{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T18:40:10Z","timestamp":1771699210443,"version":"3.50.1"},"reference-count":12,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"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":[[2014]]},"abstract":"<jats:p>Mittal and Rhoades (1999, 2000) and Mittal et al. (2011) have initiated a study of error estimates<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:msub><mml:mrow><mml:mi>E<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><\/mml:math>does not have monotone rows. In this paper, the first author continues the work in the direction for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><\/mml:math>to be a<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-matrix. We extend two theorems on summability matrix<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>of Deger et al. (2012) where they have extended two theorems of Chandra (2002) using<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03bb<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-method obtained by deleting a set of rows from Ces\u00e0ro matrix<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>. Our theorems also generalize two theorems of Leindler (2005) to<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-matrix which in turn generalize the result of Chandra (2002) and Quade (1937).<\/jats:p>","DOI":"10.1155\/2014\/267383","type":"journal-article","created":{"date-parts":[[2014,4,9]],"date-time":"2014-04-09T21:22:11Z","timestamp":1397078531000},"page":"1-6","source":"Crossref","is-referenced-by-count":8,"title":["Approximation of Signals (Functions) by Trigonometric Polynomials in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-Norm"],"prefix":"10.1155","volume":"2014","author":[{"given":"M. L.","family":"Mittal","sequence":"first","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mradul Veer","family":"Singh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"issue":"1-2","key":"1","first-page":"195","volume":"9","year":"1989","journal-title":"Analysis"},{"issue":"1","key":"11","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1524\/anly.2000.20.1.35","volume":"20","year":"2000","journal-title":"Analysis"},{"issue":"4","key":"6","first-page":"259","volume":"9","year":"1999","journal-title":"International Journal of Mathematics, Game Theory, and Algebra"},{"issue":"1","key":"7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1023\/A:1010193314045","volume":"2","year":"2000","journal-title":"Journal of Computational Analysis and Applications"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-247X(02)00211-1"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-37-00342-9"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1017\/S144678870002317X"},{"issue":"1","key":"2","first-page":"9","volume":"42","year":"1990","journal-title":"Matemati\u010dki Vesnik"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2004.07.049"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.03.053"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2010.10.051"},{"issue":"2","key":"4","first-page":"203","volume":"15","year":"2012","journal-title":"Proceedings of the Jangjeon Mathematical Society"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2014\/267383.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2014\/267383.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2014\/267383.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,17]],"date-time":"2020-08-17T14:26:24Z","timestamp":1597674384000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2014\/267383\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":12,"alternative-id":["267383","267383"],"URL":"https:\/\/doi.org\/10.1155\/2014\/267383","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}