Skip to main content

Eli Passow

Followers

9

Following

1

Public Views

Interests

Uploads

Papers by Eli Passow

Approximation with interpolatory constraints

Illinois Journal of Mathematics, Mar 1, 1976

In this article, we are interested in questions of the existence of approximating functions which... more In this article, we are interested in questions of the existence of approximating functions which have certain approximation properties with respect to a given function f and, at the same time, interpolate the values off and/or its derivatives. By "approximation properties" we refer to questions of uniform approximation and/or the degree of approximation. A common feature of the results presented here is the method of proof. In each case a set of auxiliary approximating func- tions which "surround" f is considered. The approximations to these auxiliary functions by the given approximating functions are then found to have the desired approximation properties and to have the given f in their convex hull.

Restricted approximation and interpolation

Bulletin of the American Mathematical Society, 1974

Piecewise Monotone Polynomial Approximation

Transactions of the American Mathematical Society, 1972

Piecewise monotone interpolation and approximation with Muntz polynomials

Transactions of the American Mathematical Society, 1976

The possibility (subject to certain restrictions) of solving the following approximation and inte... more The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of “Muntz polynomials” on a real interval is demonstrated: (i) approximation of a continuous function by a “copositive” Muntz polynomial; (ii) approximation of a continuous function by a “comonotone” Muntz polynomial; (iii) approximation of a continuous function with a monotone kth difference by a Muntz polynomial with a monotone kth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs.

Monotone and Convex Spline Interpolation

SIAM Journal on Numerical Analysis, 1977

For a set of monotone (and/or convex) data, we consider the possibility of finding a spline inter... more For a set of monotone (and/or convex) data, we consider the possibility of finding a spline interpolant, of pre-determined smoothness, which is monotone (and/or convex). The investigation is carried out by constructing an auxiliary set of points and using the well-known monotonicity and convexity preserving properties of Bernstein polynomials. In § 3 we consider the problem of piecewise monotone interpolation.

$Research paper thumbnail of Extended Chebyshev Systems on $( - \infty ,\infty )$$

Extended Chebyshev Systems on $( - \infty ,\infty )$

SIAM Journal on Mathematical Analysis, 1974

Let 0 =< < < < be a sequence of integers. Necessary and sufficient conditions are obtained for {x... more

Chebyshev Systems of Minimal Degree

SIAM Journal on Mathematical Analysis, 1984

Let $F = \{ {f_i } \}_{i = 0}^n $ be a set of continuous functions on $[a,b]$, and let $F^ * = \{... more Let $F = \{ {f_i } \}_{i = 0}^n $ be a set of continuous functions on $[a,b]$, and let $F^ * = \{ {f_i f_j } \}_{i,j = 0}^n $. We determine conditions on F which are necessary and sufficient for the set $F^ * $ to be a Chebyshev system on $[a,b]$ consisting of exactly $2n + 1$ distinct functions. The results have applications in the field of experimental design.

The degree of piecewise monotone interpolation

Proceedings of the American Mathematical Society, 1975

on the degree of an algebraic polynomial p{x) that interpolates the given

$Research paper thumbnail of Polynomial approximation on $y=x\sp{\alpha }$$

Polynomial approximation on $y=x\sp{\alpha }$

Proceedings of the American Mathematical Society, 1970

We show that on the curve y = x α , α y = {x^\alpha },\alpha any irrational, 0 ≦ x ≦ 1 0 \leqq x ... more

Monotone and comonotone approximation

Proceedings of the American Mathematical Society, 1974

Jackson type theorems are obtained for monotone and comonotone approximation. Namely (i) If/(;r) ... more Jackson type theorems are obtained for monotone and comonotone approximation. Namely (i) If/(;r) is a function such that the kth difference of / is =ï0 on [a, b] then the degree of approximation of/by nth degree polynomials with kth derivative ^0 is 0[a>(/; l/«1-*)] for any e>0, where a>(f; <5) is the modulus of continuity oí fon [a, b]. (ii) If f(x) is piecewise monotone on [a, b) then the degree of approximation of/ by nth degree polynomials comonotone with / is 0[o(/; I/«1"*)] for any e>0.

Negative theorems on generalized convex approximation

Pacific Journal of Mathematics, 1976

Algorithms for Computing Shape Preserving Spline Interpolations to Data

Mathematics of Computation, 1977

Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserve... more

For All Practical Purposes: Interactive Study Guide (CD ROM Plus Book)

The Mathematical Gazette, 1999

Hermite-Birkhoff interpolation: A class of nonpoised matrices

Journal of Mathematical Analysis and Applications, 1978

Comonotone polynomial approximation

Journal of Approximation Theory, 1974

Let P, be the set of all algebraic polynomials of degree II or less. For f~ C[u, b], the degree o... more Let P, be the set of all algebraic polynomials of degree II or less. For f~ C[u, b], the degree of approximation to f by polynomials in P, is En(f) = inf{llf-p 11: p E PJ, where the norm is the uniform norm. Jackson's theorem [l] states that there exists C > 0 such that En(f) < Cw(f; l/n), where w(f; 6) is the modulus of continuity off. f is said to be piecewise monotone if it has only a finite number of local maxima and minima in [a, b]. The local maxima and minima in (a, b) are called the peaks off. Wolibner [5] has shown that for any E > 0 there exists a polynomial, p, such that IIfp II < E and p is comonotone with f; i.e., p increases and decreases simultaneously with $ Let E,*(f) = inf{llf-p 11: p E P, , p comonotone with f}. Clearly E,*cf) 3 E,df). We seek an upper bound on E,*(f). Forfmonotone, Lorentz and Zeller [3] have shown that there exists C, > 0 such that E,*(f) < C,w(f; I/n). Newman, Passow, and Raymon [4] have obtained results of a modified nature. They have shown that there exists p E P, satisfying IIfp II < C+(f; l/n), C, an absolute constant, such that f and p are comonotone except in certain neighborhoods (whose diameters tend to zero with n) of the peaks. In this note we obtain a comonotone approximation on the entire interval [a, b], but at a sacrifice in the accuracy of approximation. LEMMA 1. Let fe C(j+l)[a, b] and suppose that f(u) = 0, u E (a, b). Let g(x) =f[x, u], the divided dzfirence off, where we dejine f[u, U] =f'(u). Then g E Cj[u, b] and II g(j) 11 < (j + 1)-l /lf(j+') 11.

An improved estimate of the degree of monotone interpolation

Journal of Approximation Theory, 1976

Monotone quadratic spline interpolation

Journal of Approximation Theory, 1977

An alternation theory for copositive approximation

Journal of Approximation Theory, 1977

Deficient Bernstein polynomials

Journal of Approximation Theory, 1989

On the Divergence of Lagrange Interpolation to |x|

Journal of Approximation Theory, 1995