Document Type
Lecture
Publication Date
3-10-2010
Abstract
We study remainder maps R_f(z) = Σₖ₌₁^∞ f(z + k), associated to convergent series defined by complex rational maps f ∈ ��(X), with coefficients in an arbitrary subfield �� ⊂ ℂ. All such maps are meromorphic on ℂ, form a vector space over ��, and generate what we call the remainder field ��(R_F) of ground ��. For remainder maps, we prove a criterion of algebraic independence over ��(X), stated in terms of ��-linear combinations. It turns out that any R_f is either a member of ��(X) or is transcendental, and so the field extension ��(R_F)/��(X) is purely transcendental. We show that any nonzero ϕ from the remainder field has a degree d(ϕ) ∈ ℤ, for which ϕ(x) / x^(d(ϕ)) has a nonzero limit (as x → ∞) in ��. This yields a natural asymptotic expansion of ϕ. Consequently, the remainder field can be extended to the field ��((X⁻¹)) of reverse formal Laurent series, and rational approximants of any order exist. For remainder maps, finding rational approximants reduces to the same problem for the fundamental case of f = 1 / X². We thus find the explicit formal series of any remainder map. In the general case, the difference 1 / R_f(x) - W_f(x) tends to 0 for some unique W_f ∈ ��[X], called here the inverse polynomial of R_f. In the real case, the convergence is monotonic and leads to sharp estimates. Iterating the construction leads to higher order inverse polynomials, with exponentially increasing degrees. In our example, the error made by replacing the sum of the series Σₙ≥1 1/n² (the fundamental case) by a corrected nth partial sum, is less than 10⁻⁴ for n = 1, less than 10⁻¹⁵ for n = 5, and less than 10⁻³² for n = 20. Our theory also includes the alternating remainder maps R̂_f(z) := Σₖ₌₁^∞ (-1)^(k-1) f(2z + k). Students are welcome.
Relational Format
presentation
Recommended Citation
Timofte, Vlad, "Remainder Maps" (2010). Analysis Seminar. 24.
https://egrove.olemiss.edu/math_analysis/24
