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# Is it zero?

If is an even continuous function defined on and all its midpoint Riemann sums are zero ( i.e   for every ), then is ?

Solution

Let be an absolutely summable sequence, and use it to define a function as

It is evident that is even. By the Weierstrass M-Test converges uniformly on to . Hence, is continuous, and using
the fact that we have that

Next, we make two observations:

• If for every then by the definition of the Riemann integral we have that

which yields .

• Using the identity

we obtain the following string of bi-implications for every

In order for all of the mid-point Riemann sums of to be zero, it is thus
necessary and sufficient that

(i) and
(ii)

For this reason we choose the sequence as follows

Finally, is continuous , even , non zero and its Riemann mid-points are zero. To see that is non zero we simply calculate ;