Smoothness of Function in Terms of Kernel and Its Relation With <i>K</i> ‐Derivative

We define a kernel function and K ‐derivative of a function w.r.t. this Kernel is introduced and studied, this new derivative is obtained by replacing e − s t in Laplace derivative with K ( s , t ), the newly defined kernel. Symmetric K ‐derivative is defined in a way analogous to symmetric Laplace derivative. Here, in this particular piece of work, a new kind of smoothness, called K ‐smoothness, is defined for a function f using the kernel and the inter‐relation of characteristics of f , K ‐derivative as well as symmetric K ‐derivative of f w.r.t. this kernel and its K ‐smoothness is discovered.