In this paper a determinant identity is established, from which a simple proof of the multivariate Lagrange–Good inversion formula follows directly. Further discussion on a discrete analogue of the Lagrange–Good inversion formula is also presented.

Let ${{F}_{2n,2}}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf{P}$ denote the affine automorphism group of ${{F}_{2n,2}}$. In this article the theory of continuous wavelet transform on ${{F}_{2n,2}}$ associated with $\mathbf{P}$ is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on ${{F}_{2n,2}}$ is studied, and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform; the others are from the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if $n\,=\,1$, ${{F}_{2,2}}$ is the 3-dimensional Heisenberg group ${{H}^{1}}$, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on ${{F}_{2,2}}$. This result cannot be extended to the case $n\,\ge \,2$.

We look forwards and backwards in the multi-allelic neutral exchangeable Cannings model with fixed population size and nonoverlapping generations. The Markov chain X is studied which describes the allelic composition of the population forward in time. A duality relation (inversion formula) between the transition matrix of X and an appropriate backward matrix is discussed. The probabilities of the backward matrix are explicitly expressed in terms of the offspring distribution, complementing the work of Gladstien (1978). The results are applied to fundamental multi-allelic Cannings models, among them the Moran model, the Wright-Fisher model, the Kimura model, and the Karlin and McGregor model. As a side effect, number theoretical sieve formulae occur in these examples.

We study cosine and sine Fourier transforms defined by F(t):= (2/π) and (t):= (2/π), where f is L1-integrable over[0, ∞]. We also assume than F are locally absolutely continuous over [0, ∞). In particular, this is the case if both f(x) and xf(x) are (L1-integrable over [0, ∞). Motivated by the inversion formulas, we consider the partial integras Sν (f, x):= and ν(f, x):= , the modified partial integrals uν (f, x):= sν(f, x) - F(ν)(sin νx)/x and ũν(f, x):= ν(f, x) + (ν) (cos νx)/x, where ν > 0. We give necessary and sufficient conditions for(L1 [0, ∞)-convergence of uν (f) and ũν (f) as well as for the L1 [0, X]-convergence of sν (f) and ν(f) to f as ν← ∞, where 0 < X < ∞ is fixed. On the other hand, in certain cases we conclude that sν(f) and ν(f) cannot belong to (L1 [0,∞). Conequently, it makes no sense to speak of their (L1 [0, ∞)-convergence as ν ← ∞.

As an intermediate tool, we use the Cesàro means of Fourier transforms. Then we prove Tauberian type results and apply Sidon type inequalities in order to obtain Tauberian conditions of Hardy-Karamata kind.

We extend these results to the complex Fourier transform defined by G(t):= , where g is L1- integrable over (−∞, ∞).

We obtain a single formula which, when its components are adequately chosen, transforms itself into the main formulas of the Palm theory of point processes: Little's L = λW formula [10], Brumelle's H = λG formula [5], Neveu's exchange formula [14], Palm inversion formula and Miyazawa's rate conservation law [12]. It also contains various extensions of the above formulas and some new ones.

We derive the Sobel–Uppuluri and Galambos-type extensions of the Bonferroni bounds, and further extensions of the same nature, as consequences of a single non-probabilistic inequality. The methodology follows that of Galambos.