Thesis (M.A.)--Boston University

Thesis (doctoral)--Georg-Augustus-Universität zu Göttingen, 1902. / Vita.

Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This paper discusses the following problem of approximation theory: a continuous function f(x) is to approximated over an interval alpha </= x </= B by N not necessarily connected polynomials of a given degree n, in such a way that the maximum error magnitude is a minimum. Each polynomial is associated with one subinterval [uj, uj+1]. If the end points Ui were specified in advance, the problem would reduce to N independent problems of the same type, namely, the fitting of a single polynomial in the Chebyshev sense. Here, however, the end points are taken as unknowns and the principal problem is to determine them. The paper presents a proof of the existence of the best approximation and examples showing the solution, in general, is not unique. However, in the special case of approximation of convex functions by line segments, the solution is shown to be unique. Further in this case a simple characterization of the solution is obtained and it is shown that the problem may be reduced analytically to a stage where in order to determine Ui computationally, it is only necessary to solve a system of equations rather than minimize a function. Results obtained by a dynamic programming method using a digital computer (IBM 7090) are used for illustration. / 2031-01-01

Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This paper deals with methods of Chebyshev approximation. In particular polynomial approximation of continuous functions on a finite interval are discussed. Chapter I deals with the existence and uniqueness of Chebyshev or C-polynomials. In addition, some properties of the extremal points of the error function are derived, where the error fUnction E(x) = f(x) - p(x), p(x) being the C-polynomial. Chapter II discusses a method for finding the C-polynomial of degree n--the exchange method. After choosing a set of n+2 distinct abscissas, or a reference set, the so-called levelled reference polynomial is computed by the method of divided differences or by using the approximation errors of this polynomial. A point xj of maximal error is obtained and introduced into a new reference. A new levelled reference polynomial is then computed. This process continues until a reference is gotten, whose reference deviation equals the maximal approximating error of the levelled reference polynomial. The reference deviation is the common absolute value of the levelled reference polynomial at each of the reference points. The levelled reference polynomial for this reference is then shown to be the desired C-polynomial. Chapter III deals with phase methods for constructing the a-polynomial. It is shown that under suitable restrictions, if a Pn, A and €(phi) can be found such that the basic relation f(cos phi) = Pn(cos phi) + A cos[(n+1)phi + E(phi)] is satisfied on the approximation interval, then Pn is the a-polynomial. Two methods for finding the amplitude A and the phase function €(phi) are discussed. The complex method assumes f to be analytic on a domain and uses Cauchy's integral formula to obtain new values of €(phi), starting with a set of initial values. These values in turn generate new values of Pn and A. The values of Pn as well as values of A and €(phi) at certain points are gotten through convergence of this iterative scheme. Then an interpolation formula is used to obtain Pn from its values at these points. The second method attempts to find A, €(phi) and Pn so as to satisfy the basic relation only on a discrete set of points. First, assuming €(phi) so small that cos €(phi) may be replaced by 1, an expression is obtained for Pn(cos phi). In the general case, a system of phase equations is given, from which €(phi), A and hence Pn may be obtained. Although these results are valid only on a discrete set of points in the approximation interval, the polynomial derived in this way represents a good approximation to f(x). / 2031-01-01

Thesis submitted 1970; degree awarded 1971. / In this dissertation we consider the minimax approximation of functions f(x) E"C[O, l] rotated about the origin, and the characterization of the optimal rotation, a*, of f in the sense of least minimax error over all possible rotations. The paper divides naturally into two sections: a) Existence, uniqueness, and characterization for unisolvent minimax approximation for each rotation a of f. These results are applications of Dunham (1967). b) Existence, non-uniqueness, and com.putation of a*; derivation of necessary conditions for the minimax [TRUNCATED]

No description available.

Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The problem discussed in this paper is that of finding a best approximation to a given real-valued function f(x) over a continuum by means of finding a best approximation over a discrete set of points. We are also seeking to find a numerical method of finding a best approximation over our discrete set of points. A best approximation is one which minimizes the maximum deviation of our approximation from our given function f(x). We first discuss the concept of linear programming. In this paper we are not so much concerned with the theory behind linear programming as we are with the method to solve a linear programming problem, namely the simplex method. We discuss the simplex method from the point of view of a programmer, noting how results are continually updated until an optimal solution to t he problem is found. The only theoretical aspect of linear programming which we discuss is the notion of duals and the relationship between the solution of a primal and a dual problem. This becomes very important later in the paper when we try to formulate a best a proximat ion problem as a linear programming problem. Next we discuss the theoretical aspects of best approximation over a continuum. We prove existence, uniqueness, and, most important for our purposes, characterization. Our approximating functions are assumed to form a Chebyschev set throughout this paper. Finally we discuss best approximation over a discrete set of points. We first prove that the characterization theorem holds for problems of this type. Now that we have a way to tell whether our approximation is the best that can be obtained, we turn our attention to the relationship between the best approximation problem over a continuum and a discrete set of points. We prove in a quite general context that the best approximation over a discrete set of points converges uniformly to the solution to the problem over the continuum. We then retrace our steps and establish similar results for the particular case of polynomial approximation. After this we try to find out about the rate at which this convergence takes place. In general this question has no answer for its depends on the smoothness of the functions involved; if, however, we assume the fun ctions satisfy a Holder condition we may obtain some bounds on the rate of convergence. Finally, we reformulate the best approximation problem, showing how it can be considered as a linear programming problem which we already have a means of solving. / 2031-01-01

No description available.

Thesis (doctoral)--Rheinischen Friedrich-Wilhelms-Universität, Bonn, 1980. / Includes bibliographical references (p. 252-258).

Thesis (M.Sc.)--Memorial University of Newfoundland, 2004. / Restricted until May 2005. Bibliography: leaves 75-83.