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Results tagged with integration
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user 117091
Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...
11
votes
2
answers
603
views
Function orthogonal to powers of $1/\left(1+x^2\right)$
Does there exist any continuous function $f:\mathbb{R}\to\mathbb{R}$, $f(x)/(1+x^2)\in L^1(\mathbb R)$, such that $f(0)=1$ and
$$\int_{-\infty}^{\infty}\frac{f(x)}{\left(1+x^2\right)^p}dx=0$$ for ever …
- 537
4
votes
1
answer
169
views
Functions orthogonal to powers of $1/{\left(1+x^2\right)}$
Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions with the following properties:
$f(x)$ and ${g(x)}/x$ are bounded;
${g(x)}/{\left(1+x^2\right)}\in L^1\left(\mathbb{R}\right)$;
$\lim_{x\to0} …
- 537