Hölder-Regularized Sum

The Hölder-regularized sum of order of a series is obtained by repeatedly averaging its partial sums. If , define

(1)

for , ..., . The series is Hölder-summable of order to if .

For each positive integer , Hölder summability of order , denoted , is equivalent to Cesàro summability of order , although the intermediate transformed sequences are different (Hardy 1949, p. 103). Ordinary convergence is the order-zero case.

For example, let , giving the divergent series . The partial sums are and . Their first averages are

(2)

Thus, the even and odd subsequences tend to and 0, respectively. A second averaging gives

(3)

where uses the floor function and is a harmonic number. Since , the sequence tends to , so the series is Hölder-summable of order 2 but not of order 1 (Hardy 1949, p. 94).