Quadratic pencils and least-squares piecewise-polynomial approximation

For a partition ( (O = .o m, denotes the space of piecewise-polynomials of order k and of smoothness m 1; this space can be represented as the graph of the appropriate linear operator between two finite-dimensional Hilbert spaces. It gives an approach to the C. de Boor problem, 1972, on uniform boundedness (with respect to {) in the L.-norm of the orthogonal projections onto stm, and we give the detailed analysis of a quadratic pencil (matrix-valued polynomial of the second degree) which appears in the case of a geometric mesh ( if 2m ? k. The explicit calculations and estimates of zeros of the "characteristic" polynomial show that in the case SbX), {(x) the geometric mesh with the parameter x, 0 m > 0, and any partition = ((aX,)n of the unit interval [0, 1], (01 (0. 1) ? = (0 < (1 < < (n < (n+i 1 we define the piecewise-polynomial subspace (0.2) S t km = n C(m1 )[0, 11, where P~~{feL[01VfIA ~~k-I Pk, e f EJ L [?, Il: f Ia,, iS a polnom1ial caxi of order k, 0 s a s n Each interior breakpoint (,,a 1 < a < n, generates m continuity conditions (0.3) ( IA C)=af-IA ( YA)(i) ), 0 <in, to make a functionf E Pkt be an element of Scm. Let Q = QS be the orthogonal projection onto S in L2[0, 1], i.e. with respect to the inner product (f, g) f(x) g(x) dx. We are interested in Q as a map in C[0, 1] or L' [0, 1] and we would like to get the estimates of its norm IIQIIOO = sup IIQflI/lflfloo, f where Ilf 11K0 = esssupo_x_ If(x)I ,f E L??[O 1]. Received September 29, 1980. 1980 Mathematics Subject Classification. Primary 41A1 5; Secondary 47A68. * This paper was partly supported by NSF grant 7906079. (?1983 American Mathematical Society 0025-571 8/82/0000-0750/$05.25