P. Jiménez-Rodríguez

When the Identity Theorem "Seems" to Fail

Abstract The Identity Theorem states that an analytic function (real or complex) on a connected domain is uniquely determined by its values on a sequence of distinct points that converge to a point of its domain. This result is not true in general in the real setting, if we relax the analytic hypothesis on the function to infinitely many times differentiable. In fact, we construct an algebra of functions A enjoying the following properties: (i) A is uncountably infinitely generated (that is, the cardinality of a minimal system of generators of A is uncountable); (ii) every nonzero element of A is nowhere analytic; (iii) A ⊂C∞ (ℝ); (iv) every element of A has infinitely many zeros in ℝ; and (v) for every f ε A 0 and n ε N, f(n) (the nth derivative of f) enjoys the same properties as the elements in A0 . This construction complements those made by Cater and by Kim and Kwon, and published in the American Mathematical Monthly in 1984 and 2000, respectively.

Sharp values for the constants in the polynomial Bohnenblust-Hille inequality

In this paper we prove that the complex polynomial Bohnenblust–Hille constant for 2-homogeneous polynomials in is exactly . We also give the exact value of the real polynomial Bohnenblust–Hille constant for 2-homogeneous polynomials in . Finally, we provide lower estimates for the real polynomial Bohnenblust–Hille constant for polynomials in of higher degrees.