# Tagged Questions

1answer
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### Integral of a composition of piecewise linear function with polynomial

Fix a number $k > 0$ and let $$T(x) = \begin{cases} k &: x \geq k\\ x &: |x| < k\\ -k &: x \leq -k \end{cases}.$$ Define $S(s) = \int_0^s T(|x|^{m-1}x)\;dx.$ I want to show that ...
1answer
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### Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
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1answer
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### Solution of a functional integral

I am trying to show the following integral has the following result $$-\int \nabla^2\psi \text{d} \psi^*=|\nabla\psi|^2$$ Going backwards I write ...
1answer
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### a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
2answers
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### What is $T^nf(t)$? (Question on integrals)

I am supposed to prove the following: For the operator $T$ defined by $$Tf(t)=\int_0^t(t-s)f(s)\,ds,\quad f\in C[0,1]$$ Show that $$T^nf(t)=\int_0^t\frac{(t-s)^{2n-1}}{(2n-1)!}f(s)\, ds$$ I ...
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### does the limit of the ratio of $p+1$ norm and $p$ norm equal to $\infty$ norm

Suppose that $f\in L^1(\Omega,\Sigma,\mu)\cap L^\infty(\Omega,\Sigma,\mu)$. Then I have proved that for any $1\leq p\leq \infty$, $f\in L^p(\Omega,\Sigma,\mu)$. Moreover, I have proved ...
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### Convergence of product of continuous functions and test functions

I suspect the following result is true but I"m not sure how to go about proving: It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions ...
1answer
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### derivate of indicator function

What is the derivative of the indicator function: $$f(x)=\begin{cases} 1 & x^{\min} x\leq x^{\max}\\ -\infty &\mbox{otherwise}? \end{cases}$$ thank you
1answer
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### questions about $L^p$ space with $0<p\leq 1$ parallel to the case $1<p$

Question (1). Riesz-Fischer Theorem: For $1\leq p\leq \infty$, $L^p(\mu)$ is complete. Corollary of proof: Let $1\leq p\leq \infty$. If $(f_n)_{n=1}^\infty$ is a sequence coverging to $f$ with ...
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2answers
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### On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
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1answer
93 views