A modulus of smoothness based on an algebraic addition
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The purpose of this paper is to present a new approach to smoothness of nonperiodic functions. We consider the space of continuous functions on [−1, 1] as well as the weighted Lp-space and introduce a modulus of smoothness that is based on an algebraic addition ⊕ defined on [−1, 1]. The present paper is mainly concerned with general properties and groundwork, whereas a second paper  is devoted to more complex properties, in particular to an equivalent K-functional and to the characterization of best algebraic approximation. Moreover the equivalence with the Butzer-Stens modulus will be shown there.
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