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Register a new userUncertainty product for Vilenkin groups
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We study a localization of functions defined on Vilenkin groups. To measure the localization we introduce two uncertainty products $UP_{lambda}$ and $UP_{G}$ that are similar to the Heisenberg uncertainty product. $UP_{lambda}$ and $UP_{G}$ differ from each other by the metric used for the Vilenkin group $G$. We discuss analogs of a quantitative uncertainty principle. Representations for $UP_{lambda}$ and $UP_{G}$ in terms of Walsh and Haar basis are given.
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An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported.
An inequality refining the lower bound for a periodic (Breitenberger) uncertainty constant is proved for a wide class of functions. A connection of uncertainty constants for periodic and non-periodic functions is extended to this class. A particular minimization problem for a non-periodic (Heisenberg) uncertainty constant is studied.
By a result of Schur [J. Reine Angew. Math. 1911], the entrywise product $M circ N$ of two positive semidefinite matrices $M,N$ is again positive. Vybiral [Adv. Math. 2020] improved on this by showing the uniform lower bound $M circ overline{M} geq E_n / n$ for all $n times n$ real or complex correlation matrices $M$, where $E_n$ is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 1999] and to positive definite functions on groups. Vybiral (in his original preprint) asked if one can obtain similar uniform lower bounds for higher entrywise powers of $M$, or for $M circ N$ when $N eq M, overline{M}$. A natural third question is to obtain a tighter lower bound that need not vanish as $n to infty$, i.e. over infinite-dimensional Hilbert spaces. In this note, we affirmatively answer all three questions by extending and refining Vybirals result to lower-bound $M circ N$, for arbitrary complex positive semidefinite matrices $M, N$. Specifically: we provide tight lower bounds, improving on Vybirals bounds. Second, our proof is conceptual (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchy-Schwarz inequalities. Third, we extend our tight lower bounds to Hilbert-Schmidt operators. As an application, we settle Open Problem 1 of Hinrichs-Krieg-Novak-Vybiral [J. Complexity, in press], which yields improvements in the error bounds in certain tensor product (integration) problems.
The conical function and its relativistic generalization can be viewed as eigenfunctions of the reduced 2-particle Hamiltonians of the hyperbolic Calogero-Moser system and its relativistic generalization. We prove new product formulas for these functions. As a consequence, we arrive at explicit diagonalizations of integral operators that commute with the 2-particle Hamiltonians and reduc
Let $A$ be a compact set in $mathbb{R}$, and $E=A^dsubset mathbb{R}^d$. We know from the Mattila-Sjolins theorem if $dim_H(A)>frac{d+1}{2d}$, then the distance set $Delta(E)$ has non-empty interior. In this paper, we show that the threshold $frac{d+1}{2d}$ can be improved whenever $dge 5$.
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