2
votes
2answers
38 views

Monotonically approximate $L^p$ function by step function

It is a classical fact that a $L^p(R^d)$ ($1\leq p<+\infty$) function can be approximated by step functions with compact support, but my question will be, can we require that the step function is ...
1
vote
1answer
50 views

Explanation of a passage about a smooth approximation to $L^p$ function

I'm reading J.L Vazquez "Porous Medium Equation" book. In it, he says the following: We are given a function $a:\Omega \times (0,T) \to \mathbb{R}$ such that $a \geq 0$. We find a smooth ...
0
votes
1answer
33 views

Sequence of piecewise constant functions converging to any $L^2$ function

Let $\{P_i\}_{i=1}^\infty$ be a sequence of partitions of the interval $[0,1]$ with a vanishing mesh. Additionally $H_i$ be the space of piecewise constant functions (step functions) with pieces ...
0
votes
1answer
43 views

Approximation in $L^2$ of functions with values in a convex set

Here is my problem : Let $K$ be a convex set of $\mathbb{R}^m$ ($m\in \mathbb{N}^*$), such that $0$ belongs to the interior of K, I want to approximate (in $L^2(\mathbb{R}^m,\mathbb{R}^m)$) a function ...
2
votes
1answer
302 views

Upper and lower bounds of a ratio involving vector norms

I'm working on a signal processing problem and need to analyze the following expression $$ G = \frac{n}{\sum\limits_{i=1}^n |w_i|} \frac{ \sum\limits_{i=1}^n g_i w_i^2}{\sum\limits_{i=1}^n g_i |w_i|} ...