Many different bases functions have been used to estimate the solution to differential equations, such as orthogonal bases [3, 4, 14, 15], wavelets [7–8] and hybrid [2, 13, 16–17]. The various systems of orthogonal functions may be classified into two categories. The first is piecewise continuous function (PCBF) to which the orthogonal systems of Walsh functions [5], Block-pulse functions [4, 1...

It is emphasized that a many-worlds interpretation of quantum theory exists only to the extent that the associated basis problem is solved. The core basis problem is that the robust enduring states specified by environmental decoherence effects are essentially Gaussian wave packets that form continua of non-orthogonal states. Hence they are not a discrete set of orthogonal basis states to which...

The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, ...

We are concerned with employing Model Order Reduction (MOR) to efficiently solve parameterized multiscale problems using the Localized Orthogonal Decomposition (LOD) method. Like many methods, LOD follows idea of separating problem into localized fine-scale subproblems and an effective coarse-scale system derived from solutions local problems. While Reduced Basis (RB) method has already been us...

Functional data type, which is an important data type, is widely prevalent in many fields such as economics, biology, finance, and meteorology. Its underlying process is often seen as a continuous curve. The classification process for functional data is a basic data mining task. The common method is a two-stage learning process: first, by means of basis functions, the functional data series is ...