On Optimal Filtering of a Function of Many Variables in White Gaussian Noise

The author considers asymptotic expressions for the optimum mean-square error $\delta^2$ of reproducing signals in white Gaussian noise over $\mathbf R^k$, when the signal belongs to the class of smooth periodic functions of many variables. By using the familiar general result for ellipsoids in Hilbert spaces, the author calculates the corresponding rates and constants in the example under consideration.