We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

We obtain closed expressions for weighted orthogonal polynomials and optimal approximants associated with the function f ( z ) = 1 − 1 2 ( z 1 + z 2 ) f(z)=1-\frac {1}{\sqrt {2}}(z_1+z_2) and a scale of Hilbert function spaces in the unit 2 2 -ball having reproducing kernel ( 1 − ⟨ z , w ⟩ ) − γ (1-\langle z,w\rangle )^{-\gamma } , γ > 0 \gamma >0 . Our arguments are elementary but do not rely on reduction to the one-dimensional case.

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We obtain closed expressions for weighted orthogonal polynomials and optimal approximants associated with the function) and a scale of Hilbert function spaces in the unit 2-ball having reproducing kernel (1 − z, w ) −γ , γ > 0. Our arguments are elementary but do not rely on reduction to the one-dimensional case.

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We use an observation of Bohr connecting Dirichlet series in the right half plane C + to power series on the polydisk to interpret Carlson's theorem about integrals in the mean as a special case of the ergodic theorem by considering any vertical line in the half plane as an ergodic flow on the polytorus. Of particular interest is the imaginary axis because Carlson's theorem for Lebesgue measure does not hold there. In this note, we construct measures for which Carlson's theorem does hold on the imaginary axis for functions in the Dirichlet series analog of the disk algebra A(C + ).