Thesis (Ph. D.)--York University, 2001. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 170-177). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ66344.

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references. Also available in electronic version. Access restricted to campus users.

Proefschrift--Vrije Universiteit, Amsterdam. / "Stellingen": [4] p. inserted.

Thesis--University of Pennsylvania. / "Reprint from Duke mathematical journal, 14, 4." Bibliography: p. 1104.

Thesis--University of Wisconsin--Madison. / Typescript. Vita. Includes bibliographical references (leaf 62).

Thesis (Ph. D.)--University of Wisconsin, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (M.S.)--Oregon State University, 2008. / Printout. Includes bibliographical references (leaves [51-54]). Also available on the World Wide Web.

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The fundamental theorem, as far as this work is concerned, is Weierstrass' theorem (1885) on the approximability of continuous functions by polynomials. Since the time of Weierstrass (1815-97) and his equally important contemporary Chebyshev (1821-94), the topic of approximation has grown enormously into a subject of considerable interest to both pure and applied mathematicians. The subject matter of this thesis, being exclusively concerned with polynomial approximations to a single-valued, function of one real variable, is on the side of 'applied' side of approximation theory. The first chapter lists the definitions and theorems required subsequently. Chapter is devoted to estimates for the maximum error in minimax polynomial approximations. Extensions of this are used to obtain crude error estimates for cubic spline approximations. The following chapter extends the minimax results to deal also with best L[sub]p polynomial approximations, which include beat least squares (L2) and best modulus of integral (L1) approximations as special cases. Chapter 4 is different in character. It is on the practical problem of approximating to convex or nearly convex data.

It is well known that given f there is a unique polynomial of degree at most n which interpolates f on the standard triangle with uniform nodes (i, j), i, j ≥ 0, i + j ≤n. This leads us to the study of polynomial interpolation on a "triangular" domain with the nodes, S = {([i], [j]): i, j ≥ 0, i + j ≤n}, [k] = [k][sub]q = (1-qk)/(1-q), q > 0, which includes the standard triangle as a special case. In Chapter 2 of this thesis we derive a forward difference formula (of degree at most n) in the x and y directions for the interpolating polynomial P[sub]n on S. We also construct a Lagrange form of an interpolating polynomial which uses hyperbolas (although its coefficients are of degree up to 2n) and discuss a Neville-Aitken algorithm. In Chapter 3 we derive the Newton formula for the interpolating polynomial P[sub]n on the set of distinct points {(xi, y[sub]j): i, j ≥ 0, i + j ≤n}. In particular if xi = [i][sub]p and y[sub]j = [j]q, we show that Newton's form of P[sub]n reduces to a forward difference formula. We show further that we can express the interpolating polynomial on S itself in a Lagrange form and although its coefficients Ln/ij are not as simple as those of the first Lagrange form, they all have degree n. Moreover, Ln/ij can all be expressed in terms of Lm/0,0, 0 ≤ m ≤ n. In Chapter 4 we show that P[sub]n has a limit when both p, q → 0. We then verify that the interpolation properties of the limit form depend on the appropriate partial derivatives of f(x, y). In Chapter 5 we study integration rules I[sub]n of interpolatory type on the triangle S[sub] = {(x, y): 0 ≤ x ≤y ≤ [n]). For 1 ≤ n ≤5, we calculate the weights wn/ij for I[sub]n in terms of the parameter q and study certain general properties which govern wn/ij on S[sub]n. Finally, Chapter 6 deals with the behaviour of the Lebesgue functions λ[sub]n(x, y; q) and the corresponding Lebesgue constant. We prove a property concerning where λ[sub]n takes the value 1 at points other than the interpolation nodes. We also analyse the discontinuity of the directional derivative of λ[sub]n on S[sub]n.