Coefficients of the adapted Chebyshev series

Rob Harley

June 1992

Abstract

In the report “Formalizing SML’s arithmetic by tracking cumulated error bounds” we gave a method for adapting some Chebyshev expansions for use in polynomial approximation when relative error is to be minimised. Here we give a short proof that equation (3.27) correctly defines the adapted coefficients. In particular we show that:

∑_{n=0}^{∞} a_{n}T_{2n+1}(x) = x ∑_{n=0}^{∞} b_{n}T_{2n}(x)

where:

b_{n} =^{def} (2 − δ_{0n}) ∑_{i=0}^{∞}(−)^{i}a_{n+i}

We substitute the definition of the b_{n}’s into the RHS obtaining a double series, and rearrange it into a singly infinite sum by grouping the summands along diagonals:

RHS = x ∑_{n=0}^{∞}(2 − δ_{0n}) ∑_{i=0}^{∞}(−)^{i}a_{n+i}T_{2n}(x) = x ∑_{k=0}^{∞} a_{k} ∑_{n=0}^{k}(−)^{k−n}(2 − δ_{0n})T_{2n}(x)

Then we prove the equality by considering the multiplier of each a_{k} on the left and right hand sides. On the LHS the multiplier is clearly T_{2k+1}(x), whereas on the RHS it is x ∑_{n=0}^{k}(−)^{k−n}(2 − δ_{0n})T_{2n}(x). We use the recurrence relations (3.16) to show that these two forms are in fact equal:

T_{2k+1}(x) = 2xT_{2k}(x) − T_{2k−1}(x)

= 2x(T_{2k}(x) − T_{2k−2}(x)) + T_{2k−3}(x)
= …
= 2x(∑_{n=1}^{k}(−)^{k−n}T_{2n}(x)) + (−)^{k}T_{1}(x)
= x ∑_{n=0}^{k}(−)^{k−n}(2 − δ_{0n})T_{2n}(x)

Thus the equality is proved.