N. K. Govil

Some inequalities for derivatives of polynomials

Abstract If p ( z ) is a polynomial of degree at most n having no zeros in ¦z¦ , then according to a well known result conjectured by Erdős and proved by Lax max ¦z¦ = 1 ¦p′(z)¦ ⩽ ( n 2 ) max ¦z¦ = 1 ¦p(z)¦ . On the other hand, by a result due to Turan, if p ( z ) has all its zeros in ¦z¦ ⩽ 1 , then max ¦z¦ = 1 ¦p′(z)¦ ⩾ ( n 2 ) max ¦z¦ = 1 ¦p(z)¦ . In this paper we generalize and sharpen these inequalities.

An inequality for self-inversive polynomials

Abstract A sharp Lp, p ⩾ 1, inequality for the class of self-inversive polynomials is obtained.

Certain new classes of Durrmeyer type operators

In the present paper, we estimate the rate of convergence for functions having bounded derivatives for certain Durrmeyer type generalization of Jain and Pethe operators. In the last section, we also propose a new modification of the Lupas operators and study some direct results. We also give some open problems for the readers.

On the Eneström-Kakeya theorem, II

Abstract A classical result of Enestrom and Kakeya (if a n ⩾ a n − 1 ⩾ a n − 2 … ⩾ a 1 > 0, then, for ∣z∣ > 1, ∑ k = 0 n a k z k ≠ 0) is extended to polynomials whose coefficients are monotonic but not necessarily positive.

On the maximum modulus of polynomials

Abstract Let p(z) = ∑nv = 0 avzv be a polynomial of degree n and let M(p, r) = max ¦z¦ = r ¦p(z)¦ . It has been shown that if p(z) ≠ 0 in ¦z¦ , then for 0 ⩽ r ⩽ ϱ ⩽ 1, M(p, r) ⩾ ((1 + r) (1 + ϱ)) n M(p,ϱ) . The result is best possible and for ϱ = 1 it reduces to a result of Rivlin. Besides it has been shown that if in addition p′(0) = 0, then the bound can be considerably improved.

Perturbations of the Haar wavelet

Let m ∈ Z+ be given. For any e > 0 we construct a function f{e} having the following properties: (a) f{e} has support in [−e, 1 + e]. (b) f{e} ∈ Cm(−∞,∞). (c) If h denotes the Haar function and 0 < δ <∞, then ‖f−h‖Lδ(R) ≤ (1+2δ)1/δ(2e)1/δ . (d) f{e} generates an affine Riesz basis whose frame bounds (which are given explicitly) converge to 1 as e→ 0. Let H be a Hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖ := 〈·, ·〉1/2. Let Z+ denote the natural numbers. A sequence {fn, n ∈ Z+} ⊂ H is called a frame if there are constants A and B such that for every f ∈ H A‖f‖2 ≤ ∑ n∈Z+ |〈f, fn〉| ≤ B‖f‖2. The constants A and B are called bounds of the frame. If only the right-hand inequality is satisfied for all f ∈ H, then {fn, n ∈ Z+} is called a Bessel sequence with bound B. A frame is called exact, or a Riesz basis, if upon the removal of any single element of the sequence, it ceases to be a frame. However, not every frame is a Riesz basis: as is well known, a sequence {fn, n ∈ Z+} ⊂ H is a Riesz basis if and only if it is the image of an orthonormal basis under a bounded invertible linear operator U : H → H ([1, 11]). Clearly for an orthonormal basis both frame bounds equal 1. In [5] it is shown that adding a Bessel sequence with a small bound to a Riesz basis transforms the original basis into another Riesz basis. It is also shown how frame bounds for the new basis are obtained from frame bounds of the original basis. Given an arbitrary positive integer m, in the present paper we use these results to perturb the Haar function into a function f{e} ∈ Cm(−∞,∞) with support in [−e, 1 + e] (thus having good time and frequency localization) that preserves the symmetry of the Haar wavelet and generates an affine Riesz basis in L2(−∞,∞). The lack of orthogonality precludes the use of the fast wavelet transform. On the other hand the functions f{e} are given explicitly in terms of cardinal B-splines. Other wavelets do not have a closed form representation and have to be obtained recursively, using the cascade algorithm [4, 10]. Since frame bounds are given Received by the editors March 18, 1996 and, in revised form, June 21, 1996. 1991 Mathematics Subject Classification. Primary 42C99; Secondary 41A05, 46C99.

On region containing all the zeros of a polynomial

In this paper we present a result providing an annulus containing all the zeros of a polynomial with complex coefficients. Our theorem includes as special cases several results known in this direction. Besides, we show that as corollaries this theorem can generate many more results of this type. Also, we develop MATLAB code to construct polynomials for which the bounds obtained by our results are better than obtainable from some of the known results.

On Bernstein's inequality

Abstract Let pn(z) = an Πv = 1n (z − zv), an ≠ 0 be a polynomial of degree n and let ∥p n ∥ = max ¦z¦ = 1 ¦p n (z)¦, ∥p′ n ∥ = max ¦z¦ = 1 ¦p′ n (z)¦ . It has been shown that if ¦z v ¦ ⩾ K v ⩾ 1, 1 ⩽ v ⩽ n , then ¦p ′ n ¦⩽n( ∼ v=f n 1 (K V −1) ∼ v=f n (K V +1) (K V -1) ¦p n ¦ . Our result includes as a special case Erdos conjecture first proved by Lax. Also it sharpens and includes as a special case the inequality due to Malik that if p n (z) ≠ 0 for ¦z¦ then ∥p′ n ∥ ⩽ ( n (1 + K) ) ∥p n ∥ .

On growth of polynomials

paper, we generalize and sharpen this and some other related inequalities by considering polynomials having no zeros in Izl 1.

Simultaneous approximation for the Phillips operators

We study the simultaneous approximation properties of the well-known Phillips operators. We establish the rate of convergence of the Phillips operators in simultaneous approximation by means of the decomposition technique for functions of bounded variation.