### Abstract

In many cases the smoothness of splines is important (for qualitative approximation, for the calculation of a number of functionals, etc.). In the case of discontinuity of approximated functions it is difficult to use ordinary splines. It is desirable to have splines with similar properties of the approximated function. The purpose of this paper is to introduce the concept of general smoothness with the aid of linear functionals having a definite location of supports. 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. A discrete flow often appears as a result of analog signal sampling, representing the values of a function, and in this case, the splines of the Lagrange type are used. In all cases, it is highly desirable that the generalized smoothness of the resulting spline coincides with the generalized smoothness of the original signal. Here we formulate the necessary and sufficient conditions for general smoothness of splines, and also a toolkit is being developed to build mentioned splines. The proposed scheme allows us to consider splines generated by functions from different spaces and to apply the obtained result to sources which can appear in physics, chemistry, biology, etc.

Original language | English |
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Pages (from-to) | 304-310 |

Number of pages | 7 |

Journal | WSEAS Transactions on Mathematics |

Volume | 17 |

Publication status | Published - 1 Jan 2018 |

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### Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

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*WSEAS Transactions on Mathematics*, vol. 17, pp. 304-310.

**On general smoothness of minimal splines of the Lagrange type.** / Демьянович, Юрий Казимирович; Evdokimova, Tatjana O.; Prozorova, Evelina V.

Research output

TY - JOUR

T1 - On general smoothness of minimal splines of the Lagrange type

AU - Демьянович, Юрий Казимирович

AU - Evdokimova, Tatjana O.

AU - Prozorova, Evelina V.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In many cases the smoothness of splines is important (for qualitative approximation, for the calculation of a number of functionals, etc.). In the case of discontinuity of approximated functions it is difficult to use ordinary splines. It is desirable to have splines with similar properties of the approximated function. The purpose of this paper is to introduce the concept of general smoothness with the aid of linear functionals having a definite location of supports. 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. A discrete flow often appears as a result of analog signal sampling, representing the values of a function, and in this case, the splines of the Lagrange type are used. In all cases, it is highly desirable that the generalized smoothness of the resulting spline coincides with the generalized smoothness of the original signal. Here we formulate the necessary and sufficient conditions for general smoothness of splines, and also a toolkit is being developed to build mentioned splines. The proposed scheme allows us to consider splines generated by functions from different spaces and to apply the obtained result to sources which can appear in physics, chemistry, biology, etc.

AB - In many cases the smoothness of splines is important (for qualitative approximation, for the calculation of a number of functionals, etc.). In the case of discontinuity of approximated functions it is difficult to use ordinary splines. It is desirable to have splines with similar properties of the approximated function. The purpose of this paper is to introduce the concept of general smoothness with the aid of linear functionals having a definite location of supports. 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. A discrete flow often appears as a result of analog signal sampling, representing the values of a function, and in this case, the splines of the Lagrange type are used. In all cases, it is highly desirable that the generalized smoothness of the resulting spline coincides with the generalized smoothness of the original signal. Here we formulate the necessary and sufficient conditions for general smoothness of splines, and also a toolkit is being developed to build mentioned splines. The proposed scheme allows us to consider splines generated by functions from different spaces and to apply the obtained result to sources which can appear in physics, chemistry, biology, etc.

KW - Approximate relations

KW - Chains of vectors

KW - General smoothness

KW - Splines

UR - http://www.scopus.com/inward/record.url?scp=85060617887&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85060617887

VL - 17

SP - 304

EP - 310

JO - WSEAS Transactions on Mathematics

JF - WSEAS Transactions on Mathematics

SN - 1109-2769

ER -