On Weighted Average Interpolation with Cardinal Splines

Given a sequence of data ({ y_{n} } _{n in mathbb{Z}}) with polynomial growth and an odd number (d), Schoenberg proved that there exists a unique cardinal spline (f) of degree (d) with polynomial growth such that (f ( n ) =y_{n}) for all (nin mathbb{Z}). In this work, we show that this result also holds if we consider weighted average data (fast h ( n ) =y_{n}), whenever the average function (h) satisfies some light conditions. In particular, the interpolation result is valid if we consider cell-average data (int_{n-a}^{n+a}f ( x ) dx=y_{n}) with (0< aleq 1/2). The case of even degree (d) is also studied.