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1. Sparse Polynomial Hermite Interpolation

   https://doi.org/10.1145/3476446.3535501  

   Kaltofen, Erich L. (July 2022, ISSAC '22 Proc. 2022 ACM Internat. Symp. Symbolic Algebraic Comput.)

   Amir Hashemi (Ed.)

   We present Hermite polynomial interpolation algorithms that for a sparse univariate polynomial f with coefficients from a field compute the polynomial from fewer points than the classical algorithms. If the interpolating polynomial f has t terms, our algorithms, require argument/value triples (w^i, f(w^i), f'(w^i)) for i=0,...,t + ceiling( (t+1)/2 ) - 1, where w is randomly sampled and the probability of a correct output is determined from a degree bound for f. With f' we denote the derivative of f. Our algorithms generalize to multivariate polynomials, higher derivatives and sparsity with respect to Chebyshev polynomial bases. We have algorithms that can correct errors in the points by oversampling at a limited number of good values. If an upper bound B >= t for the number of terms is given, our algorithms use a randomly selected w and, with high probability, ceiling( t/2 ) + B triples, but then never return an incorrect output. The algorithms are based on Prony's sparse interpolation algorithm. While Prony's algorithm and its variants use fewer values, namely, 2t+1 and t+B values f(w^i), respectively, they need more arguments w^i. The situation mirrors that in algebraic error correcting codes, where the Reed-Solomon code requires fewer values than the multiplicity code, which is based on Hermite interpolation, but the Reed-Solomon code requires more distinct arguments. Our sparse Hermite interpolation algorithms can interpolate polynomials over finite fields and over the complex numbers, and from floating point data. Our Prony-based approach does not encounter the Birkhoff phenomenon of Hermite interpolation, when a gap in the derivative values causes multiple interpolants. We can interpolate from t+1 values of f and 2t-1 values of f'. 

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2. Certified Hermite matrices from approximate roots

   https://doi.org/10.1016/j.jsc.2022.12.001  

   Ayyildiz Akoglu, Tulay; Szanto, Agnes (July 2023, Journal of Symbolic Computation)

   Let I = f1,..., fm ⊂ Q[x1,..., xn] be a zero dimensional radical ideal defined by polynomials given with exact rational coefficients. Assume that we are given approximations {z1,..., zk} ⊂ Cn for the common roots {ξ1,..., ξk} = V (I) ⊆ Cn. In this paper we show how to construct and certify the rational entries of Hermite matrices for I from the approximate roots {z1,..., zk}. When I is non-radical, we give methods to construct and certify Hermite matrices for √ I from the approximate roots. Furthermore, we use signatures of these Hermite matrices to give rational certificates of non-negativity of a given polynomial over a (possibly positive dimensional) real variety, as well as certificates that there is a real root within an ε distance from a given point z ∈ Qn. 

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3. Leapfrog Time-Stepping for Hermite Methods

   https://doi.org/10.1007/s10915-019-00938-x  

   Vargas, Arturo; Hagstrom, Thomas; Chan, Jesse; Warburton, Tim (July 2019, Journal of Scientific Computing)
4. Hermite multiwavelets for manifold-valued data

   https://doi.org/10.1007/s10444-023-10042-2  

   Cotronei, Mariantonia; Moosmüller, Caroline; Sauer, Tomas; Sissouno, Nada (June 2023, Advances in Computational Mathematics)
5. Scalable Hermite–Gaussian mode-demultiplexing hybrids

   https://doi.org/10.1364/OL.387460  

   Wen, He; Liu, Huiyuan; Zhang, Yuanhang; Sampson, Rachel; Fan, Shengli; Li, Guifang (April 2020, Optics Letters)

   We propose a Hermite–Gaussian (HG) mode-demulti-plexing hybrid (MDH) for coherent detection of mode-division multiplexed signals. The MDH, which performs multiple functionalities, including demultiplexing, local oscillator splitting, and optical 90-deg mixing, is realized based on the multi-plane light conversion technique. An isosceles right triangle output layout is employed to reduce the number of phase masks to fewer than the number of modes, significantly simplifying the construction of the MDH. A 10-Hermite–Gaussian (HG) mode MDH with only five phase masks is demonstrated by numerical simulation, achieving an insertion loss (IL) and mode dependent loss as low as $-<\#comment/>2.3$and 1.7 dB, respectively. The IL was further reduced to $-<\#comment/>1.5\mathrm{d}\mathrm{B}$through optimization of MDH parameters, such as the beam waists of the input HG modes and the output spots. 

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