On possible deterioration of smoothness under the operation of convolution (Q5906342)

Let \(EL^+_1\) denote the set of non-negative functions in \(L_1 (R)\) and \(S(f)\) the convolution of \(f(z)\) with \(f(-x)\). Inspired by results of \textit{D. A. Raikov}, Dokl. Akad. Nauk SSSR 23, 511-514 (1939; Zbl 0021.30605), the author constructs a function \(f\) in \(EL^+_1\), which is the restriction of an entire function, such that \(S(f)\) has infinite essential supremum in any interval. Moreover the order of \(f\) can be made not to exceed 3. The proofs depend on the theory of `tangential' approximation on \(R\) by entire functions. Further constructions include \(g\) in \(EL^+_1\) of order one such that \(S(g)\) is not analytic at the origin and for each natural number \(n\) a function \(h\) in \(EL^+_1\) of order not exceeding \(1+{1\over n}\) such that \(S(h)\) is not \((2n+2)\)-times differentiable at the origin.