Volume 35, Issue 4
Regularized Interpolation Driven by Total Variation

Anal. Theory Appl., 35 (2019), pp. 335-354.

Published online: 2020-01

[An open-access article; the PDF is free to any online user.]

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• 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.

• Keywords

Interpolation, minimization problems, functions of bounded variation, relaxed functional.

26B30, 49J45, 65D05

brezis@math.rutgers.edu (Haim Brezis)

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@Article{ATA-35-335, author = {Brezis , Haim}, title = {Regularized Interpolation Driven by Total Variation}, journal = {Analysis in Theory and Applications}, year = {2020}, volume = {35}, number = {4}, pages = {335--354}, 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.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0017}, url = {http://global-sci.org/intro/article_detail/ata/13616.html} }
TY - JOUR T1 - Regularized Interpolation Driven by Total Variation AU - Brezis , Haim JO - Analysis in Theory and Applications VL - 4 SP - 335 EP - 354 PY - 2020 DA - 2020/01 SN - 35 DO - http://doi.org/10.4208/ata.OA-0017 UR - https://global-sci.org/intro/article_detail/ata/13616.html KW - Interpolation, minimization problems, functions of bounded variation, relaxed functional. AB -

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.

Haim Brezis. (2020). Regularized Interpolation Driven by Total Variation. Analysis in Theory and Applications. 35 (4). 335-354. doi:10.4208/ata.OA-0017
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