Convergence of univariate multi-level Gaussian convolution

In this paper we consider the approximation of smooth univariate functions f using multilevel convolution with the Gaussian basis function whose initial width parameter, denoted by h, scales by one half at each stage of the multilevel algorithm. We restrict attention to approximating entire functions (restricted to the real line) and, in this setting, we are able to provide a precise closed form series expansion for the approximation error at each stage. For entire functions of exponential type a >0 that are bounded on we show that the approximation error at the pth level is bounded by a multiple of the product ∏ₖ₌₀ᵖ(exp(a²h² / 2^(2k+1))-1. Using f (x) = sin(ax) as a prototype example we derive an analytic formula for the approximation error and use this to demonstrate the accuracy of the bounds. We derive similar results for functions of exponential type a >0 that are unbounded on . Here we use f (x) = exp(ax) as the prototype example. The bounds derived in the paper show that for an appropriately chosen starting width h the proposed algorithm delivers approximations whose errors decay like, O(1/2(p+1)2 ), an exponentially fast rate with respect to the level p.