Integral method from even to odd order for trigonometric B-spline basis

Mei Li; Wanqiang Shen; Mei Li; Wanqiang Shen

doi:10.3934/math.20241729

AIMS Mathematics

2024, Volume 9, Issue 12: 36470-36492. doi: 10.3934/math.20241729

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Research article

Integral method from even to odd order for trigonometric B-spline basis

Mei Li ,
Wanqiang Shen ^,

School of Science, Jiangnan University, Wuxi 214122, China

Received: 11 November 2024 Revised: 23 December 2024 Accepted: 26 December 2024 Published: 31 December 2024
MSC : 65D10, 65D17

The conventional trigonometric B-spline basis of odd order for piecewise trigonometric polynomial space possesses a lot of good modeling properties. However, its order cannot be increased by the integral method like B-spline because of the particularity of the trigonometric polynomials. In the paper, a basis in an even-order trigonometric polynomial space is defined, and its integral relation with the existing odd-order trigonometric B-spline basis is obtained. First, the condition of the knot sequence is improved to ensure the nonnegativity of the prior odd-order trigonometric B-spline basis. Under the revised condition, a set of truncation functions is given and used to build a basis for piecewise trigonometric polynomial space without constant terms, which is also known as the direct current (DC) component-free space, secondly. The basis fulfills local support and continuity properties like B-spline of even order, and each basis function is unique under a constant multiple. Thirdly, the integral formula from the even-order to odd-order trigonometric B-spline basis is proved.
- trigonometric B-spline,
- nonnegativity,
- knot sequence,
- truncation function,
- integral formula
Citation: Mei Li, Wanqiang Shen. Integral method from even to odd order for trigonometric B-spline basis[J]. AIMS Mathematics, 2024, 9(12): 36470-36492. doi: 10.3934/math.20241729

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Abstract

The conventional trigonometric B-spline basis of odd order for piecewise trigonometric polynomial space possesses a lot of good modeling properties. However, its order cannot be increased by the integral method like B-spline because of the particularity of the trigonometric polynomials. In the paper, a basis in an even-order trigonometric polynomial space is defined, and its integral relation with the existing odd-order trigonometric B-spline basis is obtained. First, the condition of the knot sequence is improved to ensure the nonnegativity of the prior odd-order trigonometric B-spline basis. Under the revised condition, a set of truncation functions is given and used to build a basis for piecewise trigonometric polynomial space without constant terms, which is also known as the direct current (DC) component-free space, secondly. The basis fulfills local support and continuity properties like B-spline of even order, and each basis function is unique under a constant multiple. Thirdly, the integral formula from the even-order to odd-order trigonometric B-spline basis is proved.

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