The Giri Approximant: Functional Approximation Using Real Reciprocal Terms

We study a real partial-fraction representation obtained by temporarily modifying the linearcoefficient of a power series, forming a diagonal Pad´e approximant, and then reversing themodification with a linear correction. We call the resulting representation a Giri approximant.Its evaluation uses additions, reciprocal evaluations, multiplication by fixed constants, and onelinear term, without explicitly forming successive positive powers of the argument. For thetype-[2/2] construction, we derive the denominator coefficients and an explicit condition for twodistinct real poles. Under stated nondegeneracy assumptions, infinitely many modified linearcoefficients satisfy this condition. The first unmatched series coefficient supplies a local parameter-selection criterion, which must be supplemented by pole-location and conditioning constraintson the intended evaluation interval. Five examples compare the resulting type-[3/2] formswith degree-4 Taylor polynomials and ordinary type-[2/2] and type-[3/2] Pad´e approximants.The comparisons quantify the tradeoff between an unconstrained rational approximation and adecomposition into simple real reciprocal terms.