2025/09/09 by Nikiforova, Tatiana M.
#26A51 #41A50 #41A52 #42A15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.07776
In this paper, we study sums of translates on the real axis. These functions generalize logarithms of weighted algebraic polynomials. Namely, we are dealing with the following functions F(y,t) := J(t) + ∑ j=1n Kj(t-yj), y := (y1,…,yn), y1 ≤ … ≤ yn, where the field function J is a function defined on ℝ, which is "admissible" for the kernels K1,…,Kn concave on (-∞,0) and on (0,∞) and having a singularity at 0. We consider "local maxima" \beginaligned m0(y) amp; := sup t ∈ (-∞, y1] F(y, t), mn(y) := sup t ∈ [yn, ∞) F(y, t),
mj(y) amp; := sup _t ∈ [yj, yj+1] F(y, t), j = 1,…,n-1, \endaligned and the difference function D(y) := (m1(y)-m0(y), m2(y)-m1(y),…,mn(y)-mn-1(y)). We prove that, under certain assumptions on monotonicity of the kernels, D is a homeomorphism between its domain and ℝn.