Regularized Interpolation Driven by Total Variation

Abstract

We explore minimization problems of the form

$$\text{Inf} \left\{ \int^1_0 |u'|+ \sum^k_{i=1} |u(a_i) - f_i|^2 + \alpha \int^1_0 |u|^2\right\},$$

where $u$ is a function defined on $(0,1)$, $(a_i)$ are $k$ given points in $(0,1)$, with $k\geq 2$,  $(f_i)$ are $k$ given real numbers, and $\alpha \geq0$ is a parameter taken to be $0$ or $1$ for simplicity. The natural functional setting is the Sobolev space $W^{1,1}(0,1)$. When $\alpha=0$ the Inf is achieved in $W^{1,1}(0,1)$. However, when $\alpha =1$, minimizers need not exist in $W^{1,1} (0,1)$. One is led to introduce a relaxed functional defined on the space $BV(0,1)$, whose minimizers always exist and can be viewed as generalized solutions of the original ill-posed problem.