in this paper, we consider orthogonal stability of mixed type additive and cubic functional equation of the form $$f(2x+y)+f(2x-y)-f(4x)=2f (x+y)+2f(x-y)-8f(2x) +10f(x)-2f(-x),$$ with $xbot y$, where $bot$  is orthogonality in the sense of ratz.

In this paper, we solve the additive ρ-functional inequalities ‖f(x+ y)− f(x)− f(y)‖ ≤ ∥∥∥∥ρ(2f (x+ y 2 ) − f(x)− f(y) )∥∥∥∥ , (1) ∥∥∥∥2f (x+ y 2 ) − f(x)− f(y) ∥∥∥∥ ≤ ‖ρ (f(x+ y)− f(x)− f(y))‖ , (2) where ρ is a fixed non-Archimedean number with |ρ| < 1 or ρ is a fixed complex number with |ρ| < 1. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ-functional inequalit...

We propose an extensive framework for additive regression models for correlated functional responses, allowing for multiple partially nested or crossed functional random effects with flexible correlation structures for, e.g., spatial, temporal, or longitudinal functional data. Additionally, our framework includes linear and nonlinear effects of functional and scalar covariates that may vary smo...

We introduce the functional generalized additive model (FGAM), a novel regression model for association studies between a scalar response and a functional predictor. We model the link-transformed mean response as the integral with respect to t of F{X(t), t} where F(·,·) is an unknown regression function and X(t) is a functional covariate. Rather than having an additive model in a finite number ...

We suggest a new method, called Functional Additive Regression, or FAR, for efficiently performing high dimensional functional regression. FAR extends the usual linear regression model involving a functional predictor, X(t), and a scalar response, Y , in two key respects. First, FAR uses a penalized least squares optimization approach to efficiently deal with high dimensional problems involving...

In commonly used functional regression models, the regression of a scalar or functional response on the functional predictor is assumed to be linear. This means the response is a linear function of the functional principal component scores of the predictor process. We relax the linearity assumption and propose to replace it by an additive structure. This leads to a more widely applicable and mu...