Cesàro summability of the Fourier integral representations of the Bloch-Nordsieck distribution and its derivatives

Summary

The Fourier integral representations for the Bloch-Nordsieck distribution ρ(ω) and its derivatives ϱ⁽ⁿ⁾(ω)(n=1,2,...) are not Cauchy convergent. The theory therefore contains ambiguities. Consequently the equations, characterising the theory and which relate these derivatives to the discrete translations ρ(ω−mE)(m=0, 1, 2, ...,n) of ρ(ω) are not satisfied by these representations as they stand. The equations are consistently satisfied only by the regularised forms of the corresponding integrals. This regularization is carried out in this paper using the method of Cesàro summability. This allows to compute these derivatives ϱ⁽ⁿ⁾(ω)(n=0,1,2,...) and to investigate their behaviour as functions of ω in the entire half-line 0≤ω<∞.