{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:57:49Z","timestamp":1776790669509,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"267","license":[{"start":{"date-parts":[[2010,1,9]],"date-time":"2010-01-09T00:00:00Z","timestamp":1262995200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    For high dimensional problems, such as approximation and integration, one cannot afford to sample on a grid because of the curse of dimensionality. An attractive alternative is to sample on a low discrepancy set, such as an integration lattice or a digital net. This article introduces a multivariate fast discrete Walsh transform for data sampled on a digital net that requires only\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper O left-parenthesis upper N log upper N right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {O}(N \\log N)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    operations, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the number of data points. This algorithm and its inverse are digital analogs of multivariate fast Fourier transforms.\n                  <\/p>\n                  <p>This fast discrete Walsh transform and its inverse may be used to approximate the Walsh coefficients of a function and then construct a spline interpolant of the function. This interpolant may then be used to estimate the function\u2019s effective dimension, an important concept in the theory of numerical multivariate integration. Numerical results for various functions are presented.<\/p>","DOI":"10.1090\/s0025-5718-09-02202-9","type":"journal-article","created":{"date-parts":[[2009,4,27]],"date-time":"2009-04-27T13:47:33Z","timestamp":1240840053000},"page":"1573-1591","source":"Crossref","is-referenced-by-count":2,"title":["A multivariate fast discrete Walsh transform with an application to function interpolation"],"prefix":"10.1090","volume":"78","author":[{"given":"Kwong-Ip","family":"Liu","sequence":"first","affiliation":[]},{"given":"Josef","family":"Dick","sequence":"additional","affiliation":[]},{"given":"Fred","family":"Hickernell","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,9]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"337","DOI":"10.2307\/1990404","article-title":"Theory of reproducing kernels","volume":"68","author":"Aronszajn, N.","year":"1950","journal-title":"Trans. Amer. Math. 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