Lidstone interpolation III. Several variables

Abstract

A polynomial in a single variable is uniquely determined by its derivatives of even order at 0 and 1. More precisely, such an univariate polynomial can be written and a finite sum of f ( 2 n ) ( 0 ) Λ n ( 1 − z ) f^{(2n)}(0) \Lambda _n(1-z) and f ( 2 n ) ( 1 ) Λ n ( z ) f^{(2n)}(1) \Lambda _n(z) , ( n ≥ 0 n\ge 0 ), where the Λ n ( z ) \Lambda _n(z) are the Lidstone polynomials defined by the conditions ( d d z ) 2 k Λ n ( 0 ) = 0 and ( d d z ) 2 k Λ n ( 1 ) = δ k , n , k ≥ 0 , n ≥ 0. \begin{equation*} \left (\frac {\mathrm {d}}{\mathrm {d}z}\right )^{2k} \Lambda _n(0)=0\text { and } \left (\frac {\mathrm {d}}{\mathrm {d}z}\right )^{2k} \Lambda _n(1)=\delta _{k,n},\quad k\ge 0, \; n\ge 0. \end{equation*} We generalize this theory to n n variables, replacing the two points 0 0 , 1 1 in C \mathbb {C} with n + 1 n+1 points e _ 0 , e _ 1 , … , e _ n {\underline {e}}_0,{\underline {e}}_1,\dots ,{\underline {e}}_n in C n \mathbb {C}^n , where e _ 0 {\underline {e}}_0 is the origin of C n \mathbb {C}^n and e _ 1 , … , e _ n {\underline {e}}_1,\dots ,{\underline {e}}_n the canonical basis of C n \mathbb {C}^n . By selecting a suitable subset of even order derivatives at these n + 1 n+1 points, we show that any polynomial in n n variables has a unique expansion. We obtain generating series for these sequences of polynomials and we deduce an expansion for entire functions in C n \mathbb {C}^n of exponential type > π >\pi . We extend to several variables results due to Lidstone [Proceedings Edinburgh Math. Soc., II. Ser. (2)2, 16-19 (1930)], Poritsky [Trans. Amer. Math. Soc. 34 (1932), pp. 274–331], Whittaker [Proc. London Math. Soc. (2) 36 (1934), pp. 451–469], Schoenberg [Bull. Amer. Math. Soc. 42 (1936), pp. 284–288], Buck [Proc. Amer. Math. Soc. 6 (1955), pp. 793–796]. We also show that our results are, to a certain extent, best possible.