### A criterion for subharmonicity of a function of the spectrum

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We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a δ-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on δ. We apply the obtained estimates...

We determine the norm in ${L}^{p}\left(\mathbb{R}\u208a\right)$, 1 < p < ∞, of the operator $I-{\mathcal{F}}_{s}{\mathcal{F}}_{c}$, where ${\mathcal{F}}_{c}$ and ${\mathcal{F}}_{s}$ are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the ${L}^{p}$-norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real a,b. Best...

The spectral synthesis theorem for Sobolev spaces of Hedberg and Wolff [7] has been applied in combination with duality, to problems of Lq approximation by analytic and harmonic functions. In fact, such applications were one of the main motivations to consider spectral synthesis problems in the Sobolev space setting. In this paper we go the opposite way in the context of the BMO-H1 duality: we prove a BMO approximation theorem by harmonic functions and then we apply the ideas in its proof to produce...

Let $f$ be a mapping from an open set in ${\mathbf{R}}^{p}$ into ${\mathbf{R}}^{q}$, with $p\>q$. To say that $f$ preserves Brownian motion, up to a random change of clock, means that $f$ is harmonic and that its tangent linear mapping in proportional to a co-isometry. In the case $p=2$, $q=2$, such conditions signify that $f$ corresponds to an analytic function of one complex variable. We study, essentially that case $p=3$, $q=2$, in which we prove in particular that such a mapping cannot be “inner” if it is not trivial. A similar result for $p=4$, $q=2$ would solve...