TY - JOUR AU - MITTAL, M. L. AU - PRASAD, G. AU - KUMAR, RAJESH PY - 1991/03/01 Y2 - 2024/03/29 TI - ON THE $||T||\cdot C_1$ SUMMABILITY OF A SEQUENCE OF FOURIER COEFFICIENTS JF - Tamkang Journal of Mathematics JA - Tamkang J. Math. VL - 22 IS - 1 SE - Papers DO - 10.5556/j.tkjm.22.1991.4566 UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/4566 SP - 25-29 AB - <div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-weight: bold;">Mohanty and Nanda </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">(1959) </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-weight: bold;">were the first to establish a result for the $</span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPSMT';">(C,</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-weight: bold;">1)$ i.e. $C_1$-summability of the sequence $\</span><span style="font-size: 10.000000pt; font-family: 'ArialMT';">{n </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">B_n(x)\}$. </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-weight: bold;">Varshney (1959) improved the result for $</span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPS'; font-weight: bold; font-style: italic;">(N, \frac{1}{n+1}</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-weight: bold;"> ) \cdot C_1$&nbsp; summability which was generalised by several investiga- tors such as Sharma (1970), Singh </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">(1963), </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-weight: bold;">Lal (1971), Khare and Singh (1988) etc. </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">In </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-weight: bold;">this note, we haveĀ·</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">discussed $||</span><span style="font-size: 14.000000pt; font-family: 'ArialMT';">T||\cdot </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-weight: bold;">C_1$-summability of the sequence $\</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">{n </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">B_x(x)\}$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-weight: bold;">which includes the result due to Khare and Singh (1988). </span></p></div></div></div> ER -