TY - GEN

T1 - Uniqueness of Space of Hermite Type Splines

AU - Dem'Yanovich, Yuri K.

AU - Belyakova, Olga V.

AU - Le, Bich T.N.

PY - 2018/10

Y1 - 2018/10

N2 - The smoothness of functions is quite essential in applications. This smoothness can be used in functional calculations, in the construction of the finite element method, in the approximation of those or other numerical data, etc. The interest in smooth approximate spaces is supported by the desire to have a coincidence of smoothness of exact and approximate solutions. A lot of papers have been devoted to this problem. The continuity of the function at a point means equality of the limits on the right and left; generalization of this situation is the equality of values of two linear functionals (at the prescribed function) with supports located on opposite sides of the mentioned point. Such generalization allows us to introduce the concept of generalized smoothness, which gives the ability to cover various cases of singular behavior functions at some point. The generalized smoothness is called pseudo-smoothness, although, of course, we can talk about the different types of pseudo-smoothness depending on the selected functionals mentioned above. Splines are often used for processing numerical information flows; a lot of scientific papers are devoted to these investigations. Sometimes spline treatment implies to the filtration of the mentioned flows or to their wavelet decomposition. Often a discrete flow appears as a result of analog signal sampling, representing the values of a function, and in this case, the splines of Lagrange type are used. In some cases, there are two interconnected analog signals, one of which represents the values of some function, and the second one represents the values of its derivative. In this case, it is convenient to use splines of the Hermite type of the first height for processing. In all cases, it is highly desirable that the generalized smoothness of the resulting spline coincides with the generalized smoothness of original signal. The concepts, which are introduced in this paper, and the theorems, which are proved here, allow to achieve this result. The paper discusses the existence and uniqueness of spline spaces of the Hermite type of the first height (under condition of fixing the spline grid and the type of generalized smoothness). The purpose of this paper is to prove the uniqueness of the Hermite type spline space of the first height (not necessarily polynomial) having the maximum pseudosmoothness. In this paper we use the necessary and sufficient criterion of the pseudo-smoothness obtained earlier.

AB - The smoothness of functions is quite essential in applications. This smoothness can be used in functional calculations, in the construction of the finite element method, in the approximation of those or other numerical data, etc. The interest in smooth approximate spaces is supported by the desire to have a coincidence of smoothness of exact and approximate solutions. A lot of papers have been devoted to this problem. The continuity of the function at a point means equality of the limits on the right and left; generalization of this situation is the equality of values of two linear functionals (at the prescribed function) with supports located on opposite sides of the mentioned point. Such generalization allows us to introduce the concept of generalized smoothness, which gives the ability to cover various cases of singular behavior functions at some point. The generalized smoothness is called pseudo-smoothness, although, of course, we can talk about the different types of pseudo-smoothness depending on the selected functionals mentioned above. Splines are often used for processing numerical information flows; a lot of scientific papers are devoted to these investigations. Sometimes spline treatment implies to the filtration of the mentioned flows or to their wavelet decomposition. Often a discrete flow appears as a result of analog signal sampling, representing the values of a function, and in this case, the splines of Lagrange type are used. In some cases, there are two interconnected analog signals, one of which represents the values of some function, and the second one represents the values of its derivative. In this case, it is convenient to use splines of the Hermite type of the first height for processing. In all cases, it is highly desirable that the generalized smoothness of the resulting spline coincides with the generalized smoothness of original signal. The concepts, which are introduced in this paper, and the theorems, which are proved here, allow to achieve this result. The paper discusses the existence and uniqueness of spline spaces of the Hermite type of the first height (under condition of fixing the spline grid and the type of generalized smoothness). The purpose of this paper is to prove the uniqueness of the Hermite type spline space of the first height (not necessarily polynomial) having the maximum pseudosmoothness. In this paper we use the necessary and sufficient criterion of the pseudo-smoothness obtained earlier.

KW - approximate relations

KW - smoothness

KW - splines

KW - uniqueness of spline spaces

UR - http://www.scopus.com/inward/record.url?scp=85078889529&partnerID=8YFLogxK

U2 - 10.1109/ICAMCS.NET46018.2018.00037

DO - 10.1109/ICAMCS.NET46018.2018.00037

M3 - Conference contribution

AN - SCOPUS:85078889529

T3 - Proceedings - 2018 International Conference on Applied Mathematics and Computational Science, ICAMCS.NET 2018

SP - 178

EP - 183

BT - Proceedings - 2018 International Conference on Applied Mathematics and Computational Science, ICAMCS.NET 2018

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2018 International Conference on Applied Mathematics and Computational Science

Y2 - 6 October 2018 through 8 October 2018

ER -