Construction of Pairs of Reproducing Kernel Banach Spaces

Abstract

We extend the idea of reproducing kernel Hilbert spaces (RKHS) to Banach spaces, developing a theory of pairs of reproducing kernel Banach spaces (RKBS) without the requirement of existence of semi-inner product (which requirement is already explored in another construction of RKBS). We present several natural examples, which involve RKBS of functions with supremum norm and with l p -norm (1 ≤ p ≤ ∞). Special attention is devoted to the case of a pair of RKBS \((B,{B}^{\sharp })\) in which B has sup-norm and \({B}^{\sharp }\) has l 1-norm. Namely, we show that if \((B,{B}^{\sharp })\) is generated by a universal kernel and B is furnished with the sup-norm, then \({B}^{\sharp }\), furnished with the l 1-norm, is linearly isomorphically embedded in the dual of B. We reformulate the classical classification problem (support vector machine classifier) to RKBS and suggest that it will have sparse solutions when the RKBS is furnished with the l 1-norm.