of the restriction of x(lej ) taken on a hyperplane does not exceed the norm of x(lej ) on the whole Rd− , then the second condition is also satisfied. By the assumption of induction we obtain that on every such a hyperplane (we choose the hyperplane t1 = 0) the inequality Φl (b; (t̂1 , 0)) ≤ x((t̂1 , 0)) is true. Consider the line t̂1 = t̂10 parallel to the first coordinate axis. If it does not intersect the support of the function Φl (b; t), then at every point t of this line such that t1 < 0, (3.4) is true thanks to non-negativeness of the function x(t). If it does have a nonempty intersection with the support of the function Φl (b; ·), then for the functions x((t̂10 , t1 )) and Φl (b; (t̂10 , t1 )) of variable t1 conditions of Lemma 1 are satisfied. The inequality kx((t̂10 , ·))k∞ ≥ kΦl (b; (t̂10 , ·))k∞ is true due to the fact that (3.4) is true on the hyperplane t1 = 0 and due to monotonicity of x with respect to the variable t1 . The inequality for the derivatives follows from (3.3) and from the fact that in this case ° ° ° ° ° ° (le ) ° ° (lej ) = °Φl j (b; ·)° . °Φl (b; (t̂10 , ·))° L∞ (R− ) By Lemma 1 we obtain that at every point of the such line t̂1 = t̂10 inequality Φl (b; t) ≤ x(t) is true. Hence, lemma is proved. ¤ The main result of this paper is the following theorem. Theorem 3. Let k ∈ Nm , r ∈ N, |k| := m X i=1 ki ≤ r − 1 and let function φr be defined by (1.3). Assume that function x ∈ L∞ (Rm − ) satisfies the following conditions: 1) For any i = 1, ..., m and any fixed t̂i ∈ Rm − 1− x((t̂i, ti )) → 0 when ti → −∞. m \ i (Rm 2) x(k) ∈ L(r−|k|)e ∞ −) i=1 9 Vladislav Babenko, Yuliya Babenko 3) For l = 0, 1, ..., 1 + (r − |k| − 3)+ and i = 1, ..., m the derivatives x(k+lei ) are nonnegative on Rm −. Then ° ° m ° ° ki ° ° °φr−|k| ° r−|k| Y ° (k+(r−|k|)ei ) ° r ° (k) ° ∞ r kxk∞ (3.5) ° . °x °x ° ≤ r−|k| ∞ kφr k∞r i=1 Inequality (3.5) is sharp. It becomes an equality for functions x(t) = Φr (b; t). Proof. Let Φ(t) = Φr (b; t) be as in (3.1). Choose the parameters b0 , b1 , · · · , bm to satisfy the system of equations (° ° ° ° °Φ(k+(r−|k|)ei ) ° = °x(k+(r−|k|)ei ) ° , i = 1, · · · , m ∞ ° (k) ° ° ° (3.6) °Φ ° = °x(k) ° . ∞ This will provide that functions x(k) and Φ(k) satisfy conditions of Lemma 2. Let X us show that such choice of parameters is always possible. Let lj := r− ki . System (3.6) is equivalent to the following system: i6=j ° °  l1 k2 b0 b1 b2 · · · bkmm kφ0 k∞ = °x(k+(r−|k|)e1 ) °    · · · ° ° km−1 lm b0 bk11 · · · bm−1 bm kφ0 k∞ = °x(k+(r−|k|)em ) °   ° ° ° °  k1 b0 b1 · · · bkmm °φr−|k| °∞ = °x(k) ° . (3.7) To solve this system of m + 1 equations, divide each of the first m equations by the last one, and then find b1 , · · · , bm . Then plug these values in the last equation to obtain b0 , taking into account that by definition of φr , kφ0 k∞ = 1. Thus, 1 ð ° ! r−|k| °° °x(k+(r−|k|)ei ) ° °φr−|k| ° ∞ ° ° bi = , i = 1, . . . , m, (3.8) °x(k) ° and r ° (k) ° r−|k| °x ° b0 = m ° ° ki ° r Y ° ° (k+(r−|k|)ei )) ° r−|k| °φr−|k| ° r−|k| ° °x ∞ (3.9) i=1 Therefore, the needed for Lemma 2 choice of parameters is always possible. Applying now Lemma 2 gives us that for all elements t ∈ Rm −, x(k) (t) ≥ Φ(k) (t). (3.10) kxk∞ ≥ kΦk∞ . (3.11) We now show that 10 Multivariate Olovyanishnikov Inequality k j 1 j j For t ∈ Rm − and h > 0 set ∆j;h x(t) := x(t̂ , tj ) − x(t̂ , tj − h) and ∆j;h x(t) := k k −1 j j of order x(t)). By this we have defined the difference operator ∆j;h ∆1j;h (∆j;h m kj with respect to the variable tj . For k ∈ N we set now 1 m . ◦ ... ◦ ∆k1;h ∆kh x := ∆km;h Taking into account (2.8) it is not hard to check that ∀ t ∈ Rm − Z x(k) (s)Θ̄k (s − t)ds, ∆kh x(t) = Rm − where Θ̄k (s) = m Y Θkj (sj ). j=1 Qm Θ̄k (s) is a nonnegative function and suppΘ̄k = j=1 [−kj h, 0]. Because of (3.10) we obtain Z k x(k) (s)Θ̄k (s − t)ds ∆h x(t) = Rm − ≥ Z Rm − Φ(k) (s)Θ̄k (s − t)ds = ∆kh Φ(t). (3.12) Note that by assumption 1) of the theorem ∆kh x(t) = km X ··· αm =0 k1 X α1 =0 (−1)α1 +...+αm µ ¶ µ ¶ k1 km x(t1 − α1 h, · · · , tm − αm h) ··· α1 αm → x(t) when h → ∞. Therefore, from (3.12) we obtain that for any t ∈ Rm − x(t) ≥ Φ(t), and, hence, kxk∞ ≥ kΦk∞ . Note that kΦk∞ = b0 kφr k∞ . Taking this into account, inequality (3.11) becomes b0 kφr k∞ ≤ kxk∞ . (3.11) Vladislav Babenko, Yuliya Babenko 11 Substituting the value b0 from (3.9), we obtain ° ° r kφr k∞ °x(k) ° r−|k| m ° ° ki ≤ kxk∞ ° ° r Y ° (k+(r−|k|)ei )) ° r−|k| °φr−|k| ° r−|k| ° °x ∞ i=1 This inequality is equivalent to (3.5). Inequality (3.5) is sharp. It becomes an equality for functions of type (3.1). ¤ References [1] Arestov, V. V., On the exact inequalities between the norms of functions and their derivatives, Acta Sci. Math. 33, 3-4, 1972, pp. 243–267. [2] Babenko, V., On Sharp Inequalities of Kolmogorov Type for Functions of Two Variables, Dopovidi Nac. Akad. Nauk Ukrainy, 2000, N. 5, pp. 7–11 (in Russian). [3] Babenko V., Babenko Yu., About Kolmogorov Type Inequalities for Functions Defined on a Half Line, Constructive Theory of Functions, Varna 2002 (B. Bojanov, Ed.), DARBA, Sofia, 2003, pp. 205–208. [4] Buslaev, A. P., Tikhomirov, V. M., On inequalities for derivatives in multivariate case, Mat. Zametki, 25, 1, 1979, pp. 59–74 (in Russian). [5] Dinh Dung, Tikhomirov, V.M., On inequalities for derivatives in L2 -norm., Vestnik MGU, Ser.mat.meh., 1979, N.2, pp. 7–11. [6] Kolmogorov, A.N., On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Uchenye zapiski MGU. Math.,1939, 30, N.3, pp. 3–13. [7] Kolmogoroff, A., Une generalisation de J.Hadamard entre les bornes superier des derivees seccesives d’une fonction, C. R. Acad. Sci. 1938. V.207, pp. 764–765. [8] Konovalov, V. N., Exact inequalities for norms of functions, third partial and second mixed derivatives, Mat. Zametki, 23, 1, 1978, pp. 67–78 (in Russian). [9] Landau, E., Einige Ungleichungen fur zweimal differenzierbare Funktion, Proc. London Math. Soc. 1913. V.13., pp. 43–49. [10] Matorin, A. P., On inequalities between greatest of absolute values of function and its derivatives on the half-line, Ukrainian Mathematical Journal. 1955. N7, pp. 262–266 (in Russian). [11] Olovyanishnikov, V. M., To the question on inequalities between upper bounds of consecutive derivatives on a half-line, Uspehi mat. nauk., 1951, V. 6, N2(42), pp. 167–170 (in Russian). [12] Timoshin, O. A., Exact inequality between norms of derivatives of second and third order, Dokl. Ross. Acad. Nauk 344, 1, 1995, pp. 20–22 (in Russian). [13] Shoenberg, I.J., Cavaretta, A., Solution of Landau’s problem, concerning higher derivatives on halfline, M.R.C. Technical Summary Report. 1970. 12 Multivariate Olovyanishnikov Inequality [14] Shoenberg, I. J., Cavaretta, A., Solution of Landau’s problem, concerning higher derivatives on halfline, Proc. of Conference on Approximation theory. Varna. 1970. pp.297–308. Sofia. 1972 [15] Subbotin, Yu. N., Extremal functional interpolation and approximation by splines, Doctoral thesis, Sverdlovsk, 1973, (in Russian). Vladislav Babenko Department of Mathematics and Mechanics Dnepropetrovsk National University ul. Nauchnaya, 13 Dnepropetrovsk 49050 Ukraine E-mail: v babenko@dp.ukrtel.net Yuliya Babenko Center for Constructive Approximation Department of Mathematics Vanderbilt University 1326 Stevenson Center Nashville, TN 37240 USA E-mail: yuliya.v.babenko@vanderbilt.edu