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We consider processes of the form μ(t) = μ((0, t]), where μ is a σ-additive in probability stochastic set function. Convergence of a random Fourier series to μ(t) is proved, and the approximation of integrals with respect to μ using Fejer sums is obtained. For this approximation, we prove the convergence of solutions of the heat equation driven by μ. [ABSTRACT FROM AUTHOR]

Fourier extension is an approximation scheme in which a function on an arbitrary bounded domain is approximated using classical Fourier series on a bounding box. On the domain of interest the Fourier series exhibits redundancy, and it has the mathematical structure of a frame rather than a basis. It is not trivial to construct approximations in this frame using function evaluations in points that belong to the domain only, but one way to do so is through a discrete least squares approximation. The corresponding system is extremely ill-conditioned, due to the redundancy in the frame, yet its solution via a regularized singular value decomposition is known to be accurate to very high (and nearly spectral) precision. Still, this computation requires Q (N3) operations. In this paper we describe an algorithm to compute such Fourier extension frame approximations in only Q (N2 log2 N) operations for general 2D domains. The cost improves to Q (N log2 N) operations for simpler tensor-product domains. The algorithm exploits a phenomenon called the plunge region in the analysis of time-frequency localization operators, which manifests itself here as a sudden drop in the singular values of the least squares matrix. It is known that the size of the plunge region scales like Q (logN) in 1D problems. In this paper we show that for most 2D domains in the fully discrete case the plunge region scales like Q (N logN), proving a discrete equivalent of a result that was conjectured by Widom for a related continuous problem. The complexity estimate depends on the Minkowski or box-counting dimension of the domain boundary, and as such it is larger than Q (N logN) for domains with fractal shape. [ABSTRACT FROM AUTHOR]

In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm ∥⋅∥ on Rd, and entropy numbers of the embedding id : ℓ∥⋅∥d → ℓ∞d. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in L2 and H¹, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers of certain Gevrey type spaces behave preasymptotically almost identical to approximation numbers of spaces of dominating mixed smoothness. This observation can be exploited, for instance, for Galerkin schemes for the electronic Schrödinger equation, where mixed regularity is present. [ABSTRACT FROM AUTHOR]

Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary differential equations. Differences from the nonperiodic Chebyshev case are highlighted. [ABSTRACT FROM AUTHOR]

In this article we study the large-time behavior of the solution to a general Rosenau type approximation to the heat equation [P. Rosenau, Phys. Rev. A (3), 46 (1992), pp. 12-15], by showing that the solution to this approximation approaches the fundamental solution of the heat equation (the heat kernel), but at a slower rate than the usual heat equation. This result is valid in particular for the central difference scheme approximation of the heat equation, a property which to our knowledge has never been observed before. [ABSTRACT FROM AUTHOR]