In this section we give the Levinson type generalization for converses of Jensen's inequality in discrete case.
For convenience, three basic modulations which belong to different modulation families are considered here in discrete case.
Similar to the definition of semi-cycle in discrete case (see [30]), we give the definition of a semi-cycle on time scales.
Since the permanence of this system has been obtained by the comparison theorem in the continuous case, while it has been obtained by semi-cycle concept in discrete case.
In discrete case the regularizing operators of the inverse Laplace transformation are used for transformation of experimental data into a form that is more suitable for determination of decay rates and amplitudes.
In discrete case, by simplifying we can write Eq. (14) as, Y i k + 1 = Y i k - 4 w M Y i k - Y o i k + 32 w S Y i, l - 1 k - 2 Y i, l k + Y i, l + 1 k (15).
Alternatively in discrete case, Cong et al. first established a linear relationship between the internal source distribution and the surface measurement, and adopted a modified Newton algorithm with the active set strategy to solve the problem [ 13].
By Remark 5.1 and Example 5.2, our result in Theorem 3.1 not only extends the results in the discrete cases but also contains more complicated time scales.
A general method for finding sharp constants in interpolation inequalities of L ∞ - L 2 - L 2 -type was developed in [1, 11, 18], which was also used in the discrete case in the present paper.
