Abraham Lempel

Systematic derivation of spline bases

Abstract In this paper we present an explicit derivation of general spline bases over function spaces closed under differentiation. We determine the necessary and sufficient conditions for the existence of B-splines in such a function space and prove that whenever such a basis exists it is essentially unique. Rather than following the common practice of presenting a mathematical definition of B-splines and then proceeding to prove some of their desired properties, we begin by stipulating a set of design objectives and then proceed to derive functions that meet these objectives. We stipulate the following standard design objectives; a given degree of continuity, least feasible order for the given continuity requirement, and shape invariance under translation. As it turns out, these objectives form a complete set in the sense that no other requirement can be imposed without it being already implied by the ones listed. In other words, when the listed objectives are translated into algebraic constraints, the resulting equations have an essentially unique solution with no remaining degrees of freedom. Clearly, when other desirable objectives such as convexity and the variation diminishing property are attainable, they follow as a by-product by virtue of uniqueness.

On the Derivation of Spline Bases

In this paper we present an explicit derivation of general spline bases over function spaces closed under differentiation. We determine the necessary and sufficient conditions for the existence of B-splines in such a function space and prove that whenever such a basis exists it is essentially unique. Rather than following the common practice of presenting a mathematical definition of B-splines and then proceeding to prove some of their desired properties, we begin by stipulating a set of design objectives and then proceed to derive functions that meet these objectives. We stipulate the following standard design objectives: a given degree of continuity, least feasible order for the given continuity requirement, and shape invariance under translation. As it turns out, these objectives form a complete set in the sense that no other requirement can be imposed without it being already implied by the ones listed. In other words, when the listed objectives are translated into algebraic constraints, the resulting equations have an essentially unique solution with no remaining degrees of freedom. Clearly, when other desirable objectives such as convexity and the variation diminishing property are attainable, they follow as a byproduct by virtue of uniqueness.

Application of Finte Fields to Memory Interleaving

A hashing scheme for memory interleaving, based on the properties of finite field exponentiation is presented. The scheme maps sequences of l-bit addresses to sequences of m-bit memory module numbers, with m<l. For input sequences that are in arithmetic progression, the output sequence has a provably uniform distribution on the average, and no “pathologies” for a prescribed range of strides in the input sequence. We prove bounds on the lengths of runs in the output sequence, and prove the surprising result that when m/l is bounded away from both 0 and 1, the run length can be bounded by a constant. The proposed scheme is highly amenable to fast systolic implementation.