Whitney Approximation: domains and bounds

We investigate properties of holomorphic extensions in the one-variable case of Whitney's Approximation Theorem on intervals. Improving a result of Gauthier-Kienzle, we construct tangentially approximating functions which extend holomorphically to domains of optimal size. For approximands on unbounded closed intervals, we also bound the growth of holomorphic extensions, in the spirit of Arakelyan, Bernstein, Keldych, and Kober.