Communications of the
Korean Mathematical Society CKMS

In this paper, we use a generalized Brownian motion process to define a generalized Feynman integral and a generalized Fourier-Feynman transform. We also define the concepts of the multiple $L_p$ analytic generalized Fourier-Feynman transform and the generalized convolution product of functionals on function space $\cab$. We then verify the existence of the multiple $L_p$ analytic generalized Fourier-Feynman transform for functionals on function space that belong to a Banach algebra $\sab$. Finally we establish some relationships between the multiple $L_p$ analytic generalized Fourier-Feynman transform and the generalized convolution product for functionals in $\sab$.

Keywords: generalized Brownian motion process, generalized analytic Feynman integral, generalized analytic Fourier-Feynman transform, generalized convolution product, multiple $L_p$ analytic generalized Fourier-Feynman transform

MSC numbers: 60J65,28C20