arXiv subjects

J. Marshall Ash

Publications and source records attributed to J. Marshall Ash.

5 recordsLinked to original sources

arXiv2024

Counterexamples to the Gaussian vs. MZ derivatives Conjecture

J. Marcinkiewicz and A. Zygmund proved in 1936 that the special $n$-th generalized Riemann derivative ${_2}D_nf(x)$ with nodes $0,1,2,2^2,\ldots, 2^{n-1}$, is equivalent to the $n$-th Peano derivative $f_{(n)}(x)$, for all $n-1$ times Peano differentiable functions $f$ at~$x$. Call every $n$-th generalized Riemann derivative with this property an MZ derivative. The recent paper Ash, Catoiu, and Fejzić [Israel J. Math. {255} (2023):177--199] introduced the $n$-th Gaussian derivatives as the $n$-th generalized Riemann derivatives with nodes either $0,1,q,q^2,\ldots ,q^{n-1}$ or $1,q,q^2,\ldots ,q^{n}$, where~$q\neq0,\pm 1$, proved that the Gaussian derivatives are MZ derivatives, and conjectured that these are \emph{all} MZ derivatives. In this article, we invalidate this conjecture by means of two counterexamples. The order in which these are presented allows an update of the conjecture after each counterexample. The proof of the first counterexample is simple, by scales of generalized Riemann derivatives. The proof of the second involves the classification of generalized Riemann derivatives of Ash, Catoiu, and Chin [Proc. Amer. Math. Soc {146} (2018):3847--3862]. Symmetric versions of all the results are also~included.

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arXiv2024

Two Pointwise Characterizations of the Peano Derivative

We provide the first two examples of sets of generalized Riemann derivatives of orders up to $n$, $n\geq 2$, whose simultaneous existence for all functions~$f$ at~$x$ is equivalent to the existence of the $n$-th Peano derivative $f_{(n)}(x)$. In this way, we begin to understand how the theory of Peano derivatives can be explained exclusively in terms of generalized Riemann derivatives, a bold new principle in generalized differentiation. In 1936, J. Marcinkiewicz and A. Zygmund showed that the existence of $f_{(n)}(x)$ is equivalent to the existence of both $f_{(n-1)}(x)$ and the $n$th generalized Riemann derivative $\widetilde{D}_nf(x)$, based at $x,x+h,x+2h,x+2^2h,\ldots ,x+2^{n-1}h$. Our first characterization of $f_{(n)}(x)$ is that its existence is equivalent to the simultaneous existence of $\widetilde{D}_1f(x),\ldots,\widetilde{D}_nf(x)$. Our second characterization is that the existence of $f_{(n)}(x)$ is equivalent to the existence of $\widetilde{D}_1f(x)$ and of all $n(n-1)/2$ forward shifts, \[ D_{k,j}f(x)=\lim_{h\rightarrow 0} h^{-k}\sum_{i=0}^k(-1)^i\binom ki f(x+(k+j-i)h), \] for $j=0,1,\ldots,k-2$, of the $k$-th Riemann derivatives $D_{k,0}f(x)$, for $k=2,\ldots ,n$. The proof of the second result involves an interesting combinatorial algorithm that starts with consecutive forward shifts of an arithmetic progression and yields a geometric progression, using two set-operations: dilation and combinatorial Gaussian elimination. This result proves a variant of a 1998 conjecture by Ginchev, Guerragio and Rocca, predicting the same outcome for backward shifts instead of forward shifts. The conjecture has been recently settled in [5], with a proof that has this variant's proof as a prerequisite.

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arXiv2015

The classification of generalized Riemann derivatives

We characterize all pairs $(\mathcal{A}$,$\mathcal{B})$ of generalized Riemann differences for which $\mathcal{A}$-differentiability implies $\mathcal{B}$-differentiability. Two generalized Riemann derivatives $\mathcal{A}$ and $\mathcal{B}$ are equivalent if a function has a derivative in the sense of $\mathcal{A}$ at a real number $x$ if and only if it has a derivative in the sense of $\mathcal{B}$ at $x$. We determine the equivalence classes for this equivalence relation.

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arXiv2007

Exponential sums with coefficients 0 or 1 and concentrated L^{p} norms

Let f be a sum of exponentials of the form exp(2 pi i N x), where the N are distinct integers. We call f an idempotent trigonometric polynomial (because the convolution of f with itself is f) or, simply, an idempotent. We show that for every p > 1, and every set E of the torus T = R/Z with |E| > 0, there are idempotents concentrated on E in the Lp sense. More precisely, for each p > 1, there is an explicitly calculated constant Cp > 0 so that for each E with |E| > 0 and epsilon > 0 one can find an idempotent f such that the pth root of the ratio of the integral over E of the pth power of |f| to the integral over T of the pth power of |f| is greater than Cp - epsilon. This is in fact a lower bound result and, though not optimal, it is close to the best that our method gives. We also give both heuristic and computational evidence for the still open problem of whether the Lp concentration phenomenon fails to occur when p = 1.

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arXiv2000

Some spherical uniqueness theorems for multiple trigonometric series

We prove that if a multiple trigonometric series is spherically Abel summable everywhere to an everywhere finite function $f(x)$ which is bounded below by an integrable function, then the series is the Fourier series of $f(x)$ if the coefficients of the multiple trigonometric series satisfy a mild growth condition. As a consequence, we show that if a multiple trigonometric series is spherically convergent everywhere to an everywhere finite integrable function $f(x)$, then the series is the Fourier series of $f(x)$. We also show that a singleton is a set of uniqueness. These results are generalizations of a recent theorem of J. Bourgain and some results of V. Shapiro.

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