UNIFORM APPROXIMATION BY INCOMPLETE POLYNOMIALS

For any e with 0 < e < i, it is known that, for the set of all n incomplete polynomials of type , i.e [p(x) > akxk s ’n}, to have k=s the Weierstrass property on [a@, I], it is necessary that 2 --< ae --< 1. In this paper, we show that the above inequalities are essentially sufficient as wel I.

At the Conference on Rational Approximation with Emphasis on Appli- cations of Pad Approximants, held December 15-17, 1976 in Tampa, Florida, Professor G. G. Lorentz introduced new results and open questions for incomplete polynomials, defined as follows.Let e be any given real number with 0 <_ e < I.Then, a real or complex polynomial of the form n p(x) ak xk, k=s is said to be an .incompletepolynomial of type if s-> 'n.Note that the set of all incomplete polynomials of type contains polynomials of arbitrary degree, and that when > 0, this collection is not closed under ordinary addition.This set, however, is closed under ordinary multiplication.
(1.2) Furthermore, (1.2) is best possible in the sense that, for each @ with 0 < <_ I, there is a sequence [n. (x)}i=l of incomplete polynomials of 2 type satisfying (I.I) and a sequence [i}i= I with lim i for which ln. (i)l M for all i _> I. Hence, the interval [0,82) of convergence to zero in (1.2) cannot be replaced by any larger interval [0, t2+) for >0.
In Lorentz [2], the set of all incomplete polynomials of fixed type (0 < < I) is said to have the Weierstrass property on [a,l] if, for every continuous function f defined on [ate,l] there exists a sequence Pn.l(x)}i=l' with pni an incomplete polynomial of type for all i----_ I, which converges uniformly to f on [a,l].Evidently, from (1.2), a necessary condition that the set of all incomplete polynomials of a fixed type , 0 < < I, has the Weierstrass property on [a@,l] is that 82 _< a@ < i. (1. 3) The main purpose of this paper is to show that the condition (1.3) is essentially sufficient as well.The outline of the paper is as follows.
In 2, we state our new results and comment on their sharpness and their relation to known results in the literature.The proofs of these new results are then given in 3.

STATEMENTS OF NEW RESULTS.
As our first result, we have THEOREM 2.1.For any fixed @ with 0 < 8 < I, let F be any continuous function on [0, I] which is not an incomplete polynomial of type 8.Then, a necessary and sufficient condition that F be the uniform limit on [0, I] of a sequence of incomplete polynomials of type 8, is that F(x) 0 for all 0 _< x _< 82. (2.1) As an application of Theorem 2.1, fix any @ with 0 < 8 < I and consider any continuous function on [0, i] with ]IIIL=[0,1 ] i and with vanishing on [0, 2] and on [2+c, I], where 0 < c < I 2. For > 0, there exists, using l'neorem 2.1, an incomplete polynomial n of type e with assumes lln IIL[O,I] < which implies for sufficiently small, that n its maximum absolute value on [0, I] in the interval [e2, 2 + ].Thus, the sequence [(n(X)/llnl]L=[0,1])J}j=l of incomplete polynomials, each of type t, cannot tend uniformly to zero in [t 2 2 + ] for any with 0 < e < 1-e2.This observation then gives a different proof of the sharpness portion (cf.[4]) of Theorem I.I.We also remark that the sufficiency of Theorem 2.1 improves a related result of Roulier [3, Theorem 4] concerning Bernstein polynomials.
From Theorem 2.1, the following is deduced.
THEOREM 2.2.For any @ with 0 < @ < I, let [@i}i= I be any sequence of real numbers such that 0 < .< t for all i 1.Then, for any continuous function f on [ ,I], there exists a sequence Pn (x)}i=l' with each P an n.
incomplete polynomial of type t i, such that P (x) f(x), uniformly on [@2 I] n. (2.2) and such that the sequence [Pn. (x)}i=l is uniformly bounded on [0,i].
In the case of major interest in Theorem 2.2, i.e., when t. t as i ', we remark that the result of Theorem 2.2 is best possible in the following sense If [a,b D [t 2 1 with [a b [-t 2 l, then there are, continuous functions on [a,b] which cannot be uniformly approximated on [a,b] by a sequence [en. (x)}i=l, with each e an incomplete polynomial n.
of type i' where t. e as i .
As other consequences of Theorems 2.1 and 2.2, we have COROLLARY 2.3.For any @ with 0 < < I, consider any continuous function f on [@2,1].Then, for any q with i q < =, there exists a sequence [P (x)} with each P an incomplete polynomial of type n.
i=l n.
and such that the sequence [Pn. (x)izl is uniformly bounded on [0, i].
COROLLARY 2.4.For any @ with 0 < @ < I, the set of incomplete polynomialB of type @ is dense in the Banach space L "'[@2,1j (with respect to the norm q If'liE L[@2 l)j for each q with i _< q < =. q COROLLARY 2.5.For any @ with 0 < @ < I, the set of incomplete polynomials of type @ is dense in the space of continuous functions on [@2 + C, I] (with respect to the norm 11.[1.==[2+, 3) for every 0<<I e 2 The sharpness remarks following Theorem 2.2 similarly apply to the results of Corollaries 2.3-2.5.
To conclude this section, we remark that Corollary 2.5 leaves as an open question whether or not each continuous function f on [@2 I] with f(e2) # 0 is the uniform limit of incomplete polynomials of type @.In I attempting to settle this question, consider the special case of @ I and f(x) 3. PROOFS.
PROOF OF THEOREM 2.1.Let F be any continuous function [0,I] which is not an incomplete polynomial of type 8, and assume that F is the uniform limit of a sequence of incomplete polynomials of type 8.Then, ( follows from (1.2) of Theorem I.I, establishing the necessity of (2.1).
For sufficiency, let n o be any positive integer with n o _ Note that Qn' which is of degree at most n, is an incomplete polynomial type 0 for all n-> no, since (n0 + i) _> From the bf6ntz theory of best L2-approximation on [0,I], it is known (cf.Cheney [i, p. 196]) that n-n0 qj (I +qj)' n j=l (3.3)where qj n0 + j I, j I, 2, "'', n n0.

(
I-)-I.If y denotes the integer part of the real number y, let nbe the (unique) least squares approximation to the constant function 1 on [0, I], i.e., n(t) Idt, fx E [02 I]Applying the Cauchy-Schwarz inequality to the last integral,