The Weyl–Mellin quantization map

In this paper, we present a quantization of the functions of spacetime, i.e. a map, analog to Weyl map, which reproduces the [Formula: see text]-Minkowski commutation relations, and it has the desirable properties of mapping square integrable functions into Hilbert–Schmidt operators, as well as real functions into symmetric operators. The map is based on Mellin transform on radial and time coordinates. The map also defines a deformed ∗ product which we discuss with examples.

On the Non-Uniqueness of Statistical Ensembles Defining a Density Operator and a Class of Mixed Quantum States with Integrable Wigner Distribution

It is standard to assume that the Wigner distribution of a mixed quantum state consisting of square-integrable functions is a quasi-probability distribution, i.e., that its integral is one and that the marginal properties are satisfied. However, this is generally not true. We introduced a class of quantum states for which this property is satisfied; these states are dubbed “Feichtinger states” because they are defined in terms of a class of functional spaces (modulation spaces) introduced in the 1980s by H. Feichtinger. The properties of these states were studied, giving us the opportunity to prove an extension to the general case of a result due to Jaynes on the non-uniqueness of the statistical ensemble, generating a density operator.

Hermite Functions and Fourier Series

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

A New Upper Bound for Sampling Numbers

We provide a new upper bound for sampling numbers $$(g_n)_{n\in \mathbb {N}}$$ ( g n ) n ∈ N associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $$C,c>0$$ C , c > 0 (which are specified in the paper) such that $$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$ g n 2 ≤ C log ( n ) n ∑ k ≥ ⌊ c n ⌋ σ k 2 , n ≥ 2 , where $$(\sigma _k)_{k\in \mathbb {N}}$$ ( σ k ) k ∈ N is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding $$\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)$$ Id : H ( K ) → L 2 ( D , ϱ D ) . The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $$H^s_{\text {mix}}(\mathbb {T}^d)$$ H mix s ( T d ) in $$L_2(\mathbb {T}^d)$$ L 2 ( T d ) with $$s>1/2$$ s > 1 / 2 . We obtain the asymptotic bound $$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$ g n ≤ C s , d n - s log ( n ) ( d - 1 ) s + 1 / 2 , which improves on very recent results by shortening the gap between upper and lower bound to $$\sqrt{\log (n)}$$ log ( n ) . The result implies that for dimensions $$d>2$$ d > 2 any sparse grid sampling recovery method does not perform asymptotically optimal.

On some smooth approximations on thick periodic multi-structures

In this paper we study the approximation properties of measurable and square-integrable functions. In particular we show that any $L^2$-bounded function can be approximated in $L^2$-norm by smooth functions defined on a highly oscillating boundary of thick multi-structures in ${\mathbb{R}}^n$. We derive the norm estimates for the approximating functions and study their asymptotic behaviour.

Sampling in $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators on $ L^2(\mathbb{R}^d) $

<p style='text-indent:20px;'>The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define <inline-formula><tex-math id="M3">\begin{document}$ \Lambda $\end{document}</tex-math></inline-formula>-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice <inline-formula><tex-math id="M4">\begin{document}$ \Lambda $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{R}^{2d} $\end{document}</tex-math></inline-formula>. These spaces can be seen as a generalization of classical shift-invariant subspaces of square integrable functions. Obtaining sampling results for these subspaces appears as a natural question that can be motivated by the problem of channel estimation in wireless communications. These sampling results are obtained in the light of the frame theory in a separable Hilbert space.</p>

Stochastic Recovery of Square-Integrable Functions

The purpose of the study was to solve the problem of stochastic recovery of square-integrable (with respect to the Lebesgue measure) functions defined on the real line from observations with additive white Gaussian noise, for the case of discrete time. The problem is a nonparametric, i.e., infinite-dimensional, estimation problem. The study substantiates the procedure of optimal recovery, in the mean-square sense, with respect to the product of the Lebesgue measure and the Gaussian measure, and describes an algorithm for recovering such square-integrable functions. Findings of research show that the constructed procedure for nonparametric recovery of a square-integrable function gives an unbiased and consistent recovery of an unknown function. This result has not been previously described. In addition, for smooth reconstructed functions, an almost optimal reconstruction procedure is introduced and substantiated, which gives an unimprovable (in order of magnitude) estimate of the dependence of the number of orthogonal functions on the number of observations. The error of the constructed almost optimal recovery procedure in relation to the optimal recovery procedure is no more than 50 %

Toeplitz and slant Toeplitz operators on the polydisk

For n ≥ 1, let Dn be the polydisk in ℂn, and let Tn be the n-torus. L2(Tn) denotes the space of Lebesgue square integrable functions on Tn. In this paper we define slant Toeplitz operators on L2(Tn). Besides giving a necessary and sufficient condition for an operator on L2(Tn) to be slant Toeplitz, we also establish several properties of slant Toeplitz operators.

SOME RECURRENCE FORMULAS FOR A NEW CLASS OF SPECIAL POLYNOMIALS AND SPECIAL FUNCTIONS

In this paper we used a new class of special functions and special polynomials which are solutions different Sturm Liouvile differential equations of second order. These functions form a basis of a space of square integrable functions over set of a real numbers. We investigated some properties of these polynomials and established some recurrence formulas. Using a new class of special functions, we obtained some useful summation formulas and recurrence formulas.

Translates of Functions on the Heisenberg Group and the HRT Conjecture

We prove that the HRT (Heil, Ramanathan, and Topiwala) Conjecture is equivalent to the conjecture that co-central translates of square-integrable functions on the Heisenberg group are linearly independent.