Abstract: The existence of sine and cosine series as a Fourier
series, their L1-convergence seems to be one of the difficult question
in theory of convergence of trigonometric series in L1-metric norm.
In the literature so far available, various authors have studied the
L1-convergence of cosine and sine trigonometric series with special
coefficients. In this paper, we present a modified cosine and sine sums
and criterion for L1-convergence of these modified sums is obtained.
Also, a necessary and sufficient condition for the L1-convergence of
the cosine and sine series is deduced as corollaries.
Abstract: In this paper, the results of Kano from one dimensional
cosine and sine series are extended to two dimensional cosine and sine
series. To extend these results, some classes of coefficient sequences
such as class of semi convexity and class R are extended from
one dimension to two dimensions. Further, the function f(x, y) is
two dimensional Fourier Cosine and Sine series or equivalently it
represents an integrable function or not, has been studied. Moreover,
some results are obtained which are generalization of Moricz’s
results.