{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T12:58:46Z","timestamp":1774875526080,"version":"3.50.1"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"214","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>This paper gives a practical method of extending an <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n times r\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo>\u00d7<\/mml:mo>\n      <mml:mi>r<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">n\\times r<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> matrix <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P left-parenthesis z right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>P<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>z<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">P(z)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r less-than-or-equal-to n\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>r<\/mml:mi>\n      <mml:mo>\u2264<\/mml:mo>\n      <mml:mi>n<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">r \\leq n<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, with Laurent polynomial entries in one complex variable <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"z\">\n  <mml:semantics>\n    <mml:mi>z<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">z<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, to a square matrix also with Laurent polynomial entries. If <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P left-parenthesis z right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>P<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>z<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">P(z)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> has orthonormal columns when <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"z\">\n  <mml:semantics>\n    <mml:mi>z<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">z<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is restricted to the torus <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper T\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:mi mathvariant=\"bold\">T<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbf {T}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, it can be extended to a paraunitary matrix. If <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P left-parenthesis z right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>P<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>z<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">P(z)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> has rank <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r\">\n  <mml:semantics>\n    <mml:mi>r<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">r<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> for each <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"z element-of bold upper T\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>z<\/mml:mi>\n      <mml:mo>\u2208<\/mml:mo>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"bold\">T<\/mml:mi>\n      <\/mml:mrow>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">z\\in \\mathbf {T}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, it can be extended to a matrix with nonvanishing determinant on <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper T\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:mi mathvariant=\"bold\">T<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbf {T}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.<\/p>","DOI":"10.1090\/s0025-5718-96-00714-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:14:28Z","timestamp":1027721668000},"page":"723-737","source":"Crossref","is-referenced-by-count":93,"title":["An algorithm for matrix extension and wavelet construction"],"prefix":"10.1090","volume":"65","author":[{"given":"W.","family":"Lawton","sequence":"first","affiliation":[]},{"given":"S.","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Zuowei","family":"Shen","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"903","DOI":"10.2307\/2153941","article-title":"On compactly supported spline wavelets and a duality principle","volume":"330","author":"Chui, Charles K.","year":"1992","journal-title":"Trans. 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