EXTREMAL PROPERTIES OF THE DERIVATIVES OF THE NEWMAN POLYNOMIALS

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1 EXTREMAL PROPERTIES OF THE DERIVATIVES OF THE NEWMAN POLYNOMIALS Tamás Erdélyi Abstract. Let Λ n := {λ,λ,...,λ n } be a set of n distinct positive numbers. The span of {e λ t,e λ t,...,e λ nt } over R will be denoted by Our main result of this note is the following. E(Λ n ):=span{e λ t,e λ t,...,e λ nt }. Theorem. Suppose 0 < q p. Let µ be a non-negative integer. Then there are constants c (p, q, µ) > 0 and c (p, q, µ) > 0 depending only on p, q, andµsuch that c (p, q, µ) X n A µ+ q p Q (µ) Lp [0, ) sup Q E(Λ n ) Q Lq [0, ) c (p, q, µ) X n A µ+ q p where the lower bound holds for all 0 <q p and for all µ 0, while the upper bound holds when µ =0and 0 <q p and when µ, p, and0<q p., of. Introduction and Notation Let Λ n := {λ 0,λ,...,λ n } be a set of n + distinct non-negative numbers. The span over R will be denoted by {x λ 0,x λ,...,x λ n } M(Λ n ):=span{x λ 0,x λ,...,x λ n }. Elements of M(Λ n ) are called Müntz polynomials of n + terms. The span of {e λ 0t,e λ t,...,e λ nt } 99 Mathematics Subject Classification. Primary: 4A7, Secondary: 30B0, 6D5. Key words and phrases. Müntz polynomials, exponential sums, Markov-type inequality, Nikolskii-type inequality, Newman s inequality. Research is supported, in part, by NSF under Grant No. DMS Typeset by AMS-TEX

2 over R will be denoted by E(Λ n ):=span{e λ 0t,e λ t,...,e λ nt }. Elements of E(Λ n ) are called exponential sums of n + terms. For a function f defined on a set A let f A := sup{ f(x) : x A}, and let ( f Lp A := A f(x) p dx) /p, p > 0, whenever the Lebesgue integral exists. Newman s beautiful inequality (see [] and [4]) is an essentially sharp Markov-type inequality for M(Λ n )on[0,] in the case when each is non-negative. Theorem. (Newman s Inequality). Let Λ n := {λ 0,λ,...,λ n } be a set of n + distinct non-negative numbers. Then 3 xq (x) [0,] sup 9 0 Q M(Λ n ) Q [0,], or equivalently 3 Q [0, ] sup 9 0 Q E(Λ n ) Q [0, ]. An L p version of this is established in [], [], and [3]. Theorem.. Let p.letletλ n := {λ 0,λ,...,λ n } be a set of n + distinct real numbers greater than /p. Then xq (x) Lp [0,] /p +9 ( +/p) Q Lp [0,] for every Q M(Λ n ). This follows from the fact that if Λ n := {λ 0,λ,...,λ n } is a set of n + distinct non-negative numbers, then Q Lp [0, ) 9 γ j Q Lp [0, ) for every Q E(Λ n ). A simple consequence of Theorem. is the following.

3 Theorem.3 (Nikolskii-Type Inequality). Suppose 0 <q p. Let Λ n := {λ 0,λ,...,λ n } be a set of n + distinct real numbers greater than /q. Then /q /p x /q /p Q(x) Lp [0,] c(p, q) ( +/q) Q Lq [0,] for every Q M(Λ n ). Equivalently, if Λ n := {λ 0,λ,...,λ n } is a set of n + distinct non-negative numbers, then /q /p Q Lp [0, ) c(p, q) for every Q E(Λ n ). In both inequalities is a suitable choice. c(p, q) :=(8 q ) /q /p Q Lq [0, ) The purpose of this note is to show that both Theorems. and.3 are essentially sharp.. New Results The upper bound in our main theorem below follows as a combination of Theorems. ans.3. The novelty of this note is the establishment of the lower bound. Theorem.. Let Λ n := {λ,λ,...,λ n } be a set of n distinct positive numbers. Suppose 0 <q p. Let µ be a non-negative integer. Then there are constants c (p, q, µ) > 0 and c (p, q, µ) > 0 depending only on p, q, andµsuch that µ+ q p Q c (p, q, µ) (µ) µ+ Lp q p [0, ) sup c (p, q, µ), Q E(Λ n ) Q Lq [0, ) where the lower bound holds for all 0 <q p and for all µ 0, while the upper bound holds when µ =0and 0 <q p and when µ, p, and0<q p. 3. Proofs Proof of Theorem.. As we have already remarked, we need to prove only the lower bound. To this end without loss of generality we may assume that the elements of Λ n satisfy n = ; the general result follows by a linear scaling. Then the Newman polynomial T n E(Λ n ) is defined by (3.) T n (t) := πi Γ 3 e zt B n (z) dz,

4 where By the residue theorem Γ:={z C : z =} and B n (z) := T n (t) = (B n(λ j )) e λjt, and hence T n E(Λ n ). We claim that n z z +. (3.) B n (z) 3, z Γ. Indeed, it is easy to see that 0 implies z z + = + +, z Γ. So, for z Γ, n B n (z) + = + = 3, where the inequality x +x y +y = (x+y) +(x+y) + xy(x + y) (x + y), x,y 0, +(x+y) was used. We will examine T n,k E(Λ n ) defined by ( + x)( + y)( + (x + y)) T n,k (t) :=T n (k) (z)= ( z) k exp( zt) dz. πi Γ B n (z) Note that the circle Γ can be parametrized as Γ:={ exp(iu)+, u [, π)}, where for z = exp(iu)+,u [, π), we have z = exp(iu)+ u and exp( zt) exp(re( zt)) = exp(( +cosu)t) exp 4 ) ( tu.

5 Using the above inequalities together with (3.), we obtain that T n,k (t) πi π 3 π 3 π 0 ( ( exp(iu)+)) k exp( ( exp(iu)+)t) B n ( exp(iu)+) iexp(iu) du exp(iu)+ k exp( ( exp(iu)+)t) exp(iu) du B n ( exp(iu)+) ) u k exp ( tu du ( ) k ( ( ) ) ( ) k+ u t u t t exp du t 3 ( ) ( ) k+ v k exp( v ) dv π 0 t c(k) (k+)/ t (k+)/ with a constant c(k) depending only on k. Also, T n,k (t) πi π 3 π πk 3 k. So with k := 4/q we have ( ( exp(iu)+)) k exp( ( exp(iu)+)t) B n ( exp(iu)+) exp(iu)+ k exp( ( exp(iu)+)t) B n ( exp(iu)+) iexp(iu) du exp(iu) du (3.3) ( T n,k Lq [0, ) T n,k (t) q dt + 0 ) /q T n,k (t) q dt ( ) /q c(k) (k+)q/ t (k+)q/ dt +3 q kq ) /q (c(k) ( 4/q +)/ t dt +3 q 4 C(q) < with a constant C(q) depending only on q, and (3.4) T n,k [0, ) C( ) <. Now observe that T (µ) n,k (0) = πi Γ 5 ( z) k+µ B n (z) dz.

6 Here, for all z C with z > max j n,wehave ( z k+µ n + /z n B n (z) =zk+µ /z = zk+µ + = z k+µ (z ++ with a constant A m /m!. Therefore m= A m z m (3.5) T (µ) n,k (0) (k+µ)!. Pick a point y [0, ) sothat ) T (µ) (µ) n,k (y) = T n,k [0, ). m= ( λj z ) m ) Note that T (µ) n,k E(Λ n ). Combining the upper bound of Theorem. (Newman s inequality) with the Mean Value Theorem, we obtain that T (µ) n,k (t) T(µ) n,k [0, ), t I := Hence (3.5) and k := 4/q implies that [ y, y + 8 ]. (3.6) ( T (µ) n,k L p [0, ) 8 ( 8 ( ( c(p, q, µ) > 0 ) p ) /p (µ) T n,k [0, ) ) p ) /p (k + µ)! and (3.7) T (µ) n,k [0, ) Combining (3.3) (3.7) finishes the proof. (k + µ)! c(,q,µ) Λ References. P. B. Borwein and T. Erdélyi, Polynomials and Polynomials Inequalities, Springer-Verlag, New York, P. B. Borwein and T. Erdélyi, The L p version of Newman s inequality for lacunary polynomials, Proc. Amer. Math. Soc. 4 (996), T. Erdélyi, Markov- and Bernstein-type inequalities for Müntz polynomials and exponential sums in L p, J. Approx. Theory 04 (000), D. J. Newman, Derivative bounds for Müntz polynomials, J. Approx. Theory 8 (976), Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA address: terdelyi@math.tamu.edu 6