# Tagged Questions

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### Convergence of the Double Integral of a Polynomial Reciprocal

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions: (i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$; (ii) $f$ is non-degenerate, in the sense that there isn't a ...
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### Is there an example where the error of Gauss-Laguerre quadrature does not vanish?

The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum $$\sum_{i=1}^n f(x_i) w_i$$ where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
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### Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...
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### Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? [closed]

Let $f\in C([a,b],\mathbb{R})$ such that $\displaystyle\int_{a}^{b} t^nf(t)dt=0$ for all integer n. We know that $f\equiv 0$. It's call Hausdorff theorem. This theorem is wrong on $\mathbb{R^+}$, a ...
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### Inverse of matrix-valued function

Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by ...
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### Estimating a quantity from an estimate in its integral

I am reading a paper in which the following argument is made. We have two positive real valued functions $f(x)$ and $g(x)$. We know that $$\int_0^x \int_0^y f(z) \ dz \ dy \leq g(x).$$ It is then ...
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### Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable $f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
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### “Values” of divergent integrals

Are there existing theories of integration in which $I_0 = \int_0^{\infty} dx$ and $I_1 = \int_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can ...
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### Convergence of the integral of step functions

This is a question about the proof of Lemma A in §16 of the book Functional Analysis by F. Riesz and B. Sz.-Nagy. Lemma A: For every sequence of step functions $\{\varphi_n\}$ which decreases to ...
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### Is there a probability density function satisfying the following conditions?

I find myself in need of the solution of this problem in finding a probability density function. I had asked this question in Math Stack Exchange but I did not get an answer so I am posting it here. ...
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### How to get an expression for this integral(Numerically/Analytically)

I have the following problem: I need to evaluate the integral $$\int_{\cos(\alpha)}^{1} P_l(t)P_{l'}(t) dt$$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the l-th ...
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### Real analytic functions

I am quite confused with some ideas regarding the Real analytic functions. Just to introduce my questions: A function $f$ is real analytic on an open set $D$ of the real line if for any $x_0\in D$ ...
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### Gauge integral of the derivative of a function except on a set of measure 0.

For the entire question, the interval I am integrating over is $[0,1]$. Background: In order to exhibit an isometry from $L^2[0,1]$ into $l^2$, I need to either assume absolute continuity over some ...