For a Hermite-Biehler function E of mean type τ we determine the optimal (with respect to the de Branges measure of E) majorant M + E and minorant M − E of exponential type τ for the truncation of x → (x 2 + a 2 ) −1 . We prove thatwhere K is the reproducing kernel for the de Branges space H(E). As an application we determine the optimal majorant and minorant for the Heaviside function that vanish at a fixed point α = ia on the imaginary axis. We show that the difference of majorant and minorant has integral value (πa − tanh(πa)) −1 πa.
Let E = A−iB be a Hermite-Biehler entire function of exponential type τ /2 where A and B are real entire, and consider dµ(x) = dx/|E(x)| 2 . We show that the sign of the product AB is an extremal signature for the space of functions of exponential type τ with respect to the norm of L 1 (µ). This allows us to find best approximations by entire functions of exponential type τ in L 1 (µ)-norm to certain special functions (e.g., the Gaussian and the Poisson kernel).for all F ∈ B 1 (µ, τ ). We denote by A(µ, τ ) the class of high pass functions for B 1 (µ, τ ). (We use the letter A since these functions correspond to the class of integration functionals on L 1 (µ) that annihilate B 1 (µ, τ ).) Of particular interest to us is the subclassThe elements of S(µ, τ ) will be called extremal signatures for B 1 (µ, τ ) or simply extremal signatures, if measure and type are clear from the context. The following connection between extremal signatures and best approximation is well known and explains why the study of high pass signatures is relevant. For w = re iθ with r > 0 and 0 ≤ θ < 2π we define sgn(w) = e iθ , and we set sgn(0) = 0.Date: October 24, 2018. 2000 Mathematics Subject Classification. 30D10 and 30D55 and 42A10. Key words and phrases. Best approximation and extremal signature and bandlimited function and exponential type and Hardy space and Hermite-Biehler function.
Abstract. The Euler-Lagrange equations associated to the problem of minimizing a power-law functional acting on symmetrized gradients are identified. A formal derivation of the limiting system of partial differential equations stemming from these equations as p tends to infinity is provided. Our computations are reminiscent of the derivation of the infinity Laplace equation starting from the p-Dirichlet integral.
We consider weighted uniform convergence of entire analogues of the Grünwald operator on the real line. The main result deals with convergence of entire interpolations of exponential type τ > 0 at zeros of Bessel functions in spaces with homogeneous weights. We discuss extensions to Grünwald operators from de Branges spaces.
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