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Let (xn) be a sequence of real numbers. (xn) corresponds to a number Lim xn called the Banach limit of (xn) satisfying the following conditions:(1)Lim (axn + byn) = a Lim xn + b Lim yn(2)If xn ≥ 0 for every n, then Lim xn ≥ 0(3)Lim xn+1 = Lim xn(4)If xn = 1 for every n, then Lim xn = 1The existence of such limits is proved by Banach [1].

Abstract. The main results are the local representation theorems associating the local weak «-Taylor series expansion of a function defined on a Banach space to a local n-Taylor series expansion of the coefficients. These theorems are used to prove a converse of Taylor's theorem which uses weaker hyptohesis than used by others. Another useful application of the above results is done in [2] to study a class of functions called «-convex functions.1. Introduction. The first main result (Theorem 3.2) states that if a function / defined on a Banach space has a weak «-Taylor series expansion (see definition in §2) throughout an open set and if the nth order coefficient is continuous at one point, then the coefficients have strong Taylor series expansion (see definition in §2) about every point near to that point. The proof of this theorem uses a sort of mean value theorem (Theorem 3.1). A particular case of Theorem 3.1 is proved by McLeod [11] and Dieudonné [5]. Theorem 3.3 generalizes Theorem 3.2 in the sense that the nth order coefficient has a strong m-Taylor series expansion about a point instead of being continuous at that point. A useful application of Theorem 3.2 is done by the author [2] to study the question of existence of higher differentials of n-convex functions which contain a class of functions called subconvex functions.The second main result (Theorem 4.1) states that under the situation of Theorem 3.2 if the coefficients are bounded then the coefficients of the weak Taylor series are in fact iterated Fréchet differentials. This is a converse of Taylor's theorem which uses weaker hypothesis than used by Nashed [12] and Abraham and Robin [1]. In fact it does not use the superfluous condition of continuity of lower order coefficients which is used by Abraham and Robin.

Abstract.By observing a convex property of discrete differences, one-sided /c-discrete, A:-discrete Gâteaux and /c-discrete Fréchet differentials are introduced. It is proved that a locally bounded n-convex function has /c-discrete Fréchet differentials for 1 < k < n -2 and one-sided (n -l)-discrete differentials at every point of its domain. Various properties of discrete differentials of an n-convex function are studied. As an application of these results the author proves that an n-convex function has a strong (n -2)-Taylor series expansion and an (n -l)th Fréchet differential provided it has a strong n-Taylor series expansion about the point.