Approximation of sign-regular kernels

Parameterized integrals, qy=∫Kx,ydPx, are common in economic applications. To optimize a parameterized integral, or to relate one such integral to others, it is helpful to reduce the rank of the kernel K while controlling approximation error. A bound on the uniform approximation error of Kx,y≈∑i=1nfixgiy also bounds the uniform approximation error of qy≈∑i=1ncigiy=∑i=1n∫fixdPxgiy over all y and all signed measures P whose total variation does not exceed a fixed bound (such as probability measures), which may be useful in optimizing q or in relating q to other parameterized integrals. Bounding the mean squared error or the local error in approximating K generally does not bound the uniform approximation error of q. Many economically interesting kernels of parameterized integrals, including expcxy, the Gaussian probability density function, and a wide class of Green’s functions, satisfy a condition known as strict sign-regularity.