Results on the associated classical orthogonal polynomials

AbstractLet {Pk(x)} be any system of the classical orthogonal polynomials, and let {Pk(x; c)} be the corresponding associated polynomials of order c (c ∈ N). Second-order recurrence relation (in k) is given for the connection coefficient an−1,k(c) in Pn−1(x;c)=σk=0n−1 an−1,k(c)Pk(x). This result is obtained thanks to a new explicit form of the fourth-order differential equation satisfied by Pn−1(·;c)

Efficient merging of multiple segments of B\'ezier curves

This paper deals with the merging problem of segments of a composite B\'ezier curve, with the endpoints continuity constraints. We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis (P. Wo\'zny, S. Lewanowicz, Comput. Aided Geom. Design 26 (2009), 566--579) to compute the control points of the merged curve. Thanks to using fast schemes of evaluation of certain connections involving Bernstein and dual Bernstein polynomials, the complexity of our algorithm is significantly less than complexity of other merging methods

B\'ezier representation of the constrained dual Bernstein polynomials

Explicit formulae for the B\'ezier coefficients of the constrained dual Bernstein basis polynomials are derived in terms of the Hahn orthogonal polynomials. Using difference properties of the latter polynomials, efficient recursive scheme is obtained to compute these coefficients. Applications of this result to some problems of CAGD is discussed.Comment: 10 page

Minimal projective operators

MR0549984The author reviews results (without proofs) from the theory of minimal projective operators. As he remarks, an excellent introduction to this theory is the survey paper by E. W. Cheney and K. H. Price [Approximation theory (Proc. Sympos., Lancaster, 1969), pp. 261–289, Academic Press, London, 1970; MR0265842]. The author is motivated by a number of papers in this topic published after 1970, bringing essentially new results, e.g., an existence theorem for the minimal operators in the class of all projective operators from a linear normed space onto its subspace due to P. D. Morris and Cheney [J. Reine Angew. Math. 270 (1974), 61–76; MR0358188]. The paper consists of the following chapters: (0) Introduction; (1) Projective operators; (2) Fundamental properties of minimal projective operators; (3) Existence and characterization of minimal projective operators; (4) Some polynomial projective operators in the space C[−1,1]; References (45 items)

A fast algorithm for the construction of recurrence relations for modified moments

A new approach is presented for constructing recurrence relations for the modified moments of a function with respect to the Gegenbauer polynomials