On interpolation by analytic functions with special properties and some weak lower bounds on the size of circuits with symmetric gates

The author investigates the question of whether or not a specific Boolean function in n variables can be interpolated by an analytic function in the same variables whose partial derivatives of all orders span a subspace of low dimension in the space of analytic functions. The upper and lower bounds for this dimension yield some weak circuit lower bounds. For a particular function, an Ω(n/log n)-size lower bound is obtained for its computation by a circuit whose gates are symmetric. For the same function an Ω(n) lower bound is obtained for the circuit with mod_k gates