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

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.