Some Formulas for Legendre Functions Induced by the Poisson Transform

Integration formulas for the conditional transform involving the first variation

In this paper, we show that the conditional transform with respect to the Gaussian process involving the first variation can be expressed in terms of the conditional transform without the first variation. We then use this result to obtain various integration formulas involving the conditional $diamond$-product and the first variation.

The Legendre transform

f((1− t)x1 + tx2) ≤ (1− t)f(x1) + tf(x2), x1, x2 ∈ C, 0 ≤ t ≤ 1, then f : X → R is convex. Proof. Let (x1, α1), (x2, α2) ∈ epi f and 0 ≤ t ≤ 1. The fact that the pairs (xi, αi) belong to epi f means in particular that f(xi) < ∞, and hence that xi ∈ C, as otherwise we would have f(xi) =∞. But (1− t)(x1, α1) + t(x2, α2) = ((1− t)x1 + tx2, (1− t)α1 + tα2), and, as x1, x2 ∈ C, f((1− t)x1 + tx2) ≤ (...

Some Asymptotic Formulas for Multiplicative Functions

Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in themathematics literature, the PSF is commonly stated and proven in the pointwise sense for various t...

integration formulas for the conditional transform involving the first variation

in this paper, we show that the conditional transform with respect to the gaussian process involving the first variation can be expressed in terms of the conditional transform without the first variation. we then use this result to obtain various integration formulas involving the conditional $diamond$-product and the first variation.