On certain discrete inequalities and their continous analogues

In a 1955 paper, Ky Fan, Olga Taussky, and John Todd presented discrete analogues of inequalities of Wirtinger type, and by taking limits they were able to recover the continuous inequalities. We generalize their techniques to mixed and higher derivatives and inequalities with weight functions in the integrals. We have also considered analogues of inequalities of Muller and Redheffer and have used these inequalities to derive a necessary and sufficient condition on ordered pairs of numbers so that the first number is the square norm of the k th derivative of some periodic function and the second number is the square norm of the m th derivative of the same periodic function.