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Tamás Erdélyi

Publications and source records attributed to Tamás Erdélyi.

At least 19 recordsLinked to original sources
arXiv2026

The Zero Set of an Electric Field from a Finite Number of Point Charges

We consider the structure of the zero set in ${\mathbb R}^3$ of the electric vector field ${\mathbf F}=(X,Y,Z)$ from a finite set of point charges. We are most interested in the case where the point charges all lie in a plane, and we consider just the zero set in ${\mathbb R}^2$ of the electric vector field ${\mathbf F}=(X,Y)$ from the finite set of point charges. The conjecture is that the zero set of ${\mathbf F}=(X,Y)$ in ${\mathbb R}^2$ is finite. We show fairly easily that this conjecture is true in a Special Case: when the point charges for ${\mathbf F}=(X,Y)$ lie on a line, and we consider the possible zeros throughout ${\mathbb R}^2$. However, even in this Special Case, it is hard to get complete structural information about the zero sets of $X$ and $Y$ separately. We describe structural information about the asymptotic directions at infinity of these two zero sets, and relate this to the interlacing of the zero sets of sequences of polynomials. Then we consider the General Case where the point charges can be anywhere in the plane. As in the Special Case, we construct sequences of polynomials whose zero sets include the asymptotic directions at infinity of the zero sets of $X$ and $Y$ separately. But now the asymptotic directions are not necessarily interlacing, and the structure of the zero sets of these polynomials is less evident. Nonetheless, using these polynomials it might be possible to show that the zeros of ${\mathbf F}=(X,Y)$ are bounded, and then perhaps also, as a result, confirm the conjecture that the zero set of ${\mathbf F}=(X,Y)$ is finite.

math.CA↗
arXiv2025

On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges

Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. The graphs of the trigonometric polynomials $R_k$ on the period suggest many zeros of $R_k(t)-n$ in a dense fashion on the period. Let ${\Cal N}(I,R_k-n)$ denote the number of zeros, counted with multiplicities, of the trigonometric polynomial $R_k-n$ in an interval $I := [α,β] \subset [0,2π)$. Improving earlier results proved only for the interval $I := [0,2π)$, in this paper we show that $$\frac{n|I|}{8π} - \frac{2}π (2n\log n)^{1/2} - 1 \leq N(I,R_k-n) \leq \frac{n|I|}π + \frac{8}π(2n\log n)^{1/2}\,,\qquad k \geq 2\,,$$ for every interval $I := [α,β] \subset [0,2π)$, where $|I| = β-α$ denotes the length of the interval $I$.

math.CA↗
arXiv2024

Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials (II)

We prove that there is an absolute constant $c > 0$ such that for every $$a_0,a_1, \ldots,a_n \in [1,M]\,, \qquad 1 \leq M \leq \frac 14 \exp \left( \frac n9 \right)\,,$$ there are $$b_0,b_1,\ldots,b_n \in \{-1,0,1\}$$ such that the polynomial $P$ of the form $\displaystyle{P(z) = \sum_{j=0}^n{b_ja_jz^j}}$ has at least $\displaystyle{c \left( \frac{n}{\log(4M)} \right)^{1/2}-1}$ distinct sign changes in $I_a := (1-2a,1-a)$, where $\displaystyle{a := \left( \frac{\log(4M)}{n} \right)^{1/2} \leq 1/3}$. This improves and extends earlier results of Bloch and Pólya and Erdélyi and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov.

math.NT↗
arXiv2024

The number of real zeros of polynomials with constrained coefficients

We prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the form $$P(z) = \sum_{j=0}^{n}{a_jz^j}\,, \quad |a_0| = 1\,, \quad |a_j| \leq M\,, \quad a_j \in \Bbb{C}\,, \quad M \geq 1\,,$$ has at most $cn^{1/2}(1+\log M)^{1/2}$ zeros in the interval $[-1,1]$. This result is sharp up to the multiplicative constant $c > 0$ and extends an earlier result of Borwein, Erdélyi, and Kós from the case $M=1$ to the case $M \geq $1. This has also been proved recently with the factor $(1+\log M)$ rather than $(1+\log M)^{1/2}$ in the Appendix of a recent paper by Jacob and Nazarov by using a different method. We also prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the above form has at most $(c/a)(1+\log M)$ zeros in the interval $[-1+a,1-a]$ with $a \in (0,1)$. Finally we correct a somewhat incorrect proof of an earlier result of Borwein and Erdélyi by proving that there is a constant $η> 0$ such that every polynomial $P$ of the above form with $M = 1$ has at most $ηn^{1/2}$ zeros inside any polygon with vertices on the unit circle, where the multiplicative constant $η> 0$ depends only on the polygon.

math.CA↗
arXiv2023

The $L_q$ norm of the Rudin-Shapiro polynomials on subarcs of the unit circle

Littlewood polynomials are polynomials with each of their coefficients in $\{-1,1\}$. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro polynomials. Let $P_k$ and $Q_k$ denote the Rudin-Shapiro polynomials of degree $n-1$ with $n:=2^k$. For polynomials $S$ we define $$M_q(S,[α,β]) := \left( \frac{1}{β-α} \int_α^β {\left| S(e^{it}) \right|^q\,dt} \right)^{1/q}\,, \qquad q > 0\,.$$ Let $γ:= \sin^2(π/8)$. We prove that $$\fracγ{4π}(γn)^{q/2} \leq M_q(P_k,[α,β])^q \leq (2n)^{q/2}$$ for every $q > 0$ and $32π/n \leq β-α$. The same estimates hold for $P_k$ replaced by $Q_k$.

math.CA↗
arXiv2021

Fourier Sparse Leverage Scores and Approximate Kernel Learning

We prove new explicit upper bounds on the leverage scores of Fourier sparse functions under both the Gaussian and Laplace measures. In particular, we study $s$-sparse functions of the form $f(x) = \sum_{j=1}^s a_j e^{i λ_j x}$ for coefficients $a_j \in \mathbb{C}$ and frequencies $λ_j \in \mathbb{R}$. Bounding Fourier sparse leverage scores under various measures is of pure mathematical interest in approximation theory, and our work extends existing results for the uniform measure [Erd17,CP19a]. Practically, our bounds are motivated by two important applications in machine learning: 1. Kernel Approximation. They yield a new random Fourier features algorithm for approximating Gaussian and Cauchy (rational quadratic) kernel matrices. For low-dimensional data, our method uses a near optimal number of features, and its runtime is polynomial in the $statistical\ dimension$ of the approximated kernel matrix. It is the first "oblivious sketching method" with this property for any kernel besides the polynomial kernel, resolving an open question of [AKM+17,AKK+20b]. 2. Active Learning. They can be used as non-uniform sampling distributions for robust active learning when data follows a Gaussian or Laplace distribution. Using the framework of [AKM+19], we provide essentially optimal results for bandlimited and multiband interpolation, and Gaussian process regression. These results generalize existing work that only applies to uniformly distributed data.

cs.DS↗
arXiv2020

Do flat skew-reciprocal Littlewood polynomials exist?

Polynomials with coefficients in $\{-1,1\}$ are called Littlewood polynomials. Using special properties of the Rudin-Shapiro polynomials and classical results in approximation theory such as Jackson's Theorem, de la Vallée Poussin sums, Bernstein's inequality, Riesz's Lemma, divided differences, etc., we give a significantly simplified proof of a recent breakthrough result by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba stating that there exist absolute constants $η_2 > η_1 > 0$ and a sequence $(P_n)$ of Littlewood polynomials $P_n$ of degree $n$ such that $$η_1 \sqrt{n} \leq |P_n(z)| \leq η_2 \sqrt{n}\,, \qquad z \in \mathbb{C}\,, \, \, |z| = 1,,$$ confirming a conjecture of Littlewood from 1966. Moreover, the existence of a sequence $(P_n)$ of Littlewood polynomials $P_n$ is shown in a way that in addition to the above flatness properties a certain symmetry is satisfied by the coefficients of $P_n$ making the Littlewood polynomials $P_n$ close to skew-reciprocal.

math.CA↗
arXiv2019

Turán-type reverse Markov inequalities for polynomials with restricted zeros

Let ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For integers $0 \leq k \leq n$ let ${\mathcal F}_{n,k}^c$ be the set of all polynomials $P \in {\mathcal P}_n^c$ having at least $n-k$ zeros in $D^+$. Let $$\|f\|_A := \sup_{z \in A}{|f(z)|}$$ for complex-valued functions defined on $A \subset {\Bbb C}$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \left(\frac{n}{k+1}\right)^{1/2} \leq \inf_{P}{\frac{\|P^{\prime}\|_{[-1,1]}}{\|P\|_{[-1,1]}}} \leq c_2 \left(\frac{n}{k+1}\right)^{1/2}$$ for all integers $0 \leq k \leq n$, where the infimum is taken for all $0 \not\equiv P \in {\mathcal F}_{n,k}^c$ having at least one zero in $[-1,1]$. This is an essentially sharp reverse Markov-type inequality for the classes ${\mathcal F}_{n,k}^c$ extending earlier results of Turán and Komarov from the case $k=0$ to the cases $0 \leq k \leq n$.

math.CA↗
arXiv2019

Arestov's theorems on Bernstein's inequality

We give a simple, elementary, and at least partially new proof of Arestov's famous extension of Bernstein's inequality in $L_p$ to all $p \geq 0$. Our crucial observation is that Boyd's approach to prove Mahler's inequality for algebraic polynomials $P_n \in {\mathcal P}_n^c$ can be extended to all trigonometric polynomials $T_n \in {\mathcal T}_n^c$.

math.CV↗
arXiv2019

Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

Let $n_1 < n_2 < \cdots < n_N$ be non-negative integers. In a private communication Brian Conrey asked how fast the number of real zeros of the trigonometric polynomials $T_N(θ) = \sum_{j=1}^N {\cos (n_jθ)}$ tends to $\infty$ as a function of $N$. Conrey's question in general does not appear to be easy.Let ${\mathcal P}_n(S)$ be the set of all algebraic polynomials of degree at most $n$ with each of their coefficients in $S$. For a finite set $S \subset {\mathbb C}$ let $M = M(S) := \max\{|z|: z \in S\}$. It has been shown recently that if $S \subset {\mathbb R}$ is a finite set and $(P_n)$ is a sequence of self-reciprocal polynomials $P_n \in {\mathcal P}_n(S)$ with $|P_n(1)|$ tending to $\infty$, then the number of zeros of $P_n$ on the unit circle also tends to $\infty$. In this paper we show that if $S \subset {\mathbb Z}$ is a finite set, then every self-reciprocal polynomial $P \in {\mathcal P}_n(S)$ has at least $$c(\log\log\log|P(1)|)^{1-\varepsilon}-1$$ zeros on the unit circle of ${\mathbb C}$ with a constant $c > 0$ depending only on $\varepsilon > 0$ and$M = M(S)$. Our new result improves the exponent $1/2 - \varepsilon$ in a recent result by Julian Sahasrabudhe to $1 - \varepsilon$. Sahasrabudhe's new idea [66] is combined with the approach used in [34] offering an essencially simplified way to achieve our improvement. We note that in both Sahasrabudhe's paper and our paper the assumption that the finite set $S$ contains only integers is deeply exploited.

math.NT↗
arXiv2019

The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocal

Let $${\mathcal K}_n := \left\{p_n: p_n(z) = \sum_{k=0}^n{a_k z^k}, \enspace a_k \in {\mathbb C}\,,\enspace |a_k| = 1 \right\}\,.$$ A sequence $(P_n)$ of polynomials $P_n \in {\mathcal K}_n$ is called ultraflat if $(n + 1)^{-1/2}|P_n(e^{it})|$ converge to $1$ uniformly in $t \in {\mathbb R}$. In this paper we prove that $$\frac{1}{2π} \int_0^{2π}{\left| (P_n - P_n^*)(e^{it}) \right|^q \, dt} \sim \frac{{2}^q Γ\left(\frac{q+1}{2} \right)}{Γ\left(\frac q2 + 1 \right) \sqrtπ} \,\, n^{q/2}$$ for every ultraflat sequence $(P_n)$ of polynomials $P_n \in {\mathcal K}_n$ and for every $q \in (0,\infty)$, where $P_n^*$ is the conjugate reciprocal polynomial associated with $P_n$, $Γ$ is the usual gamma function, and the $\sim$ symbol means that the ratio of the left and right hand sides converges to $1$ as $n \rightarrow \infty$. Another highlight of the paper states that $$\frac{1}{2π}\int_0^{2π}{\left| (P_n^\prime - P_n^{*\prime})(e^{it}) \right|^2 \, dt} \sim \frac{2n^3}{3}$$ for every ultraflat sequence $(P_n)$ of polynomials $P_n \in {\mathcal K}_n$. We prove a few other new results and reprove some interesting old results as well.

math.CA↗
arXiv2018

Improved results on the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle

Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant $A>0$ such that the equation $R_k(t) = (1+η)n$ has at least $An^{0.5394282}$ distinct zeros in $[0,2π)$ whenever $η$ is real and $|η| < 2^{-11}$. In this paper we show that the equation $R_k(t)=(1+η)n$ has at least $(1/2-|η|-\varepsilon)n/2$ distinct zeros in $[0,2π)$ for every $η\in (-1/2,1/2)$, $\varepsilon > 0$, and sufficiently large $k \geq k_{η,\varepsilon}$.

math.CA↗
arXiv2018

The sharp Remez-type inequality for even trigonometric polynomials on the period

We prove that $$\max_{t \in [-π,π]}{|Q(t)|} \leq T_{2n}(\sec(s/4)) = \frac 12 ((\sec(s/4) + \tan(s/4))^{2n} + (\sec(s/4) - \tan(s/4))^{2n})$$ for every even trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m(\{t \in [-π,π]: |Q(t)| \leq 1\}) \geq 2π-s\,, \qquad s \in (0,2π)\,,$$ where $m(A)$ denotes the Lebesgue measure of a measurable set $A \subset {\Bbb R}$ and $T_{2n}$ is the Chebysev polynomial of degree $2n$ on $[-1,1]$ defined by $T_{2n}(\cos t) = \cos(2nt)$ for $t \in {\Bbb R}$. This inequality is sharp. We also prove that $$\max_{t \in [-π,π]}{|Q(t)|} \leq T_{2n}(\sec(s/2)) = \frac 12 ((\sec(s/2) + \tan(s/2))^{2n} + (\sec(s/2) - \tan(s/2))^{2n})$$ for every trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m(\{t \in [-π,π]: |Q(t)| \leq 1\}) \geq 2π-s\,, \qquad s \in (0,π)\,.$$

math.CA↗
arXiv2018

Reverse Markov- and Bernstein-type inequalities for incomplete polynomials

Let ${\mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${\mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$\|f\|_A := \sup_{x \in A}{|f(x)|}$$ for real-valued functions $f$ defined on a set $A \subset {\Bbb R}$. Let $$V_a^b(f) := \int_a^b{|f^{\prime}(x)| \, dx}$$ denote the total variation of a continuously differentiable function $f$ on an interval $[a,b]$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \frac nk\leq \min_{P \in {\mathcal P}_{n,k}}{\frac{\|P^{\prime}\|_{[0,1]}}{V_0^1(P)}} \leq \min_{P \in {\mathcal P}_{n,k}}{\frac{\|P^{\prime}\|_{[0,1]}}{|P(1)|}} \leq c_2 \left( \frac nk + 1 \right)$$ for all integers $n \geq 1$ and $k \geq 1$. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \left(\frac nk\right)^{1/2} \leq \min_{P \in {\mathcal P}_{n,k}}{\frac{\|P^{\prime}(x)\sqrt{1-x^2}\|_{[0,1]}}{V_0^1(P)}} \leq \min_{P \in {\mathcal P}_{n,k}}{\frac{\|P^{\prime}(x)\sqrt{1-x^2}\|_{[0,1]}}{|P(1)|}} \leq c_2 \left(\frac nk + 1\right)^{1/2}$$ for all integers $n \geq 1$ and $k \geq 1$.

math.CA↗
arXiv2018

On the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle

In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. In this paper we study the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle. We also show that the Rudin-Shapiro polynomials $P_k$ and $Q_k$ of degree $n-1$ with $n := 2^k$ have $o(n)$ zeros on the unit circle. This should be compared with a result of B. Conrey, A. Granville, B. Poonen, and K. Soundararajan stating that for odd primes $p$ the Fekete polynomials $f_p$ of degree $p-1$ have asymptotically $κ_0 p$ zeros on the unit circle, where $0.500813>κ_0>0.500668$. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by B. Rodgers. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that the $k$-th Rudin-Shapiro polynomials $P_k$ and $Q_k$ of degree $n-1 = 2^k-1$ have at least $c_2n$ zeros in the annulus $$\left \{z \in {\Bbb C}: 1 - \frac{c_1}{n} < |z| < 1 + \frac{c_1}{n} \right \}\,.$$

math.CV↗