UNIVARIATE MULTIQUADRIC APPROXIMATION - QUASI-INTERPOLATION TO SCATTERED DATA

The univariate multiquadric function with center x(j) is-an-element-of R has the form {phi(j)(x) = [(x - x(j))2 + c2]1/2, x is-an-element-of R} where c is a positive constant. We consider three approximations, namely, L(A)f, L(B)f, and L(C)f, to a function {f(x), x0 less-than-or-equal-to x less-than-or-equal-to x(N)) from the space that is spanned by the multiquadrics {phi(j):j = 0,1,...,N} and by linear polynomials, the centers {x(j):j = 0,1,...,N} being given distinct points of the interval [x0, x(N)]. The coefficients of L(A)f and L(B)f depend just on the function values {f(x(j)):j = 0,1,...,N). while L(C)f also depends on the extreme derivatives f'(x0) and f'(x(N)). These approximations are defined by quasi-interpolation formulas that are shown to give good accuracy even if the distribution of the centers in [x0, x(N)] is very irregular. When f is smooth and c = O(h), where h is the maximum distance between adjacent centers, we find that the error of each quasi-interpolant is O(h2\log h\) away from the ends of the range x0 less-than-or-equal-to x less-than-or-equal-to x(N). Near the ends of the range, however, the accuracy of L(A)f and L(B)f is only O(h), because the polynomial terms of these approximations are zero and a constant, respectively. Thus, some of the known accuracy properties of quasi-interpolation when there is an infinite regular grid of centers {x(j) = jh: j is-an-element-of L}, given by Buhmann (1988), are preserved in the case of a finite range x0 less-than-or-equal-to x less-than-or-equal-to x(N), and there is no need for the centers {x(j):j = 0,...,N} to be equally spaced.