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Results tagged with ca.classical-analysis-and-odes
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user 8257
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
23
votes
5
answers
5k
views
Existence of a smooth function with nowhere converging Taylor series at every point
By Borel's theorem, for any sequence of real numbers $a_n,$ there is a $C^{\infty}$-function
$f:\mathbb{R}\to\mathbb{R}$ whose Taylor series at 0 is $\sum a_nx^n.$ In particular, there are $C^{\infty} …
- 689
1
vote
Does this recursion preserve monotonicity? (was: A nice problem that I am unable to solve)
First, notice that $f_n(x)=f_n(1-x).$ Therefore,
$f_{n+1}(x)=1/2(f_n(x^2)+f_n(1-(1-x)^2))=1/2(f_n(x^2)+f_n(2x-x^2))$, but both
x^2, 2x-x^2 are increasing functions on [0, 1/2]. So, by induction, you …
- 689