Peter B. BorweinWhat Peter has to say about himself
In recent years, my primary research interests have revolved around the interplay of classical analysis and approximation theory, computational complexity, number theory and symbolic computation. Analytic problems whose attack and proof lend themselves to extensive computational experimentation have attracted me most.
Substantial symbolic and numeric calculation has led to the discovery of some rather beautiful analytic objects (series, iterations, etc.), and in many cases, such as the derivation of Ramanujan type series considerably aids the proofs. The questions that arise impinge on issues in approximation theory, a number theory and computational complexity (not to mention the obvious Computational issues).
This research has also led to some of the most efficient known algorithms for various elementary functions and constants (a number of the recent record calculations of pi have used one of our algorithms). A pleasant by-product has the been the detection of various phenomena of a kind probably not visible without substantial interactive computation.Specific on going lines of inquiry include:
Iterations and expansions related to special functions, emphasizing low complexity expansions.
Analytic inequalities and expansions for rational functions and polynomials, including lacunary polynomials.
Approximation issues concerning Chebyshev systems, their Chebyshev polynomials, Muntz systems and incomplete rationals. The related conjectures of D.J. Newman.
Expansion problems of a number theoretic variety. The relationship between orthogonal expansions and irrationality proofs (such as zeta(3)). Growth problems concerning polynomials with integer coefficients associated with conjectures of Erdos and Szekeres.
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