## On the Lebesgue constant of weighted Leja points for Lagrange interpolation on unbounded domains

## Abstract

Here, this work focuses on weighted Lagrange interpolation on an unbounded domain and analyzes the Lebesgue constant for a sequence of weighted Leja points. The standard Leja points are a nested sequence of points defined on a compact subset of the real line and can be extended to unbounded domains with the introduction of a weight function w:R → [0,1]. Due to a simple recursive formulation in one dimension, such abscissas provide a foundation for high-dimensional approximation methods such as sparse grid collocation, deterministic least squares and compressed sensing. Just as in the unweighted case of interpolation on a compact domain, we use results from potential theory to prove that the Lebesgue constant for the Leja points grows subexponentially with the number of interpolation nodes.

- Authors:

- Univ. of Tennessee, Knoxville, TN (United States)
- Univ. of Tennessee, Knoxville, TN (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1559607

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Accepted Manuscript

- Journal Name:
- IMA Journal of Numerical Analysis

- Additional Journal Information:
- Journal Volume: 39; Journal Issue: 2; Journal ID: ISSN 0272-4979

- Publisher:
- Oxford University Press/Institute of Mathematics and its Applications

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; weighted Leja sequence; Lagrange interpolation; Lebesgue constant

### Citation Formats

```
Jantsch, Peter, Webster, Clayton G., and Zhang, Guannan. On the Lebesgue constant of weighted Leja points for Lagrange interpolation on unbounded domains. United States: N. p., 2018.
Web. doi:10.1093/imanum/dry002.
```

```
Jantsch, Peter, Webster, Clayton G., & Zhang, Guannan. On the Lebesgue constant of weighted Leja points for Lagrange interpolation on unbounded domains. United States. doi:10.1093/imanum/dry002.
```

```
Jantsch, Peter, Webster, Clayton G., and Zhang, Guannan. Sat .
"On the Lebesgue constant of weighted Leja points for Lagrange interpolation on unbounded domains". United States. doi:10.1093/imanum/dry002. https://www.osti.gov/servlets/purl/1559607.
```

```
@article{osti_1559607,
```

title = {On the Lebesgue constant of weighted Leja points for Lagrange interpolation on unbounded domains},

author = {Jantsch, Peter and Webster, Clayton G. and Zhang, Guannan},

abstractNote = {Here, this work focuses on weighted Lagrange interpolation on an unbounded domain and analyzes the Lebesgue constant for a sequence of weighted Leja points. The standard Leja points are a nested sequence of points defined on a compact subset of the real line and can be extended to unbounded domains with the introduction of a weight function w:R → [0,1]. Due to a simple recursive formulation in one dimension, such abscissas provide a foundation for high-dimensional approximation methods such as sparse grid collocation, deterministic least squares and compressed sensing. Just as in the unweighted case of interpolation on a compact domain, we use results from potential theory to prove that the Lebesgue constant for the Leja points grows subexponentially with the number of interpolation nodes.},

doi = {10.1093/imanum/dry002},

journal = {IMA Journal of Numerical Analysis},

number = 2,

volume = 39,

place = {United States},

year = {2018},

month = {6}

}