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The calculation of integrals involving $$B$$-splines by means of recursion relations. (English) Zbl 1088.65019
Summary: A procedure is given for constructing the exact integrals involving $$B$$-splines using recursion relations. The recursive integrals form the basic constituents for the exact evaluation of integrals that appear in the calculation of atomic and molecular properties. The method can be applied to solve differential equations as well as to produce a complete set of basis functions that may approximate a function arbitrarily well depending on the degree $$k$$ and the number of $$B$$-splines that are employed in the approximation. The advantage of this method is that the recursions developed over a fixed interval represent exact results for the particular integral involved. Several examples are also provided to show how these recursion relations can be applied to evaluate integrals involving multiple $$B$$-splines of same or different degree. Closed forms of these integrals involve complicated recursion relations.

##### MSC:
 65D20 Computation of special functions and constants, construction of tables 65D07 Numerical computation using splines 33E20 Other functions defined by series and integrals 65Q05 Numerical methods for functional equations (MSC2000) 33F05 Numerical approximation and evaluation of special functions
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