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1. Examples of Riesz Bases of Exponentials for Multi-tiling Domains and Their Duals

   https://doi.org/10.1007/s44007-023-00078-7  

   Frederick, Christina; Yacoubou Djima, Karamatou (November 2023, La Matematica)

   Abstract A well-studied problem in sampling theory is to find an expansion of a function in terms of a Riesz basis of exponentials for$$L^2(\Omega )$$ $L^2\left(\Omega \right)$, where$$\Omega $$ $\Omega$is a bounded, measurable set. For such a basis, we are guaranteed the existence of a unique biorthogonal dual basis that can be used to calculate the expansion coefficients. Much attention has been paid to the existence of Riesz bases of exponentials for various domains; however, the sampling and reconstruction problems in these cases are less understood. Recently, explicit formulas for the corresponding dual Riesz bases were introduced in Frederick and Okoudjou in [Appl Comput Harmon Anal 51:104–117, 2021; Frederick and Mayeli in J Fourier Anal Appl 27(5):1–21, 2021] for a class of multi-tiling domains. In this paper, we further this work by presenting explicit examples of a finite co-measurable union of intervals or multi-rectangles. In the higher-dimensional case, we also discuss how different sampling strategies lead to different Riesz bounds. 

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2. Approximately Hadamard Matrices and Riesz Bases in Random Frames

   https://doi.org/10.1093/imrn/rnad080  

   Dong, Xiaoyu; Rudelson, Mark (April 2023, International Mathematics Research Notices)

   Abstract An $$n \times n$$ matrix with $$\pm 1$$ entries that acts on $${\mathbb {R}}^{n}$$ as a scaled isometry is called Hadamard. Such matrices exist in some, but not all dimensions. Combining number-theoretic and probabilistic tools, we construct matrices with $$\pm 1$$ entries that act as approximate scaled isometries in $${\mathbb {R}}^{n}$$ for all $$n \in {\mathbb {N}}$$. More precisely, the matrices we construct have condition numbers bounded by a constant independent of $$n$$. Using this construction, we establish a phase transition for the probability that a random frame contains a Riesz basis. Namely, we show that a random frame in $${\mathbb {R}}^{n}$$ formed by $$N$$ vectors with independent identically distributed coordinate having a nondegenerate symmetric distribution contains many Riesz bases with high probability provided that $$N \ge \exp (Cn)$$. On the other hand, we prove that if the entries are sub-Gaussian, then a random frame fails to contain a Riesz basis with probability close to $$1$$ whenever $$N \le \exp (cn)$$, where $c<C$ are constants depending on the distribution of the entries. 

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3. On Parseval Frames of Kernel Functions in de Branges Spaces of Entire Vector Valued Functions

   https://doi.org/10.1007/978-3-030-76473-9_1  

   al-Sa'di, Sa'ud; Weber, Eric S. (May 2022, New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative)

   null; null; Shoikhet, D.; Vajiac, E. (Ed.)

   We consider the existence and structure properties of Parseval frames of kernel functions in vector valued de Branges spaces. We develop some sufficient conditions for Parseval sequences by identifying the main construction with Naimark dilation of frames. The dilation occurs by embedding the de Branges space of vector valued functions into a dilated de Branges space of vector valued functions. The embedding also maps the kernel functions associated with a frame sequence of the original space into a Riesz basis for the embedding space. We also develop some sufficient conditions for a dilated de Branges space to have the Kramer sampling property. 

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4. Schatten Classes and Commutators of Riesz Transform on Heisenberg Group and Applications

   https://doi.org/10.1007/s00041-023-09995-1  

   Fan, Zhijie; Lacey, Michael; Li, Ji (April 2023, Journal of Fourier Analysis and Applications)

   Abstract We study commutators with the Riesz transforms on the Heisenberg group$${\mathbb {H}}^{n}$$ $H^n$. The Schatten norm of these commutators is characterized in terms of Besov norms of the symbol. This generalizes the classical Euclidean results of Peller, Janson–Wolff and Rochberg–Semmes. The method in proof bypasses the use of Fourier analysis, allowing us to address not just the Riesz transforms, but also the Cauchy–Szegő projection and second order Riesz transforms on$${\mathbb {H}}^{n}$$ $H^n$among other settings. 

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5. On the Fractional Dynamics of Kinks in Sine-Gordon Models

   https://doi.org/10.3390/math13020220  

   Bountis, Tassos; Cantisán, Julia; Cuevas-Maraver, Jesús; Macías-Díaz, Jorge Eduardo; Kevrekidis, Panayotis G (January 2025, Mathematics)

   In the present work, we explored the dynamics of single kinks, kink–anti-kink pairs and bound states in the prototypical fractional Klein–Gordon example of the sine-Gordon equation. In particular, we modified the order β of the temporal derivative to that of a Caputo fractional type and found that, for 1<β<2, this imposes a dissipative dynamical behavior on the coherent structures. We also examined the variation of a fractional Riesz order α on the spatial derivative. Here, depending on whether this order was below or above the harmonic value α=2, we found, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explored the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved. 

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