{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T09:39:02Z","timestamp":1773999542708,"version":"3.50.1"},"reference-count":36,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T00:00:00Z","timestamp":1752624000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,3,26]]},"abstract":"Abstract<\/jats:title>\n \n Let\n \n \n \n \n \n \n {<\/m:mo>\n \n \n \n \u039b<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n =<\/m:mo>\n \n {<\/m:mo>\n \n \n \n \u03bb<\/m:mi>\n <\/m:mrow>\n \n 1<\/m:mn>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n \u2026<\/m:mo>\n ,<\/m:mo>\n \n \n \u03bb<\/m:mi>\n <\/m:mrow>\n \n \n \n d<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:math>\n ${\\left\\{{{\\Lambda}}_{n}=\\left\\{{\\lambda }_{1,n},\\dots ,{\\lambda }_{{d}_{n},n}\\right\\}\\right\\}}_{n}$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n be a sequence of finite multisets of real numbers such that\n d<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>\n \u2192 \u221e as\n n<\/jats:italic>\n \u2192 \u221e, and let\n \n \n \n f<\/m:mi>\n :<\/m:mo>\n \u03a9<\/m:mi>\n \u2282<\/m:mo>\n \n \n R<\/m:mi>\n <\/m:mrow>\n \n d<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n \u2192<\/m:mo>\n R<\/m:mi>\n <\/m:math>\n $f:{\\Omega}\\subset {\\mathbb{R}}^{d}\\to \\mathbb{R}$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n be a Lebesgue measurable function defined on a domain \u03a9 with 0 <\n \u03bc<\/jats:italic>\n \n d<\/jats:italic>\n <\/jats:sub>\n (\u03a9) < \u221e, where\n \u03bc<\/jats:italic>\n \n d<\/jats:italic>\n <\/jats:sub>\n is the Lebesgue measure in\n \n \n \n \n \n R<\/m:mi>\n <\/m:mrow>\n \n d<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n <\/m:math>\n ${\\mathbb{R}}^{d}$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We say that\n \n \n \n \n \n \n {<\/m:mo>\n \n \n \n \u039b<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:math>\n ${\\left\\{{{\\Lambda}}_{n}\\right\\}}_{n}$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n has an asymptotic distribution described by\n f<\/jats:italic>\n , and we write\n \n \n \n \n \n \n {<\/m:mo>\n \n \n \n \u039b<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n \u223c<\/m:mo>\n f<\/m:mi>\n <\/m:math>\n ${\\left\\{{{\\Lambda}}_{n}\\right\\}}_{n}\\sim f$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n , if (*)\n \n \n \n \n \n lim<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n \u2192<\/m:mo>\n \u221e<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n \n \n 1<\/m:mn>\n <\/m:mrow>\n \n \n \n d<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n <\/m:mfrac>\n \n \n \u2211<\/m:mo>\n <\/m:mrow>\n \n i<\/m:mi>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n \n \n \n d<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n <\/m:msubsup>\n F<\/m:mi>\n \n (<\/m:mo>\n \n \n \n \u03bb<\/m:mi>\n <\/m:mrow>\n \n i<\/m:mi>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n \n \n 1<\/m:mn>\n <\/m:mrow>\n \n \n \n \u03bc<\/m:mi>\n <\/m:mrow>\n \n d<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n \n (<\/m:mo>\n \n \u03a9<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mfrac>\n \n \n \u222b<\/m:mo>\n <\/m:mrow>\n \n \u03a9<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n F<\/m:mi>\n \n (<\/m:mo>\n \n f<\/m:mi>\n \n (<\/m:mo>\n \n x<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n d<\/m:mi>\n x<\/m:mi>\n <\/m:math>\n ${\\mathrm{lim}}_{n\\to \\infty }\\frac{1}{{d}_{n}}{\\sum }_{i=1}^{{d}_{n}}F\\left({\\lambda }_{i,n}\\right)=\\frac{1}{{\\mu }_{d}\\left({\\Omega}\\right)}{\\int }_{{\\Omega}}F\\left(f\\left(\\boldsymbol{x}\\right)\\right)\\mathrm{d}\\boldsymbol{x}$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n for every continuous function\n F<\/jats:italic>\n with bounded support. If \u039b\n \n n<\/jats:italic>\n <\/jats:sub>\n is the spectrum of a matrix\n A<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>\n , we say that\n \n \n \n \n \n \n {<\/m:mo>\n \n \n \n A<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:math>\n ${\\left\\{{A}_{n}\\right\\}}_{n}$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n has an asymptotic spectral distribution described by\n f<\/jats:italic>\n and we write\n \n \n \n \n \n \n {<\/m:mo>\n \n \n \n A<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n \n \n \n \u223c<\/m:mo>\n <\/m:mrow>\n \n \u03bb<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n \n f<\/m:mi>\n <\/m:math>\n ${\\left\\{{A}_{n}\\right\\}}_{n}{ \\sim }_{\\lambda } f$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In the case where\n d<\/jats:italic>\n = 1, \u03a9 is a bounded interval, \u039b\n \n n<\/jats:italic>\n <\/jats:sub>\n \u2286\n f<\/jats:italic>\n (\u03a9) for all\n n<\/jats:italic>\n , and\n f<\/jats:italic>\n satisfies suitable conditions, Bogoya, B\u00f6ttcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to 0 of the difference between the properly sorted vector\n \n \n \n \n [<\/m:mo>\n \n \n \n \u03bb<\/m:mi>\n <\/m:mrow>\n \n 1<\/m:mn>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n \u2026<\/m:mo>\n ,<\/m:mo>\n \n \n \u03bb<\/m:mi>\n <\/m:mrow>\n \n \n \n d<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n ]<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n $\\left[{\\lambda }_{1,n},\\dots ,{\\lambda }_{{d}_{n},n}\\right]$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n and the vector of samples\n \n \n \n \n [<\/m:mo>\n \n f<\/m:mi>\n \n (<\/m:mo>\n \n \n \n x<\/m:mi>\n <\/m:mrow>\n \n 1<\/m:mn>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n ,<\/m:mo>\n \u2026<\/m:mo>\n ,<\/m:mo>\n f<\/m:mi>\n \n (<\/m:mo>\n \n \n \n x<\/m:mi>\n <\/m:mrow>\n \n \n \n d<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n ]<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n $\\left[f\\left({x}_{1,n}\\right),\\dots ,f\\left({x}_{{d}_{n},n}\\right)\\right]$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n , i.e., (**)\n \n \n \n \n \n lim<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n \u2192<\/m:mo>\n \u221e<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n \n \n \n max<\/m:mi>\n <\/m:mrow>\n \n i<\/m:mi>\n =<\/m:mo>\n 1<\/m:mn>\n ,<\/m:mo>\n \u2026<\/m:mo>\n ,<\/m:mo>\n \n \n d<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n <\/m:msub>\n |<\/m:mo>\n f<\/m:mi>\n \n (<\/m:mo>\n \n \n \n x<\/m:mi>\n <\/m:mrow>\n \n i<\/m:mi>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \u2212<\/m:mo>\n \n \n \u03bb<\/m:mi>\n <\/m:mrow>\n \n \n \n \u03c4<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n \n (<\/m:mo>\n \n i<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n |<\/m:mo>\n =<\/m:mo>\n 0<\/m:mn>\n ,<\/m:mo>\n <\/m:math>\n ${\\mathrm{lim}}_{n\\to \\infty } {\\mathrm{max}}_{i=1,\\dots ,{d}_{n}}\\vert f\\left({x}_{i,n}\\right)-{\\lambda }_{{\\tau }_{n}\\left(i\\right),n}\\vert =0,$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n where\n \n \n \n \n \n x<\/m:mi>\n <\/m:mrow>\n \n 1<\/m:mn>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n \u2026<\/m:mo>\n ,<\/m:mo>\n \n \n x<\/m:mi>\n <\/m:mrow>\n \n \n \n d<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:math>\n ${x}_{1,n},\\dots ,{x}_{{d}_{n},n}$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a uniform grid in \u03a9 and\n \u03c4<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>\n is the sorting permutation. We extend this result to the case where\n d<\/jats:italic>\n \u2a7e 1 and \u03a9 is a Peano\u2013Jordan measurable set (i.e., a bounded set with\n \u03bc<\/jats:italic>\n \n d<\/jats:italic>\n <\/jats:sub>\n (\u2202\u03a9) = 0). We also formulate and prove a uniform convergence result analogous to (**) in the more general case where the function\n f<\/jats:italic>\n takes values in the space of\n k<\/jats:italic>\n \u00a0\u00d7\u00a0\n k<\/jats:italic>\n matrices. Our derivations are based on the concept of monotone rearrangement (quantile function) as well as on matrix analysis arguments stemming from the theory of generalized locally Toeplitz sequences and the observation that any finite multiset of numbers\n \n \n \n \n \n \u039b<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n =<\/m:mo>\n \n {<\/m:mo>\n \n \n \n \u03bb<\/m:mi>\n <\/m:mrow>\n \n 1<\/m:mn>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n \u2026<\/m:mo>\n ,<\/m:mo>\n \n \n \u03bb<\/m:mi>\n <\/m:mrow>\n \n \n \n d<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n ${{\\Lambda}}_{n}=\\left\\{{\\lambda }_{1,n},\\dots ,{\\lambda }_{{d}_{n},n}\\right\\}$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n can always be interpreted as the spectrum of a matrix\n \n \n \n \n \n A<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n =<\/m:mo>\n diag<\/m:mi>\n \n (<\/m:mo>\n \n \n \n \u03bb<\/m:mi>\n <\/m:mrow>\n \n 1<\/m:mn>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n \u2026<\/m:mo>\n ,<\/m:mo>\n \n \n \u03bb<\/m:mi>\n <\/m:mrow>\n \n \n \n d<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n ${A}_{n}=\\mathrm{diag}\\left({\\lambda }_{1,n},\\dots ,{\\lambda }_{{d}_{n},n}\\right)$<\/jats:tex-math>\n \n <\/jats:alternatives>\n <\/jats:inline-formula>\n . The theoretical results are illustrated through numerical experiments, and a reinterpretation of them in terms of vague convergence of probability measures is hinted.\n <\/jats:p>","DOI":"10.1515\/jnma-2023-0091","type":"journal-article","created":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T11:56:51Z","timestamp":1752667011000},"page":"21-60","source":"Crossref","is-referenced-by-count":2,"title":["From asymptotic distribution and vague convergence to uniform convergence, with numerical applications"],"prefix":"10.1515","volume":"34","author":[{"given":"Giovanni","family":"Barbarino","sequence":"first","affiliation":[{"name":"Department of Mathematics and Operations Research , University of Mons , Mons , Belgium"}]},{"given":"Sven-Erik","family":"Ekstr\u00f6m","sequence":"additional","affiliation":[{"name":"Division of Scientific Computing, Department of Information Technology , Uppsala University , Uppsala , Sweden"}]},{"given":"Carlo","family":"Garoni","sequence":"additional","affiliation":[{"name":"Department of Mathematics , University of Rome Tor Vergata , Roma , Italy"}]},{"given":"David","family":"Meadon","sequence":"additional","affiliation":[{"name":"Division of Scientific Computing, Department of Information Technology , Uppsala University , Uppsala , Sweden"}]},{"given":"Stefano","family":"Serra-Capizzano","sequence":"additional","affiliation":[{"name":"Division of Scientific Computing, Department of Information Technology , Uppsala University , Uppsala , Sweden"},{"name":"Department of Science and High Technology , University of Insubria , Como , Italy"}]},{"given":"Paris","family":"Vassalos","sequence":"additional","affiliation":[{"name":"Department of Informatics , Athens University of Economics and Business , Athens , Greece"}]}],"member":"374","published-online":{"date-parts":[[2025,7,16]]},"reference":[{"key":"2026030615521524879_j_jnma-2023-0091_ref_001","doi-asserted-by":"crossref","unstructured":"C. Garoni and S. Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications, vol.\u00a0I, Cham, Springer, 2017.","DOI":"10.1007\/978-3-319-53679-8"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_002","doi-asserted-by":"crossref","unstructured":"C. Garoni and S. Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications, vol.\u00a0II, Cham, Springer, 2018.","DOI":"10.1007\/978-3-030-02233-4"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_003","doi-asserted-by":"crossref","unstructured":"G. Barbarino, \u201cA systematic approach to reduced GLT,\u201d BIT Numer. Math., vol.\u00a062, pp.\u00a0681\u2013743, 2022, https:\/\/doi.org\/10.1007\/s10543-021-00896-7.","DOI":"10.1007\/s10543-021-00896-7"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_004","doi-asserted-by":"crossref","unstructured":"G. Barbarino, C. Garoni, M. Mazza, and S. Serra-Capizzano, \u201cRectangular GLT sequences,\u201d Electron. Trans. Numer. Anal., vol.\u00a055, pp.\u00a0585\u2013617, 2022, https:\/\/doi.org\/10.1553\/etna_vol55s585.","DOI":"10.1553\/etna_vol55s585"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_005","doi-asserted-by":"crossref","unstructured":"G. Barbarino, C. Garoni, and S. Serra-Capizzano, \u201cBlock generalized locally Toeplitz sequences: theory and applications in the unidimensional case,\u201d Electron. Trans. Numer. Anal., vol.\u00a053, pp.\u00a028\u2013112, 2020, https:\/\/doi.org\/10.1553\/etna_vol53s28.","DOI":"10.1553\/etna_vol53s28"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_006","doi-asserted-by":"crossref","unstructured":"G. Barbarino, C. Garoni, and S. Serra-Capizzano, \u201cBlock generalized locally Toeplitz sequences: theory and applications in the multidimensional case,\u201d Electron. Trans. Numer. Anal., vol.\u00a053, pp.\u00a0113\u2013216, 2020, https:\/\/doi.org\/10.1553\/etna_vol53s113.","DOI":"10.1553\/etna_vol53s113"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_007","doi-asserted-by":"crossref","unstructured":"A. B\u00f6ttcher, C. Garoni, and S. Serra-Capizzano, \u201cExploration of Toeplitz-like matrices with unbounded symbols is not a purely academic journey,\u201d Sb. Math., vol.\u00a0208, pp.\u00a01602\u20131627, 2017, https:\/\/doi.org\/10.1070\/sm8823.","DOI":"10.1070\/SM8823"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_008","unstructured":"U. Grenander and G. Szeg\u0151, Toeplitz Forms and Their Applications, 2nd ed. New York, AMS Chelsea Publishing, 1984."},{"key":"2026030615521524879_j_jnma-2023-0091_ref_009","doi-asserted-by":"crossref","unstructured":"F. Avram, \u201cOn bilinear forms in Gaussian random variables and Toeplitz matrices,\u201d Probab. Theory Relat. Fields, vol.\u00a079, pp.\u00a037\u201345, 1988, https:\/\/doi.org\/10.1007\/bf00319101.","DOI":"10.1007\/BF00319101"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_010","doi-asserted-by":"crossref","unstructured":"S. V. Parter, \u201cOn the distribution of the singular values of Toeplitz matrices,\u201d Linear Algebra Appl., vol.\u00a080, pp.\u00a0115\u2013130, 1986, https:\/\/doi.org\/10.1016\/0024-3795(86)90280-6.","DOI":"10.1016\/0024-3795(86)90280-6"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_011","doi-asserted-by":"crossref","unstructured":"E. E. Tyrtyshnikov, \u201cA unifying approach to some old and new theorems on distribution and clustering,\u201d Linear Algebra Appl., vol.\u00a0232, pp.\u00a01\u201343, 1996, https:\/\/doi.org\/10.1016\/0024-3795(94)00025-5.","DOI":"10.1016\/0024-3795(94)00025-5"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_012","doi-asserted-by":"crossref","unstructured":"E. E. Tyrtyshnikov and N. L. Zamarashkin, \u201cSpectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships,\u201d Linear Algebra Appl., vol.\u00a0270, pp.\u00a015\u201327, 1998, https:\/\/doi.org\/10.1016\/s0024-3795(97)80001-8.","DOI":"10.1016\/S0024-3795(97)80001-8"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_013","doi-asserted-by":"crossref","unstructured":"N. L. Zamarashkin and E. E. Tyrtyshnikov, \u201cDistribution of eigenvalues and singular values of Toeplitz matrices under weakened conditions on the generating function,\u201d Sb. Math., vol.\u00a0188, pp.\u00a01191\u20131201, 1997, https:\/\/doi.org\/10.1070\/sm1997v188n08abeh000251.","DOI":"10.1070\/SM1997v188n08ABEH000251"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_014","doi-asserted-by":"crossref","unstructured":"P. Tilli, \u201cA note on the spectral distribution of Toeplitz matrices,\u201d Linear Multilinear Algebra, vol.\u00a045, pp.\u00a0147\u2013159, 1998, https:\/\/doi.org\/10.1080\/03081089808818584.","DOI":"10.1080\/03081089808818584"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_015","doi-asserted-by":"crossref","unstructured":"P. Tilli, \u201cSome results on complex Toeplitz eigenvalues,\u201d J. Math. Anal. Appl., vol.\u00a0239, pp.\u00a0390\u2013401, 1999, https:\/\/doi.org\/10.1006\/jmaa.1999.6572.","DOI":"10.1006\/jmaa.1999.6572"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_016","doi-asserted-by":"crossref","unstructured":"A. B\u00f6ttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, New York, Springer, 1999.","DOI":"10.1007\/978-1-4612-1426-7"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_017","doi-asserted-by":"crossref","unstructured":"S. Serra-Capizzano, \u201cGeneralized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations,\u201d Linear Algebra Appl., vol.\u00a0366, pp.\u00a0371\u2013402, 2003, https:\/\/doi.org\/10.1016\/s0024-3795(02)00504-9.","DOI":"10.1016\/S0024-3795(02)00504-9"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_018","doi-asserted-by":"crossref","unstructured":"S. Serra-Capizzano, \u201cThe GLT class as a generalized Fourier analysis and applications,\u201d Linear Algebra Appl., vol.\u00a0419, pp.\u00a0180\u2013233, 2006, https:\/\/doi.org\/10.1016\/j.laa.2006.04.012.","DOI":"10.1016\/j.laa.2006.04.012"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_019","doi-asserted-by":"crossref","unstructured":"P. Tilli, \u201cLocally Toeplitz sequences: spectral properties and applications,\u201d Linear Algebra Appl., vol.\u00a0278, pp.\u00a091\u2013120, 1998, https:\/\/doi.org\/10.1016\/s0024-3795(97)10079-9.","DOI":"10.1016\/S0024-3795(97)10079-9"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_020","doi-asserted-by":"crossref","unstructured":"D. Bianchi, \u201cAnalysis of the spectral symbol associated to discretization schemes of linear self-adjoint differential operators,\u201d Calcolo, vol.\u00a058, 2021, Art. no.\u00a038, https:\/\/doi.org\/10.1007\/s10092-021-00426-5.","DOI":"10.1007\/s10092-021-00426-5"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_021","doi-asserted-by":"crossref","unstructured":"D. Bianchi and S. Serra-Capizzano, \u201cSpectral analysis of finite-dimensional approximations of 1d waves in non-uniform grids,\u201d Calcolo, vol.\u00a055, 2018, Art. no.\u00a047, https:\/\/doi.org\/10.1007\/s10092-018-0288-x.","DOI":"10.1007\/s10092-018-0288-x"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_022","doi-asserted-by":"crossref","unstructured":"C. Garoni, H. Speleers, S.-E. Ekstr\u00f6m, A. Reali, S. Serra-Capizzano, and T. J. R. Hughes, \u201cSymbol-based analysis of finite element and isogeometric B-spline discretizations of eigenvalue problems: exposition and review,\u201d Arch. Comput. Methods Eng., vol.\u00a026, pp.\u00a01639\u20131690, 2019.","DOI":"10.1007\/s11831-018-9295-y"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_023","doi-asserted-by":"crossref","unstructured":"B. Beckermann and A. B. J. Kuijlaars, \u201cSuperlinear convergence of conjugate gradients,\u201d SIAM J. Numer. Anal., vol.\u00a039, pp.\u00a0300\u2013329, 2001, https:\/\/doi.org\/10.1137\/s0036142999363188.","DOI":"10.1137\/S0036142999363188"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_024","doi-asserted-by":"crossref","unstructured":"A. B. J. Kuijlaars, \u201cConvergence analysis of Krylov subspace iterations with methods from potential theory,\u201d SIAM Rev., vol.\u00a048, pp.\u00a03\u201340, 2006, https:\/\/doi.org\/10.1137\/s0036144504445376.","DOI":"10.1137\/S0036144504445376"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_025","doi-asserted-by":"crossref","unstructured":"J. M. Bogoya, A. B\u00f6ttcher, S. M. Grudsky, and E. A. Maximenko, \u201cMaximum norm\u202fversions of the Szeg\u0151 and Avram\u2013Parter theorems for Toeplitz matrices,\u201d J. Approx. Theory, vol.\u00a0196, pp.\u00a079\u2013100, 2015.","DOI":"10.1016\/j.jat.2015.03.003"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_026","doi-asserted-by":"crossref","unstructured":"J. M. Bogoya, A. B\u00f6ttcher, and E. A. Maximenko, \u201cFrom convergence in distribution to uniform convergence,\u201d Bol. Soc. Mat. Mex., vol.\u00a022, pp.\u00a0695\u2013710, 2016, https:\/\/doi.org\/10.1007\/s40590-016-0105-y.","DOI":"10.1007\/s40590-016-0105-y"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_027","doi-asserted-by":"crossref","unstructured":"B. Fristedt and L. Gray, A Modern Approach to Probability Theory, Boston, Birkh\u00e4user, 1997.","DOI":"10.1007\/978-1-4899-2837-5"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_028","unstructured":"W. Rudin, Principles of Mathematical Analysis, 3rd ed. New York, McGraw-Hill, 1976."},{"key":"2026030615521524879_j_jnma-2023-0091_ref_029","doi-asserted-by":"crossref","unstructured":"G. Barbarino, D. Bianchi, and C. Garoni, \u201cConstructive approach to the monotone rearrangement of functions,\u201d Expo. Math., vol.\u00a040, pp.\u00a0155\u2013175, 2022, https:\/\/doi.org\/10.1016\/j.exmath.2021.10.004.","DOI":"10.1016\/j.exmath.2021.10.004"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_030","doi-asserted-by":"crossref","unstructured":"G. P\u00f3lya and G. Szeg\u0151, Problems and Theorems in Analysis I. Series. Integral Calculus. Theory of Functions, Berlin\u2013Heidelberg, Springer, 1998.","DOI":"10.1007\/978-3-642-61905-2"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_031","doi-asserted-by":"crossref","unstructured":"R. Bhatia, Matrix Analysis, New York, Springer, 1997.","DOI":"10.1007\/978-1-4612-0653-8"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_032","doi-asserted-by":"crossref","unstructured":"G. Barbarino, \u201cSpectral measures,\u201d in Structured Matrices in Numerical Linear Algebra: Analysis, Algorithms and Applications, Springer INdAM Series, vol.\u00a030, Cham, Springer, 2019, pp.\u00a01\u201324.","DOI":"10.1007\/978-3-030-04088-8_1"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_033","doi-asserted-by":"crossref","unstructured":"A. B\u00f6ttcher and S. M. Grudsky, Spectral Properties of Banded Toeplitz Matrices, Philadelphia, SIAM, 2005.","DOI":"10.1137\/1.9780898717853"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_034","doi-asserted-by":"crossref","unstructured":"D. Noutsos, S. Serra-Capizzano, and P. Vassalos, \u201cThe conditioning of FD matrix sequences coming from semi-elliptic differential equations,\u201d Linear Algebra Appl., vol.\u00a0428, pp.\u00a0600\u2013624, 2008, https:\/\/doi.org\/10.1016\/j.laa.2007.08.008.","DOI":"10.1016\/j.laa.2007.08.008"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_035","doi-asserted-by":"crossref","unstructured":"S.-E. Ekstr\u00f6m, I. Furci, C. Garoni, C. Manni, S. Serra-Capizzano, and H. Speleers, \u201cAre the eigenvalues of the B-spline isogeometric analysis approximation of \u2212\u0394u = \u03bbu known in almost closed form?\u201d Numer. Linear Algebra Appl., vol.\u00a025, 2018, Art. no.\u00a0e2198.","DOI":"10.1002\/nla.2198"},{"key":"2026030615521524879_j_jnma-2023-0091_ref_036","unstructured":"A. Klenke, Probability Theory: A Comprehensive Course, London, Springer, 2008."}],"container-title":["Journal of Numerical Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jnma-2023-0091\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jnma-2023-0091\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T15:52:41Z","timestamp":1772812361000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jnma-2023-0091\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,16]]},"references-count":36,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,7,16]]},"published-print":{"date-parts":[[2026,3,26]]}},"alternative-id":["10.1515\/jnma-2023-0091"],"URL":"https:\/\/doi.org\/10.1515\/jnma-2023-0091","relation":{},"ISSN":["1570-2820","1569-3953"],"issn-type":[{"value":"1570-2820","type":"print"},{"value":"1569-3953","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,16]]}}}