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The multiplicity of a spline zero

The multiplicity of a zero of a (univariate, polynomial) spline is defined in terms of its B-spline coefficients, thus making certain bounds trivial while, at the same time, adhering to the principle that the multiplicity of a zero indicates the number of simple zeros nearby achievable by a nearby element from the same class. In particular, the multiplicity depends on the class to which the function is assumed to belong. The resulting multiplicity turns out to coincide with that given recently in more traditional terms by T. N. T. Goodman. The variation-diminishing property of B-splines provides a ready upper bound on the number of zeros of a (univariate) spline in terms of the number of sign changes in the sequence of its B-spline coefficients. Because of the compact support of the B-spline, this bound is even local. Finally, the bound even holds if zeros are counted with certain suitable multiplicities. However, the sharpness of the resulting bound depends on just how one defines the multiplicity of a spline zero. For arbitrary real-valued functions, smooth or not, on some interval I, zeros have long been classified as nodal (aka simple, odd) or nonnodal (aka double, even) depending on whether or not the function changes sign across the zero. Here, the function f is said to change sign across ξ, or, ξ is a sign change of f , iff, for all ε > 0, f (α)f (ω) < 0 for some ξ − ε ≤ α < ξ < ω ≤ ξ + ε. For a continuous f , a sign change is necessarily also a zero of f , but need not be a zero if f fails to be continuous. Not surprisingly, such a sign change is treated, nevertheless, as a zero whenever zeros are considered in the context of the possible sign changes of the function. For smooth f , this rudimentary multiplicity concept (of simple vs double zeros) has been augmented and refined by assigning the multiplicity m + 1 to any zero of f which is a zero of multiplicity m of Df , the first derivative of f. This refinement is mainly motivated by the desire to exploit Rolle's theorem to the fullest, and is, offhand, of not much help when f is not smooth. It is one purpose of this note to advocate, instead, the notion (implicit in various definitions of multiplicity of zeros) …