{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:44:51Z","timestamp":1776725091989,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"231","license":[{"start":{"date-parts":[[2000,8,25]],"date-time":"2000-08-25T00:00:00Z","timestamp":967161600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G left-parenthesis x right-parenthesis equals Start 1 By 3 Matrix 1st Row 1st Column e Superscript i lamda x Baseline upper I Subscript m Baseline 2nd Column a m p semicolon 0 c Subscript negative 1 Baseline e Superscript minus i nu x Baseline plus c 0 plus c 1 e Superscript i alpha x Baseline 3rd Column a m p semicolon e Superscript minus i lamda x Baseline upper I Subscript m Baseline EndMatrix comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\">\n                                <mml:mtr>\n                                  <mml:mtd>\n                                    <mml:msup>\n                                      <mml:mi>e<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>i<\/mml:mi>\n                                        <mml:mi>\n                                          \u03bb\n                                          \n                                        <\/mml:mi>\n                                        <mml:mi>x<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msup>\n                                    <mml:msub>\n                                      <mml:mi>I<\/mml:mi>\n                                      <mml:mi>m<\/mml:mi>\n                                    <\/mml:msub>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mn>0<\/mml:mn>\n                                    <mml:mtext>\u00a0<\/mml:mtext>\n                                    <mml:msub>\n                                      <mml:mi>c<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mo>\n                                          \u2212\n                                          \n                                        <\/mml:mo>\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:msub>\n                                    <mml:msup>\n                                      <mml:mi>e<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mo>\n                                          \u2212\n                                          \n                                        <\/mml:mo>\n                                        <mml:mi>i<\/mml:mi>\n                                        <mml:mi>\n                                          \u03bd\n                                          \n                                        <\/mml:mi>\n                                        <mml:mi>x<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msup>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>c<\/mml:mi>\n                                      <mml:mn>0<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>c<\/mml:mi>\n                                      <mml:mn>1<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:msup>\n                                      <mml:mi>e<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>i<\/mml:mi>\n                                        <mml:mi>\n                                          \u03b1\n                                          \n                                        <\/mml:mi>\n                                        <mml:mi>x<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msup>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:msup>\n                                      <mml:mi>e<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mo>\n                                          \u2212\n                                          \n                                        <\/mml:mo>\n                                        <mml:mi>i<\/mml:mi>\n                                        <mml:mi>\n                                          \u03bb\n                                          \n                                        <\/mml:mi>\n                                        <mml:mi>x<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msup>\n                                    <mml:msub>\n                                      <mml:mi>I<\/mml:mi>\n                                      <mml:mi>m<\/mml:mi>\n                                    <\/mml:msub>\n                                  <\/mml:mtd>\n                                <\/mml:mtr>\n                              <\/mml:mtable>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">G(x)=\\left [\\begin {matrix}e^{i\\lambda x}I_m &amp; 0\\ c_{-1}e^{-i\\nu x}+c_0+c_1 e^{i\\alpha x} &amp; e^{-i\\lambda x}I_m \\end {matrix}\\right ],<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c Subscript j Baseline element-of double-struck upper C Superscript m times m\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mi>j<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>m<\/mml:mi>\n                                <mml:mo>\n                                  \u00d7\n                                  \n                                <\/mml:mo>\n                                <mml:mi>m<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">c_j\\in \\mathbb {C}^{m\\times m}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha comma nu greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03bd\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha ,\\nu &gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha plus nu equals lamda\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>\n                              \u03bd\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha +\\nu =\\lambda<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . For rational\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha slash nu\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>\n                              \u03bd\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha \/\\nu<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such matrices\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are periodic, and their Wiener-Hopf factorization with respect to the real line\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    always exists and can be constructed explicitly. For irrational\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha slash nu\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>\n                              \u03bd\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha \/\\nu<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">c_0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and commuting\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c 1 c 0 Superscript negative 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:msubsup>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">c_1c_0^{-1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c Subscript negative 1 Baseline c 0 Superscript negative 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:msubsup>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">c_{-1}c_0^{-1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    was disposed of earlier\u2014it was discovered that an almost periodic factorization of such matrices\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    does not always exist, and a necessary and sufficient condition for its existence was found. This paper is devoted mostly to the situation when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">c_0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is not invertible but the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c Subscript j\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mi>j<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">c_j<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    commute pairwise (\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"j equals 0 comma plus-or-minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">j=0,\\pm 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ). The complete description is obtained when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m less-than-or-equal-to 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">m\\leq 3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ; for an arbitrary\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , certain conditions are imposed on the Jordan structure of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c Subscript j\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mi>j<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">c_j<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Difficulties arising for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m equals 4\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">m=4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are explained, and a classification of both solved and unsolved cases is given. The main result of the paper (existence criterion) is theoretical; however, a significant part of its proof is a constructive factorization of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in numerous particular cases. These factorizations were obtained using Maple; the code is available from the authors upon request.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01161-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1053-1070","source":"Crossref","is-referenced-by-count":1,"title":["Almost periodic factorization of certain block triangular matrix functions"],"prefix":"10.1090","volume":"69","author":[{"given":"Ilya","family":"Spitkovsky","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Darryl","family":"Yong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,8,25]]},"reference":[{"issue":"8","key":"1","first-page":"859","article-title":"Positive extensions of matrix functions of two variables with support in an infinite band","volume":"323","author":"Bakonyi, Mih\u00e1ly","year":"1996","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"M. A. Bastos, Yu. I. Karlovich, I. M. Spitkovsky, and P. M. Tishin, On a new algorithm for almost periodic factorization, Operator Theory: Advances and Applications 103 (1998), 53\u201374.","DOI":"10.1007\/978-3-0348-8793-9_3"},{"key":"3","series-title":"Interscience Tracts in Pure and Applied Mathematics, No. 22","volume-title":"Almost periodic functions","author":"Corduneanu, C.","year":"1968"},{"issue":"5","key":"4","first-page":"441","article-title":"The functional equation \ud835\udf13(\ud835\udc65+\ud835\udefc)-\ud835\udc4f(\ud835\udc65)\ud835\udf13(\ud835\udc65)=\ud835\udc54(\ud835\udc65)","volume":"5","author":"Karapetjanc, N. K.","year":"1970","journal-title":"Izv. Akad. Nauk Armjan. SSR Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3043","issn-type":"print"},{"issue":"3-4","key":"5","first-page":"581","article-title":"On the Haseman problem","volume":"26","author":"Karlovich, Yu. I.","year":"1993","journal-title":"Demonstratio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0420-1213","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1070\/IM1990v034n02ABEH000646","article-title":"Factorization of almost periodic matrix functions and the Noether theory of certain classes of equations of convolution type","volume":"53","author":"Karlovich, Yu. I.","year":"1989","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0373-2436","issn-type":"print"},{"key":"7","isbn-type":"print","first-page":"122","article-title":"(Semi)-Fredholmness of convolution operators on the spaces of Bessel potentials","author":"Karlovich, Yuri","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3764350687"},{"key":"8","isbn-type":"print","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1090\/conm\/189\/02271","article-title":"Almost periodic factorization: an analogue of Chebotarev\u2019s algorithm","author":"Karlovich, Yuri","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0821803042"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1006\/jmaa.1995.1230","article-title":"Factorization of almost periodic matrix functions","volume":"193","author":"Karlovich, Yuri","year":"1995","journal-title":"J. Math. Anal. 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