Series In Terms Of Functions Research Articles

Approximation of Fourier series in terms of functions in $L_p$ Spaces for $0

Many results introduced for the absolutely convergence of Fourier series in terms of absolutely continuous functions. Here we study the convergence of Fourier series in terms of $p$-integrable functions series.

Effective approximation of piecewise smooth functions by their expansion into fast convergent series in terms of functions formed by eigenfunctions of Sturm–Liouville problems

Using the eigenfunctions of two Sturm–Liouville problems (with the same operator of a general form but with two different versions of boundary conditions), a method for constructing these specific basis functions is developed. The corresponding expansions of smooth and piecewise smooth functions in terms of such basis lead to fast convergent series. This result makes it possible to approximate the functions of the above class by a small number of terms. We also give brief information on a method for constructing multidimensional (in particular, two-dimensional) specific basis functions with the above properties. The proposed method is based on the ideas of the author's earlier works. However, it is, in essence, a new method that has substantially improved characteristics and is mainly oriented to the approximation of piecewise smooth functions. We also consider several important special cases.

Effective approximation of piecewise smooth functions by their expansion into fast convergent series in terms of functions formed by eigenfunctions of Sturm–Liouville problems

Using the eigenfunctions of two Sturm–Liouville problems (with the same operator of a general form but with two different versions of boundary conditions), a method for constructing these specific basis functions is developed. The corresponding expansions of smooth and piecewise smooth functions in terms of such basis lead to fast convergent series. This result makes it possible to approximate the functions of the above class by a small number of terms. We also give brief information on a method for constructing multidimensional (in particular, two-dimensional) specific basis functions with the above properties. The proposed method is based on the ideas of the author's earlier works. However, it is, in essence, a new method that has substantially improved characteristics and is mainly oriented to the approximation of piecewise smooth functions. We also consider several important special cases.

On one model of a transversally isotropic elastic foundation

The method of expansion of the stress and displacement components into series in terms of functions of the transverse coordinate and the method of variation with respect to the state being determined are used to construct an iterative model of a transversally isotropic elastic foundation. This model allows for all the SSS components. The order of the equations does not depend on the number of terms of the expansions, which approximate the displacements and stresses. As an example, the problem on a concentrated force applied to an elastic foundation is considered.

An algorithm for summing series and their derivatives

An algorithm for summing series in terms of functions satisfying general recurrence relations is presented. Extension to the summing of the derivatives is considered.