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  <front>
    <journal-meta>
      <journal-id journal-id-type="issn">2411-3336</journal-id>
      <journal-id journal-id-type="eissn">2541-9404</journal-id>
      <journal-title-group>
        <journal-title xml:lang="ru">Записки Горного института</journal-title>
        <journal-title xml:lang="en">Journal of Mining Institute</journal-title>
      </journal-title-group>
      <publisher>
        <publisher-name xml:lang="ru">Санкт-Петербургский горный университет императрицы Екатерины ΙΙ</publisher-name>
        <publisher-name xml:lang="en">Empress Catherine II Saint Petersburg Mining University</publisher-name>
      </publisher>
    </journal-meta>
    <article-meta>
      <article-id custom-type="pmi" pub-id-type="custom">pmi-15451</article-id>
      <article-id pub-id-type="uri">https://pmi.spmi.ru/pmi/article/view/15451</article-id>
      <article-categories>
        <subj-group subj-group-type="section-heading" xml:lang="ru">
          <subject>Статьи</subject>
        </subj-group>
        <subj-group subj-group-type="section-heading" xml:lang="en">
          <subject>Articles</subject>
        </subj-group>
      </article-categories>
      <title-group>
        <article-title xml:lang="en">On the closedness theorem in the theory of trigonometric series</article-title>
        <trans-title-group xml:lang="ru">
          <trans-title>О теореме замкнутости в теории тригонометрических рядов</trans-title>
        </trans-title-group>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <name name-style="eastern">
            <surname>Krylov</surname>
            <given-names>N. M.</given-names>
          </name>
          <name-alternatives>
            <name name-style="eastern" xml:lang="ru">
              <surname>Крылов</surname>
              <given-names>Н. М.</given-names>
            </name>
            <name name-style="western" xml:lang="en">
              <surname>Krylov</surname>
              <given-names>N. M.</given-names>
            </name>
          </name-alternatives>
          <email>pmi@spmi.ru</email>
          <xref ref-type="aff" rid="aff1"/>
        </contrib>
        <aff-alternatives id="aff1">
          <aff>
            <institution xml:lang="ru"> (Россия)</institution>
          </aff>
          <aff>
            <institution xml:lang="en"> (Russia)</institution>
          </aff>
        </aff-alternatives>
      </contrib-group>
      <pub-date pub-type="epub" iso-8601-date="1913-12-01">
        <day>01</day>
        <month>12</month>
        <year>1913</year>
      </pub-date>
      <pub-date date-type="collection">
        <year>1913</year>
      </pub-date>
      <volume>4</volume>
      <issue>4</issue>
      <fpage>256</fpage>
      <lpage>258</lpage>
      <history>
        <date date-type="received" iso-8601-date="1913-06-19">
          <day>19</day>
          <month>06</month>
          <year>1913</year>
        </date>
        <date date-type="accepted" iso-8601-date="1913-08-26">
          <day>26</day>
          <month>08</month>
          <year>1913</year>
        </date>
        <date date-type="rev-recd" iso-8601-date="1913-12-01">
          <day>01</day>
          <month>12</month>
          <year>1913</year>
        </date>
      </history>
      <permissions>
        <copyright-statement xml:lang="ru">© 1913 Н. М. Крылов</copyright-statement>
        <copyright-statement xml:lang="en">© 1913 N. M. Krylov</copyright-statement>
        <copyright-year>1913</copyright-year>
        <copyright-holder xml:lang="ru">Н. М. Крылов</copyright-holder>
        <copyright-holder xml:lang="en">N. M. Krylov</copyright-holder>
        <license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by/4.0" xml:lang="ru">
          <license-p>Эта статья доступна по лицензии Creative Commons Attribution 4.0 International (CC BY 4.0)</license-p>
        </license>
        <license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by/4.0" xml:lang="en">
          <license-p>This article is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0)</license-p>
        </license>
      </permissions>
      <self-uri xlink:type="simple" xlink:href="https://pmi.spmi.ru/pmi/article/view/15451">https://pmi.spmi.ru/pmi/article/view/15451</self-uri>
      <abstract xml:lang="ru">
        <p>Приведено весьма простое доказательство проф. В. А. Стеклова, выражаемой формулой , но все же существенно отличается от различных доказательств той же теоремы, данных проф. В. А. Стекловым и, приближается скорее по идей к доказательству проф. Hurwitz’a (основанному на методе ариеметических средних Cesaro-Fejer’a), имея за собой, мне кажется, преимущество большей простоты, т. к. оно базируется на применении теоремы о почленном интегрировании тригонометрических рядов, которая сама представляет собою частный случай теоремы замкнутости.</p>
      </abstract>
      <abstract xml:lang="en">
        <p>A very simple proof was given by Prof. V.A. Steklov, expressed by the formula, but ше still differs significantly from various proofs of the same theorem given by prof. V. A. Steklov and, rather in terms of ideas, approaches the proof of Prof. Hurwitz (based on the Cesaro-Fejer method of ariemetic averages), having, it seems to me, the advantage of greater simplicity, since it is based on the application of the theorem on term-by-term integration of trigonometric series, which itself is a special case of the closedness theorem.</p>
      </abstract>
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