{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T12:37:53Z","timestamp":1773232673026,"version":"3.50.1"},"reference-count":10,"publisher":"Wiley","license":[{"start":{"date-parts":[[2012,1,1]],"date-time":"2012-01-01T00:00:00Z","timestamp":1325376000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2012]]},"abstract":"<jats:p>Various investigators such as Khan (1974), Chandra (2002), and Liendler (2005) have determined the degree of approximation of 2<jats:italic>\u03c0<\/jats:italic>-periodic signals (functions) belonging to Lip<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mi>\u03b1<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>class of functions through trigonometric Fourier approximation using different summability matrices with monotone rows. Recently, Mittal et al. (2007 and 2011) have obtained the degree of approximation of signals belonging to Lip<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mi>\u03b1<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>- class by general summability matrix, which generalize some of the results of Chandra (2002) and results of Leindler (2005), respectively. In this paper, we determine the degree of approximation of functions belonging to Lip\u2009<jats:italic>\u03b1<\/jats:italic>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:math>(<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:msup><mml:mi>L<\/mml:mi><mml:mi>r<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math>,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi>\u03be<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>) classes by using Ces\u00e1ro-N\u00f6rlund<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:msup><mml:mi>C<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msup><mml:mo>\u00b7<\/mml:mo><mml:msub><mml:mi>N<\/mml:mi><mml:mi>p<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>summability without monotonicity condition on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:mo stretchy=\"false\">{<\/mml:mo><mml:mrow><mml:msub><mml:mi>p<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:mrow><\/mml:math>, which in turn generalizes the results of Lal (2009). We also note some errors appearing in the paper of Lal (2009) and rectify them in the light of observations of Rhoades et al. (2011).<\/jats:p>","DOI":"10.1155\/2012\/964101","type":"journal-article","created":{"date-parts":[[2012,6,27]],"date-time":"2012-06-27T10:55:09Z","timestamp":1340794509000},"page":"1-11","source":"Crossref","is-referenced-by-count":5,"title":["Trigonometric Approximation of Signals (Functions) Belonging to<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>W<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:msup><mml:mi>L<\/mml:mi><mml:mi>r<\/mml:mi><\/mml:msup><mml:mtext>,<\/mml:mtext><mml:mi>\u2009\u03be<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>Class by Matrix<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:msup><mml:mi>C<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msup><mml:mo>\u00b7<\/mml:mo><mml:msub><mml:mi>N<\/mml:mi><mml:mi>p<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>Operator"],"prefix":"10.1155","volume":"2012","author":[{"given":"Uaday","family":"Singh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India"}]},{"given":"M. L.","family":"Mittal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India"}]},{"given":"Smita","family":"Sonker","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India"}]}],"member":"311","reference":[{"issue":"9","key":"3","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1002\/(SICI)1096-9845(199609)25:9<909::AID-EQE591>3.0.CO;2-M","volume":"25","year":"1996","journal-title":"Earthquake Engineering and Structural Dynamics"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/S0965-9978(96)00027-0"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-247X(02)00211-1"},{"issue":"2","key":"5","first-page":"132","volume":"5","year":"1974","journal-title":"Indian Journal of Pure and Applied Mathematics"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2004.07.049"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.03.053"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2010.10.051"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2008.12.051"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2011.01.034"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-42-00913-X"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2012\/964101.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2012\/964101.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2012\/964101.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,20]],"date-time":"2017-06-20T16:36:45Z","timestamp":1497976605000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2012\/964101\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012]]},"references-count":10,"alternative-id":["964101","964101"],"URL":"https:\/\/doi.org\/10.1155\/2012\/964101","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012]]}}}