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Approximation of a continuous function by a smooth one on an open set

I'm interested in the following kind of theorems : Let $M$ be a real analytic manifold and $U$ an open set of $M$. Let $f : U \to \mathbb{R}$ a continuous function. Then, there is a $C_{\infty}$ ...
Noether's user avatar
  • 193
3 votes
1 answer
81 views

Estimating the Size of an Approximating Polyline

let $\gamma(s) = \left(x(s),y(s)\right), s\in[0,1]; \gamma'(s) = 1$ be a length-parameterized curve in the plane, with finite and strictly positive curvature. Questions: is it possible to estimate ...
Manfred Weis's user avatar
  • 13.2k
3 votes
0 answers
269 views

covariant derivative of manifold-valued function and logarithm map

Let $M$ be a Riemannian manifold and $f\colon \Omega\subset \mathbb{R}^d\rightarrow M$ a smooth, i.e. $C^\infty$, function. For any $p\in M$ let $T_pM$ be the tangent space at $p$ and $\log_p\colon U\...
user35593's user avatar
  • 2,286