Best One-Sided L 1-Approximation by Blending Functions of Order (2, 2) Original Research Article

Let f∈C2, 2([−1, 1]2) be a real function satisfying ∂4f/∂x2 ∂y2⩾0 on [−1, 1]2. We study the problem of best one-sided L1-approximation to f from the linear space {h∈C2, 2([−1, 1]2): ∂4h/∂x2 ∂y2=0} of all blending functions of order (2, 2). The unique best one-sided L1-approximant to f from above is characterized by transfinite Hermite interpolation on the canonical grid {(x, y)∈[−1, 1]2 : |x|=|y|}. For f even with respect to one of its variables we characterize the unique best one-sided L1-approximant to f from below by transfinite Hermite interpolation on the canonical grid {(x, y)∈[−1, 1]2 : |x|+|y|=1}. There is no canonical grid for the entire cone class of functions f with ∂4f/∂x2 ∂y2⩾0 on [−1, 1]2 when we approximate from below. The best one-sided L1-approximant from above has the smoothness of f. The best one-sided L1-approximant to f from below is a blending-spline function with two line segment knots {(x, 0): −1⩽x⩽1} and {(0, y): −1⩽y⩽1}; i.e., the best one-sided approximation to f from below possesses a saturation effect with respect to the smoothness of f.