## Congres international des mathematiciens. NICE 1970 by Comite d'Organization du Congres

By Comite d'Organization du Congres

Papers from the overseas arithmetic convention in great in 1970

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T m, t] n (t) 43 Curves and surfaces Generally, for an exact interpolation process on the polynomials of degree m, a Taylor formula is used to establish an expression of the error involving the derivative of order m + 1 of the function f (assumed to be sufficiently derivable). The following equations are obtained: where a ~ ~ ~ Rf (t) = Rf (t) = h. tim) (n n (t) (m + 1) f: Km (x, t) f{m + I) (t) dt where Km is the error core function. The convergence problem is as follows: if Pm is the interpolation polynomial of f on m + 1 points, will the interpolation error Rf equal zero when m moves towards infinity?

The common points allow the discontinuity of certall derivatives to be obtained. If m> 1 and if II never has more than m - 1 common points, then Mi m and Ni m are continuous functions. {soning is u~ed. The II elements are B-spline nodes. The following are taken from the recurrence formulae on the divided differences: if t E [tj, tj + d if not Curves and surfaces N·I,m (t) = 1- t - t· I+m-I 1- N; I I ' m- 1 (t) + 63 t; + m - t t;+m t;+1 N;+I,m-l(t) The Ni,m verify the following properties: t E ]t;, t; + m[ N I,m =0 m L t N·I,m (t) ;= 1 ¢ ]t;, t; + m[ =1 The ~; are known as the B-spline nodal points.

S is a polynomial of degree 2q - 1 in each interval ] tj, ti+ d, i = 1, ... , n - 1; 2. s is a polynomial of degree q - 1 in [a, t 1 [ and] tn' b] ; 3. s(2q-2) is continuous (derivative of order 2q - 2). S is the name given to the set of s functions. A function of this type may be expressed in the following form: q-i n s(t)=i:O ajti+i:ldi (t-ti)~q-l (2q-l)! Curves and surfaces with: (t - t j ) and: t n j= _I +- 51 t - tj if t - ti > 0 O'f 1 not dj (tj)k = 0 ; k = 0, ... , q - 1 I Note that on each intelVal, s is entirely defined by its value and the value of the derivatives up to the order q - 1 at each extremity of the intelVal.