Approximation and the Multidimensional Moment Problem
The aim of this paper is to apply polynomial approximation by sums of squares in several real variables to the multidimensional moment problem. The general idea is to approximate any element of the positive cone of the involved function space with sums whose terms are squares of polynomials. First,...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/59 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832589115121467392 |
|---|---|
| author | Octav Olteanu |
| author_facet | Octav Olteanu |
| author_sort | Octav Olteanu |
| collection | DOAJ |
| description | The aim of this paper is to apply polynomial approximation by sums of squares in several real variables to the multidimensional moment problem. The general idea is to approximate any element of the positive cone of the involved function space with sums whose terms are squares of polynomials. First, approximations on a Cartesian product of intervals by polynomials taking nonnegative values on the entire <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula>, or on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></semantics></math></inline-formula>, are considered. Such results are discussed in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></semantics></math></inline-formula> and in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mfenced separators="|"><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfenced></mrow></semantics></math></inline-formula>-type spaces, for a large class of measures, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>,</mo></mrow></semantics></math></inline-formula> for compact subsets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mo> </mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula> of the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula>. Thus, on such subsets, any nonnegative function is a limit of sums of squares. Secondly, applications to the bidimensional moment problem are derived in terms of quadratic expressions. As is well known, in multidimensional cases, such results are difficult to prove. Directions for future work are also outlined. |
| format | Article |
| id | doaj-art-5ae620caad5546e9a4a96264a188ee12 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-5ae620caad5546e9a4a96264a188ee122025-01-24T13:22:17ZengMDPI AGAxioms2075-16802025-01-011415910.3390/axioms14010059Approximation and the Multidimensional Moment ProblemOctav Olteanu0Independent Researcher, 060042 Bucharest, RomaniaThe aim of this paper is to apply polynomial approximation by sums of squares in several real variables to the multidimensional moment problem. The general idea is to approximate any element of the positive cone of the involved function space with sums whose terms are squares of polynomials. First, approximations on a Cartesian product of intervals by polynomials taking nonnegative values on the entire <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula>, or on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></semantics></math></inline-formula>, are considered. Such results are discussed in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></semantics></math></inline-formula> and in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mfenced separators="|"><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfenced></mrow></semantics></math></inline-formula>-type spaces, for a large class of measures, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>,</mo></mrow></semantics></math></inline-formula> for compact subsets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mo> </mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula> of the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula>. Thus, on such subsets, any nonnegative function is a limit of sums of squares. Secondly, applications to the bidimensional moment problem are derived in terms of quadratic expressions. As is well known, in multidimensional cases, such results are difficult to prove. Directions for future work are also outlined.https://www.mdpi.com/2075-1680/14/1/59polynomial approximationseveral dimensionssums of squaresmoment problemdeterminate measure |
| spellingShingle | Octav Olteanu Approximation and the Multidimensional Moment Problem Axioms polynomial approximation several dimensions sums of squares moment problem determinate measure |
| title | Approximation and the Multidimensional Moment Problem |
| title_full | Approximation and the Multidimensional Moment Problem |
| title_fullStr | Approximation and the Multidimensional Moment Problem |
| title_full_unstemmed | Approximation and the Multidimensional Moment Problem |
| title_short | Approximation and the Multidimensional Moment Problem |
| title_sort | approximation and the multidimensional moment problem |
| topic | polynomial approximation several dimensions sums of squares moment problem determinate measure |
| url | https://www.mdpi.com/2075-1680/14/1/59 |
| work_keys_str_mv | AT octavolteanu approximationandthemultidimensionalmomentproblem |