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We consider approximation problems for a special space of d variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show that, depending on the norm for measuring the error, the problems are strongly polynomially or quasi-polynomially tractable even in the model of computation where functional evaluations have the cost exponential in the number of active variables.
This article studies the problem of approximating functions belonging to a Hilbert space $H_d$ with an isotropic or anisotropic Gaussian reproducing kernel, $$ K_d(bx,bt) = expleft(-sum_{ell=1}^dgamma_ell^2(x_ell-t_ell)^2right) mbox{for all} bx,b
We investigate multivariate integration for a space of infinitely times differentiable functions $mathcal{F}_{s, boldsymbol{u}} := {f in C^infty [0,1]^s mid | f |_{mathcal{F}_{s, boldsymbol{u}}} < infty }$, where $| f |_{mathcal{F}_{s, boldsymbol{u}}
Most commonly used emph{adaptive} algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a user-specified ab
We propose an optimal approximation formula for analytic functions that are defined on a complex region containing the real interval $(-1,1)$ and possibly have algebraic singularities at the endpoints of the interval. As a space of such functions,we
We consider universal approximations of symmetric and anti-symmetric functions, which are important for applications in quantum physics, as well as other scientific and engineering computations. We give constructive approximations with explicit bound