{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T08:03:23Z","timestamp":1768550603952,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,14]],"date-time":"2025-01-14T00:00:00Z","timestamp":1736812800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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 R2, or on R+2, are considered. Such results are discussed in L\u03bc1R2 and in CS1\u00d7S2-type spaces, for a large class of measures, \u03bc, for compact subsets Si,\u00a0i=1,2 of the interval [0,+\u221e). 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.<\/jats:p>","DOI":"10.3390\/axioms14010059","type":"journal-article","created":{"date-parts":[[2025,1,14]],"date-time":"2025-01-14T06:13:07Z","timestamp":1736835187000},"page":"59","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Approximation and the Multidimensional Moment Problem"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7059-9387","authenticated-orcid":false,"given":"Octav","family":"Olteanu","sequence":"first","affiliation":[{"name":"Independent Researcher, 060042 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Choudary, A.D.R., and Niculescu, C.P. (2014). Real Analysis on Intervals, Springer.","DOI":"10.1007\/978-81-322-2148-7"},{"key":"ref_2","unstructured":"Rudin, W. (1987). Real and Complex Analysis, McGraw-Hill Book Company. [3rd ed.]."},{"key":"ref_3","unstructured":"Rudin, W. (1991). Functional Analysis, McGraw-Hill, Inc. International Editions. [2nd ed.]."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Schaefer, H.H., and Wolff, M.P. (1999). Topological Vector Spaces, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4612-1468-7"},{"key":"ref_5","unstructured":"Cristescu, R. (1976). Ordered Vector Spaces and Linear Operators, Academiei, Bucharest, and Abacus Press."},{"key":"ref_6","unstructured":"Boboc, N., and Bucur, G. (1976). Convex Cones of Continuous Functions on Compact Spaces, Academiei. (In Romanian)."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bucur, I., and Paltineanu, G. (2020). Topics in Uniform Approximation of Continuous Functions, Birkhauser.","DOI":"10.1007\/978-3-030-48412-5"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Schm\u00fcdgen, K. (2017). The Moment Problem. Graduate Texts in Mathematics, Springer International Publishing.","DOI":"10.1007\/978-3-319-64546-9"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/BF01420423","article-title":"A remark on the multidimensional moment problem","volume":"243","author":"Berg","year":"1979","journal-title":"Math. Ann."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/0022-1236(84)90042-9","article-title":"Moment problem on a compact subset of \n\t\t\tR2 and decomposition of polynomials of several variables","volume":"58","author":"Cassier","year":"1984","journal-title":"J. Funct. Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/BF01446568","article-title":"The K-moment problem for compact semi-algebraic sets","volume":"289","year":"1991","journal-title":"Math. Ann."},{"key":"ref_12","first-page":"969","article-title":"Positive polynomials on compact semi-algebraic sets","volume":"42","author":"Putinar","year":"1993","journal-title":"IU Math. J."},{"key":"ref_13","first-page":"787","article-title":"The moment problem on semi-algebraic compacts","volume":"323","author":"Putinar","year":"1996","journal-title":"Comptes Rendus Acad. Sci. Paris Ser. I"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2489","DOI":"10.1090\/S0002-9947-2012-05701-1","article-title":"The K-moment problem for continuous linear functionals","volume":"365","author":"Lasserre","year":"2013","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1090\/S0002-9939-1991-1059628-5","article-title":"An operator-valued moment problem","volume":"112","author":"Lemnete","year":"1991","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1137\/S0040585X97T990083","article-title":"New Checkable Conditions for Moment Determinacy of Probability Distributions","volume":"65","author":"Stoyanov","year":"2020","journal-title":"Theory Probab. Its Appl."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Inverardi, P.L.N., and Tagliani, A. (2021). Stieltjes and Hamburger Reduced Moment Problem When MaxEnt Solution Does Not Exist. Mathematics, 9.","DOI":"10.3390\/math9040309"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Inverardi, P.L.N., Tagliani, A., and Stoyanov, J.M. (2023). The Problem of Moments: A Bunch of Classical Results with Some Novelties. Symmetry, 15.","DOI":"10.3390\/sym15091743"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Novi Inverardi, P.L., and Tagliani, A. (2024). Probability Distributions Approximation via Fractional Moments and Maximum Entropy: Theoretical and Computational Aspects. Axioms, 13.","DOI":"10.3390\/axioms13010028"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"109674","DOI":"10.1016\/j.jfa.2022.109674","article-title":"Self-adjoint operators associated with Hankel moment matrices","volume":"283","author":"Berg","year":"2022","journal-title":"J. Funct. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s11785-023-01339-7","article-title":"Stability in Truncated Trigonometric Scalar Moment Problems","volume":"17","year":"2023","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s13398-020-00905-4","article-title":"From the Hahn\u2013Banach extension theorem to the isotonicity of convex functions and the majorization theory","volume":"114","author":"Niculescu","year":"2020","journal-title":"Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A Mat."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zarban, A.A., Colwell, D., and Salopek, D.M. (2024). Pricing a Defaultable Zero-Coupon Bond under Imperfect Information and Regime Switching. Mathematics, 12.","DOI":"10.3390\/math12172740"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Li, C., and Chu, W. (2024). Generating Functions for Binomial Series Involving Harmonic-like Numbers. Mathematics, 12.","DOI":"10.3390\/math12172685"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1007\/s40314-024-02946-6","article-title":"A note on a general sequence of \u03bb-Sz\u00e1sz Kantorovich type operators","volume":"43","author":"Rao","year":"2024","journal-title":"Comp. Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"100168","DOI":"10.1016\/j.kjs.2023.12.007","article-title":"Rate of approximation of blending type modified univariate and bivariate \u03bb-Schurer-Kantorovich operators","volume":"51","author":"Aslan","year":"2024","journal-title":"Kuwait J. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.aej.2024.07.015","article-title":"Approximation by a new Stancu variant of generalized (\u03bb, \u03bc)- Bern- stein operators","volume":"107","author":"Aslan","year":"2024","journal-title":"Alex. Eng. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1080\/13873954.2024.2335382","article-title":"Kantorovich-Stancu type (\u03b1,\u03bb,s)\u2014Bernstein operators and their approximation properties","volume":"30","author":"Turhan","year":"2024","journal-title":"Math. Comput. Model. Dyn. Syst."},{"key":"ref_29","first-page":"739","article-title":"Applications of theorems on extension of linear operators to the moment problem and to a generalization of Mazur-Orlicz theorem","volume":"313","author":"Olteanu","year":"1991","journal-title":"C. R. Acad. Sci. Paris"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Olteanu, O. (2021). On Markov Moment Problem and Related Results. Symmetry, 13.","DOI":"10.3390\/sym13060986"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1515\/math-2022-0001","article-title":"On Hahn-Banach theorem and some of its applications","volume":"20","author":"Olteanu","year":"2022","journal-title":"Open Math."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Olteanu, O. (2024). Applications of the Hahn-Banach Theorem, a Solution of the Moment Problem and the Related Approximation. Mathematics, 12.","DOI":"10.3390\/math12182878"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/59\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:28:24Z","timestamp":1759919304000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/59"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,14]]},"references-count":32,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010059"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010059","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,14]]}}}