# Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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### Lebesgue measure as a fixpoint: change of variables formulas

This question is inspired by several others on a similar topic: see e.g. this one and a sequence of linked questions. Let us so far focus on $\Bbb R^n$ endowed with standard Borel structure. For any ...
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### Distinct eigenvalues for integral operator?

Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator $$\mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy$$ so that it has all its non-...
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### Intuition about the Bernstein polynomials proof of the Weierstrass approximation theorem

The Weierstrass approximation theorem can be stated as follows: Let $f\in C([a,b])$, then there exists a sequence $(p_n)_{n\in \mathbb{N}}$ of polynomials in $[a,b]$ such that $(p_n)$ converges ...
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### The space of test-functions carries any other structure on it?

I'm starting to study distributions and on the lecture notes I'm reading the author defines a test-function as a function $f : U\subset \mathbb{R}^n\to \mathbb{R}$ which is infinitely differentiable ...
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### If $f$ and $g$ are analytic functions, does $f \circ g - g \circ f = g- f$ implies $f = g$?

The question is in the title. There are actually two variants, one for real analytic functions and the other one for complex analytic functions. It came up as an attempt to solve this question by ...
For all non-negative integers $i$ and $j$ such that $j\leq i$, define the array of polynomials $$p_{ij}(z):=\sum_{h=(j-1)_+}^{i-1} {i\choose h}{i-j\choose{i-h-1}}z^h,$$ where $(a)_+=\max\{a,0\}$ (we ...