Q. I. Rahman

New inequalities for polynomials

Using a recently developed method to determine bound-preserving convolution operators in the unit disk, we derive various refinements and generalizations of the well-known inequalities of S. Bernstein and M. Ricsz for polynomials. Many of these results take into account the size of one or more of the coefficients of the polynomial in question. Other results of similar nature are obtained from a new interpolation formula.

Some inequalities for polynomials with a prescribed zero

Received by the editors May 17, 1974. AMS (MOS) subject classifications (1970). Primary 30A06, 30A64; Secondary 26A82.

Characterization of the speed of convergence of the trapezoidal rule

SummaryOur aim is to determine the precise space of functions for which the trapezoidal rule converges with a prescribed rate as the number of nodes tends to infinity. Excluding or controlling odd functions in some way it is possible to establish a correspondence between the speed of convergence and regularity properties of the function to be integrated. In this way we characterize Sobolev spaces, certain spaces of infinitely differentiable functions, of functions holomorphic in a strip, of entire functions of order greater than 1 and of entire functions of exponential type by the speed of convergence.

A Bandlimited Function Simulating a Duration-Limited One

We construct a bandlimited function Φ ≢ 0 for which |Φ(x)| decreases nearly as fast as possible when x → ±∞. Such a function may be approximately considered as a duration-limited one. It is of interest in the reconstruction of entire and harmonic functions of exponential type from given values.

A quadrature formula with zeros of Bessel functions as nodes

A quadrature formula for entire functions of exponential type wherein the nodes are the zeros of the Bessel function of the first kind was recently obtained by C. Frappier and P. Olivier. Here the condition imposed on the function is relaxed. Some applications of the formula are also given

On Bernstein's inequality

1. Introduction and statement of results. If p n ( z ) is a polynomial of degree at most n, then according to a famous result known as Bernsteins inequality (for references see [ 4 ]) (1) Here equality holds if and only if p n ( z ) has all its zeros at the origin and so it is natural to seek for improvements under appropriate assumptions on the zeros of p n ( z ). Thus, for example, it was conjectured by P. Erdos and later proved by Lax [ 2 ] that if p n ( z ) does not vanish in │ z │ (2) On the other hand, Turan [ 5 ] showed that if p n ( z ) is a polynomial of degree n having all its zeros in │z│ ≦ 1, then (3)

On a problem of Turan about polynomials

It is shown that if p,(x) is a polynomial of degree n whose graph on the interval -1 < x < 1 is contained in the unit disk then the absolute value of its second derivative cannot exceed l(n 1)(2n2 4n + 3) on [-1, 1]. In the year 1889 A. A. Markoff [2] proved the following result: THEOREM A. If p (x) = En= a, xp is a polynomial of degree n and p (x) < 1 in the interval -1 < x < 1 then in the same interval (1) pn (x)I < n2 The constant n2 in (1) cannot be replaced by any lower constant. In fact, the nth Tchebycheff polynomial of the first kind n (2) Tn(x) = cos (n arc cos x) = 2n-1 I x cos((v -2)7Tn)} P=1 satisfies the conditions of Theorem A and its derivative at the point x = 1 is equal to n2. W. A. Markoff (brother of A. A. Markoff) considered the problem of determining exact bounds for the kth derivative of pn(x) at a given point xo in [-1, 1] under the conditions of Theorem A. His results appeared in a Russian journal in the year 1892; a German version of his remarkable paper was later published in Mathematische Annalen [3]. Amongst other things he proved: THEOREM B. Under the conditions of Theorem A _m<ax l lp(k) (x)I < n2 (n2 12) (n2 22)... (n2 (k 1)2) k= 1,2,...,n. The right-hand side of this inequality is exactly equal to T(k)(1), where Tn(x) is the nth Tchebycheff polynomial of the first kind (2). W. A. Markoffs proof Presented to the Society, October 27, 1973 under the title On a problem of Turdn; received by the editors March 31, 1975. AMS (MOS) subject class ifcations (1970). Primary 30A06, 30A40; Secondary 42A04.

Polynomials with a parabolic majorant and the Duffin-Schaeffer inequality

Abstract For yϵ R let I y : = { x + iy : −1 ⩽ x ⩽ 1}. It was proved by R. J. Duffin and A. C. Schaeffer that if p ( x ) : = ∑ v = 0 n a v x v is a polynomial of degree at most n with real coefficients such that ¦p( cos ( vπ n ))¦ ⩽ 1 for v = 0, 1, …, n and T n is the n th Chebyshev polynomial of the first kind then max z ϵ I y ¦p (k) (z)¦ ⩽ ¦T n (k) (1 + iy)¦ for k = 1, 2, … . To this we add that if τ n + 2 ( z ) : = (1 − z 2 ) T n ( z ) then max z ϵ I y ∥( d k dz k )((1 − z 2 ) p(z))¦ ⩽ ¦τ n + 2 (k) (1 + iy)¦ for k = 3, 4, … . The result can be looked upon as an inequality for polynomials with a parabolic majorant, analogous to that of Duffin and Schaeffer.

Quadrature formulae and functions of exponential type

We obtain certain generalizations of the trapezoidal rule and the Euler-Maclaurin formula that involve derivatives. In the case of quadrature of functions of exponential type over infinite intervals we find conditions under which existence of the (improper) integral and convergence of the approximating series become equivalent

On the L p convergence of Lagrange interpolating entire functions of exponential type

Abstract Let f: R ↦ C be a continuous, 2π-periodic function and for each n ϵ N let tn(f; ·) denote the trigonometric polynomial of degree ⩽n interpolating f in the points 2kπ (2n + 1) (k = 0, ±1, …, ±n) . It was shown by J. Marcinkiewicz that lim n → ∞ ∝ 0 2π ¦f(θ) − t n (f θ)¦ p dθ = 0 for every p > 0 . We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.