Norms Of Polynomials Research Articles

Multidimensional fractal scaling analysis using higher order moving average polynomials and its fast algorithm

The detrending moving average (DMA) analysis demonstrates excellent performance for the characterization of long-range correlations and fractal scaling and is performed in various research fields. The conventional DMA with a simple moving average can remove linear trends embedded in the observed time series. To improve the detrending ability of the DMA, higher-order DMA including a higher order polynomial detrending was also introduced using the Savitzky-Golay filter and its fast implementation algorithm was developed. However, the higher-order DMA applicable to higher dimensional data is yet to be well established. As the data dimension increases, an increase in the computational cost becomes a problem that needs to be resolved. Further, the implementation of the higher order DMA is a time-consuming procedure. To resolve this problem, we here proposed a fast algorithm for multidimensional DMA with higher order polynomial detrending. In the proposed algorithm, to reduce the computational complexity, parallel translation and recurrence techniques are introduced. Monte Carlo experiments for two-dimensional data show that the computational time of the proposed algorithm is approximately proportional to the cubic of the data length, whereas the computational time of the conventional implementation is approximately proportional to the quartic of the data length. Moreover, we evaluate the estimation accuracy of the Hurst exponent of the proposed method. Finally, we demonstrate the possible application of the proposed method by estimating the Hurst exponent of images.

Randomized weakly admissible meshes

A weakly admissible mesh (WAM) on a continuum real-valued domain is a sequence of discrete grids such that the discrete maximum norm of polynomials on the grid is comparable to the supremum norm of polynomials on the domain. The asymptotic rate of growth of the grid sizes and of the comparability constants must grow in a controlled manner. In this paper, we recognize that the notion of a WAM can be generalized to a sequence of hierarchical subspaces not necessarily of polynomial functions, and we analyze particular strategies for random sampling as a technique for generating WAM’s. Our main results show that WAM’s and their stronger variant, admissible meshes (AM’s), can be generated by random sampling, and our analysis provides concrete estimates for growth and rates of growth of both the meshes and the discrete-continuum comparability constants.

ON ONE INEQUALITY OF DIFFERENT METRICS FOR TRIGONOMETRIC POLYNOMIALS

We study the sharp inequality between the uniform norm and \(L^p(0,\pi/2)\)-norm of polynomials in the system \(\mathscr{C}=\{\cos (2k+1)x\}_{k=0}^\infty\) of cosines with odd harmonics. We investigate the limit behavior of the best constant in this inequality with respect to the order \(n\) of polynomials as \(n\to\infty\) and provide a characterization of the extremal polynomial in the inequality for a fixed order of polynomials.

Norms of polynomials of the Volterra operator

We compute the operator norm of real-quadratic polynomials of the Volterra operator. This is used to test whether the Crouzeix conjecture holds for the Volterra operator.

On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices

Let \(U^N = (U_1^N,\dots ,U^N_p)\) be a p-tuple of \(N\times N\) independent Haar unitary matrices and \(Z^{NM}\) be any family of deterministic matrices in \({\mathbb {M}}_N({\mathbb {C}})\otimes {\mathbb {M}}_M({\mathbb {C}})\). Let P be a self-adjoint non-commutative polynomial. In Voiculescu (Int Math Res Notices 1:41–63, 1998), Voiculescu showed that the empirical measure of the eigenvalues of this polynomial evaluated in Haar unitary matrices and deterministic matrices converges towards a deterministic measure defined thanks to free probability theory. Now, let f be a smooth function. The main technical result of this paper is a precise bound of the difference between the expectation of $$\begin{aligned} \frac{1}{MN} {{\,\mathrm{Tr}\,}}_{{\mathbb {M}}_N({\mathbb {C}})}\otimes {{\,\mathrm{Tr}\,}}_{{\mathbb {M}}_M({\mathbb {C}})}\left( f(P(U^N\otimes I_M,Z^{NM})) \right) , \end{aligned}$$and its limit when N goes to infinity. If f is seven times differentiable, we show that it is bounded by \(M^2 \left\| f\right\| _{{\mathcal {C}}^6} \ln ^2(N)\times N^{-2}\). As a corollary we obtain a new proof with quantitative bounds of a result of Collins and Male which gives sufficient conditions for the operator norm of a polynomial evaluated in Haar unitary matrices and deterministic matrices to converge almost surely towards its free limit. Our result also holds in much greater generality. For instance, it allows to prove that if \(U^N\) and \(Y^{M_N}\) are independent and \(M_N=o(N^{1/3}\ln ^{-2/3}(N))\), then the norm of any polynomial in \((U^N\otimes I_{M_N}, I_N\otimes Y^{M_N})\) converges almost surely towards its free limit. Previous results required that \(M=M_N\) is constant.

Multiplication operator and average characteristic polynomial associated with exceptional Jacobi polynomials

Studying the multiplication operator associated with exceptional Jacobi polynomials, the zero distribution of the corresponding average characteristic polynomials is determined. Applying this result, the location of zeros of certain self-inversive polynomials is examined.

On the unimodality of average edge cover polynomials

The edge cover polynomial of a graph G is the function E(G,x)=∑i≥1e(G,i)xi, where e(G,i) is the number of edge coverings of G with size i. In this paper, we show that the average edge cover polynomial of order n is reduced to the edge cover polynomial of complete graph Kn, based on which we establish that the average edge cover polynomial of order n is unimodal and has at least n−3 non-real roots.

On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices

Let $X^N = (X_1^N,\dots, X^N_d)$ be a d-tuple of $N\times N$ independent GUE random matrices and $Z^{NM}$ be any family of deterministic matrices in $\mathbb{M}_N(\mathbb{C})\otimes \mathbb{M}_M(\mathbb{C})$. Let $P$ be a self-adjoint non-commutative polynomial. A seminal work of Voiculescu shows that the empirical measure of the eigenvalues of $P(X^N)$ converges towards a deterministic measure defined thanks to free probability theory. Let now $f$ be a smooth function, the main technical result of this paper is a precise bound of the difference between the expectation of $$\frac{1}{MN}\text{Tr}\left( f(P(X^N\otimes I_M,Z^{NM})) \right)$$ and its limit when $N$ goes to infinity. If $f$ is six times differentiable, we show that it is bounded by $M^2\left\Vert f\right\Vert_{\mathcal{C}^6}N^{-2}$. As a corollary we obtain a new proof of a result of Haagerup and Thorbj\o rnsen, later developed by Male, which gives sufficient conditions for the operator norm of a polynomial evaluated in $(X^N,Z^{NM},{Z^{NM}}^*)$ to converge almost surely towards its free limit. Restricting ourselves to polynomials in independent GUE matrices, we give concentration estimates on the largest eingenvalue of these polynomials around their free limit. A direct consequence of these inequalities is that there exists some $\beta>0$ such that for any $\varepsilon_1<3+\beta)^{-1}$ and $\varepsilon_2<1/4$, almost surely for $N$ large enough, $$-\frac{1}{N^{\varepsilon_1}} \| P(X^N)\| - \left\Vert P(x)\right\Vert \leq \frac{1}{N^{\varepsilon_2}}.$$ Finally if $X^N$ and $Y^{M_N}$ are independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any polynomial in $(X^N\otimes I_{M_N},I_N\otimes Y^{M_N})$ converges almost surely towards its free limit. This result is an improvement of a Theorem of Pisier, who was himself using estimates from Haagerup and Thorbj\o rnsen, where $M_N$ had size $o(N^{1/4})$.

Multidimensional Fractal Scaling Analysis Using Higher Order Moving Average Polynomials and Its Fast Algorithm

Multidimensional Fractal Scaling Analysis Using Higher Order Moving Average Polynomials and Its Fast Algorithm

A sharp Bombieri inequality, logarithmic energy and well conditioned polynomials

In this paper we explore the connections between minimizers of the discrete logarithmic energy on the 2-dimensional sphere, univariate polynomials with optimal condition number in the Shub-Smale sense and a quotient involving norms of polynomials. Our main results are that polynomials with optimal condition number produce spherical points with small logarithmic energy (the reverse result was already known) and a sharp Bombieri type inequality for univariate polynomials with complex coefficients.

Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices

In this article, we consider the generalised two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. It is shown that with parameters chosen as $$1/k_i$$ , $$k_i\in {\mathbb {Z}}_{>0}$$ ( $$i=1,\,2$$ ), the average characteristic polynomials admit $$(k_1+k_2+2)$$ -term recurrence relations, which can be interpreted as spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest solvable example is given.

Spectral Curves, Variational Problems and the Hermitian Matrix Model with External Source

We consider the hermitian random matrix model with external source and general polynomial potential, when the source has two distinct eigenvalues but is otherwise arbitrary. All such models studied so far have a common feature: an associated cubic equation (“spectral curve”), one of whose solutions can be expressed in terms of the Cauchy (a.k.a. Stieltjes) transform of the limiting eigenvalue distribution \(\lambda \). This is our starting point: we show that to any such a spectral curve (not necessarily given by a random matrix ensemble) it corresponds a unique vector-valued measure with three components on the complex plane, characterized as a solution of a variational problem stated in terms of their logarithmic energy. We describe all possible geometries of the supports of these measures: the third component, if non-trivial, lives on a contour on the plane and separates the supports of the other two measures, both on the real line. This general result is applied to the random matrix model with external source, under an additional assumption of uniform boundedness of the zeros of a sequence of average characteristic polynomials, when the size of the matrices goes to infinity (equivalently, uniform boundedness of certain recurrence coefficients). It is shown that any limiting zero distribution for such a sequence can be written in terms of a solution of a spectral curve, and thus admits the variational description obtained in the first part of the paper. As a consequence of our analysis, we obtain that the density of this limiting measure can have only a handful of local behaviors: sine, Airy and their higher order type behavior, Pearcey or yet the fifth power of the cubic (but no higher order cubics can appear). We also compare our findings with the most general results available in the literature, showing that once an additional symmetry is imposed, our vector critical measure contains enough information to recover the solutions to the constrained equilibrium problem that was known to describe the limiting eigenvalue distribution in this symmetric situation.

Exact upper bounds of norms of polynomials, splines, and their derivatives on arbitrary interval

We establish sharp estimates of the $L_q$-norms on any finite interval for the polynomials and splines and their derivatives with the help of local $L_p$-norms of these polynomials and splines.

Heights of polynomials over lemniscates

We consider a family of heights defined by the $L_p$ norms of polynomials with respect to the equilibrium measure of a lemniscate for $0 \le p \le \infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler measure) and $p=\infty$ corresponds to the standard supremum norm. This special choice of the measure allows to find an explicit form for the geometric mean of a polynomial, and estimate it via certain resultant. For lemniscates satisfying appropriate hypotheses, we establish explicit polynomials of lowest height, and also show their uniqueness. We discuss relations between the standard results on the Mahler measure and their analogues for lemniscates that include generalizations of Kronecker's theorem on algebraic integers in the unit disk, as well as of Lehmer's conjecture.

Generalizations of Zygmund-type integral inequalities for the polar derivative of a complex polynomial

Some Zygmund-type integral inequalities for the polar derivatives of complex polynomials, inspired by the classical Bernstein-type inequalities that relate the uniform norms of polynomials and their derivatives on the unit circle, are investigated. The obtained results sharpen as well as generalize some already known L^{delta }-estimates between polynomials and their polar derivatives.

Ordering among the topologies induced by various polynomial norms

Let us consider the following two norms in the vector space \,$\mathcal{P}$ \,of all complex polynomials: $$ \|p\|_{D_{r}}:= \sup\{|p(z)|: |z|<r\}, \text{ and } \|p\|_{1}:=\sum_{i=0}^{n}|a_{i}|, $$ where \,$p(z) = \sum_{i=0}^{n} a_{i} z^{i}$. In this note we show that, if $0 <\varepsilon < \varepsilon' < 1 < r < r'$, then $$ \|\cdot\|_{D_{\varepsilon}}\prec\|\cdot\|_{D_{\varepsilon'}}\prec\|\cdot\|_{D_{1}}\prec \|\cdot\|_{1}\prec \|\cdot\|_{D_{r}}\prec \|\cdot\|_{D_{r'}}, $$ where \,$\prec$ \,represents the natural (strict) partial order in their corresponding induced topologies.

Combinatorial analysis of the period mapping: the topology of 2D fibres

We study the period mapping from the moduli space of real hyperelliptic curves to a Euclidean space. The mapping arises in the analysis of Chebyshev’s construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of the moduli space into polyhedra labelled by planar graphs allows us to investigate the global topology of low-dimensional fibres of the period mapping. Bibliography: 23 titles.

Tree-Based Algorithms for the Stability of Discrete-Time Switched Linear Systems Under Arbitrary and Constrained Switching

We present a direct approach to study the stability of discrete-time switched linear systems that can be applied to arbitrary switching, as well as when switching is constrained by a switching automaton. We explore the tree of possible matrix products, by pruning the subtrees rooted at contractions and looking for unstable repeatable products. Generically, this simple strategy either terminates with all contracting leafs-showing the system's asymptotic stability-or finds the shortest unstable and repeatable matrix product. Although it behaves in the worst case as the exhaustive search, we show that its performance is greatly enhanced by measuring contractiveness w.r.t. sum-of-squares polynomial norms, optimized to minimize the largest expansion among the system's modes.

Polynomial Norms

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from $d^{\text{th}}$ roots of polynomials, we prove that any norm can be approximated arbitrarily well by a polynomial norm. We then investigate the computational problem of testing whether a form gives a polynomial norm. We show that this problem is strongly NP-hard already when the degree of the form is 4, but can always be answered by testing feasibility of a semidefinite program (of possibly large size). We further study the problem of optimizing over the set of polynomial norms using semidefinite programming. To do this, we introduce the notion of r-sos-convexity and extend a result of Reznick on sum of squares representation of positive definite forms to positive definite biforms. We conclude with some applications of polynomial norms to statistics and dynamical systems.

Improving the repeatability of two-rate model parameter estimations by using autoencoder networks.

The adaptive changes elicited in visuomotor adaptation experiments are usually well explained at group level by two-rate models (Smith et al., 2006), but parameters fitted to individuals show considerable variance. Data cleaning can mitigate this problem, but the assumption of smoothness can be problematic due to fast adaptive changes with discontinuous derivatives. In this paper, we collected time-series data from an experimental paradigm involving repeated training and investigated the effect of various cleaning methods, including an autoencoder network (AE), on the parameter estimation. We compared changes in the fitted parameters across different methods and across training repetitions. The results suggest that AE performed best overall, without introducing an underestimation bias on bf like moving average or piecewise polynomials, and that it reduced the within-subject variance overall and especially that of the fast retention rate af by >50%.