Interpolation by Hankel translates of a basis function: inversion formulas and polynomial bounds

Area of Science:

• Mathematical Analysis
• Harmonic Analysis
• Approximation Theory

Background:

• Established interpolation schemes using Hankel translates of basis functions Φ in continuous function spaces Y n.
• Defined a connection between basis function Φ and weight w via a distributional identity involving the generalized Hankel transform hμ'.
• Introduced the Zemanian space ℋμ and Bessel differential operator Sμ.

Purpose of the Study:

• To derive explicit representations for derivatives of basis functions (SμmΦ) and their Hankel transforms.
• To establish polynomial bounds for these derivatives and for members of the interpolation space Y n.
• To provide inverse formulas generalizing the relationship between Hankel transforms and weights.

Main Methods:

• Utilized projection operators from a direct sum decomposition of the Zemanian space ℋμ.
• Derived formulas for derivatives SμmΦ and their Hankel transforms under conditions of continuity for SμmΦ.
• Applied these formulas to obtain polynomial bounds for the derivatives.

Main Results:

• Obtained explicit representations for the mth iterate of the Bessel differential operator (SμmΦ) and its Hankel transform.
• Established conditions for the validity of these representations based on the continuity of SμmΦ.
• Derived polynomial bounds for the derivatives SμmΦ and corresponding results for the interpolation space Y n.

Conclusions:

• The derived formulas serve as inverses to generalized equations relating Hankel transforms and weights.
• The results provide valuable tools for analyzing and bounding derivatives within interpolation spaces.
• This work extends the understanding of Hankel translates interpolation and its applications in function spaces.