Convolutions and Fourier-Feynman transforms of functionals involving multiple integrals (Q1923896)

Fourier transforms with respect to the analytic Feynman integral and the related topics: convolution product and its Fourier transform, the inversion formula, Parseval's identity, are discussed for functionals involving multiple time integrals, e.g., of the form \[ F(x)=\exp [\int^T_0 \int^T_0f(s,t,x(s), x(t))ds dt]. \] Two situations are considered: a simpler one, where \(f\) is such that \(F\) belongs to the Banach algebra consisting of stochastic Fourier transforms of complex Borel measures on \(L_2([0,T])\); a considerably more complicated one, where \(f\) is required only to be \(L_{1,\infty} ([0,T]^2\times R^2)\), i.e., \(L_1\) with respect to the last two variables, the \(L_1\)-norm being \(L_\infty\) of the first two variables.