Shifted Cubic Spline Wavelets with Two Vanishing Moments on the Interval and a Splitting Algorithm

The present paper is devoted to the application of local polynomial and non-polynomial interpolation splines of the third order of approximation for the numerical solution of the Volterra integral equation of the second kind. Computational schemes based on the use of the splines include the ability to calculate the integrals over the kernel multiplied by the basis function which are present in the computational methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra integral equations is also discussed. The results of the numerical experiments are presented.

Application Local Polynomial and Non-polynomial Splines of the Third Order of Approximation for the Construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind

Бурова

2021

Continuum Wavelets and Distributions

Dem’yanovich

Ivantsova

Ponomareva

2022

The purpose of this work is to obtain a wavelet expansion of information flows, which are distribution flows (in the terminology of Schwartz). The concept of completeness is introduced for a family of abstract functions. Using the mentioned families, nested spaces of distribution flows are constructed. The projection of the enclosing space onto the nested space generates a wavelet expansion. Decomposition and reconstruction formulas for the above expansion are derived. These formulas can be used for wavelet expansion of the original information flow coming from the analog device. This approach is preferable to the approach in which the analog flow is converted into a discrete numerical flow using quantization and digitization. The fact is that quantization and digitization lead to significant loss of information and distortion. This paper also considers the wavelet expansion of a discrete flow of distributions using the Haar type functions.

Discrete and Continuous Wavelet Expansions

Dem’yanovich

2022

This paper proposes a new approach to the construction of wavelet decomposition, which is suitable for processing a wide range of information flows. The proposed approach is based on abstract functions with values in linear topological spaces. It is defined by embedded spaces and their projections. The proposed approach allows for adaptive ways of decomposition for the initial flow depending on the speed changes of the last one. The initial information flows can be real number flows, flows of complex and p-adic numbers, as well as flows of (finite or infinite) vectors, matrices, etc. The result is illustrated with examples of spline-wavelet decompositions of discrete flows, and also with the example of the decomposition of a continuous flow.