Abstract The study of some signal processing problems within Bayesian frameworks and semigroups theory, in the case where the Banach space under consideration may be nonseparable, was discussed in the paper published in 2022 by two of the authors. For the Cauchy problem for the diffusion equation, which is a typical example of evolution equation, there was studied the question of what can be saved in the study of the problem in the case with the initial data in the nonseparable classical Morrey space. In this paper we develop this approach to an essentially more general case of generalized Morrey spaces with general weights. The study is based on emersion of such spaces into weighted Lebesgue spaces with Muckenhoupt weights.

Abstract In this note, we study maximizers for Fourier extension inequalities on the sphere. We prove that constant functions are local maximizers for the $$L^p(\mathbb {S}^{d-1})$$ L p ( S d - 1 ) to $$L^p(\mathbb {R}^d)$$ L p ( R d ) Fourier extension estimates in the same range of exponents p for which they are global maximizers for the $$L^2(\mathbb {S}^{d-1})$$ L 2 ( S d - 1 ) to $$L^p_{rad}L^2_{ang}(\mathbb {R}^d)$$ L rad p L ang 2 ( R d ) mixed-norm Fourier extension inequalities. Moreover, in the case of low dimensions, we improve the range of exponents for which constant functions are known to be the unique global maximizers for the $$L^2(\mathbb {S}^{d-1})$$ L 2 ( S d - 1 ) to $$L^p_{rad}L^2_{ang}(\mathbb {R}^d)$$ L rad p L ang 2 ( R d ) mixed-norm Fourier extension estimate on the sphere, covering, for the case of dimensions $$d=2,3$$ d = 2 , 3 , the entire Stein–Tomas range. This is achieved by establishing novel hierarchies between certain weighted norms of Bessel functions.

Abstract An integrable polygon is one whose interior angles are fractions of $$\pi $$ π ; that is to say of the form $$\frac{\pi }{n}$$ π n for positive integers n . We consider the Laplace spectrum on these polygons with the Dirichlet and Neumann boundary conditions, and we obtain new spectral invariants for these polygons. This includes new expressions for the spectral zeta function and zeta-regularized determinant as well as a new spectral invariant contained in the short-time asymptotic expansion of the heat trace. Moreover, we demonstrate relationships between the short-time heat trace invariants of general polygonal domains (not necessarily integrable) and smoothly bounded domains and pose conjectures and further related directions of investigation.

Abstract We develop an alternative approach to the study of Fourier series, based on the Short-Time-Fourier Transform (STFT) acting on $$L_{\nu }^{2}(0,1)$$ L ν 2 ( 0 , 1 ) , the space of measurable functions f in $$\mathbb {R}$$ R , square-integrable in (0, 1), and time-periodic up to a phase factor: for fixed $$\nu \in \mathbb {R}$$ ν ∈ R , $$\begin{aligned} f(t+k)=e^{2\pi ik\nu }f(t){, \ }k\in \mathbb {Z}\text {.} \end{aligned}$$ f ( t + k ) = e 2 π i k ν f ( t ) , k ∈ Z . The resulting phase space is the vertical strip $$\mathbb {C}/\mathbb {Z}=[0,1)\times \mathbb {R}$$ C / Z = [ 0 , 1 ) × R , a flat model of an infinite cylinder, which leads to Gabor frames with an interesting structure theory, allowing for a Janssen-type representation. As expected, a Gaussian window leads to a Fock space of entire functions, studied in the companion paper by the same authors [Beurling-type density theorems for sampling and interpolation on the flat cylinder]. When g is a Hermite function, we are lead to true Fock spaces of polyanalytic functions (Landau level eigenspaces) on the vertical strip $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R . We first prove a density condition for a lattice to be interpolating in this space. Furthermore, an analogue of the sufficient Wexler-Raz conditions is obtained which leads to new criteria for Gabor frames in $$L^{2}(\mathbb {R})$$ L 2 ( R ) , and to sufficient conditions for Gabor frames in $$L_{\nu }^{2}(0,1)$$ L ν 2 ( 0 , 1 ) with Hermite windows (an analogue of a theorem of Gröchenig and Lyubarskii about Gabor frames with Hermite windows) and with totally positive windows in the Feichtinger algebra (an analogue of a recent theorem of Gröchenig). We also consider a vectorial STFT in $$L_{\nu }^{2}(0,1)$$ L ν 2 ( 0 , 1 ) and, using the vector with the first Hermite functions as window, we introduce the (full) Fock spaces of polyanalytic functions on $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R and their associated Bargmann-type transforms, and prove an analogue of Vasilevski’s orthogonal decomposition into true polyanalytic Fock spaces (Landau level eigenspaces on $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R ). We conclude the paper with an analogue of Gröchenig-Lyubarskii’s sufficient condition for Gabor super-frames with Hermite functions, which is equivalent to a sufficient sampling condition on the full Fock space of polyanalytic functions on $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R . The proofs of the results about Gabor frames, involving some of Gröchenig’s most significant results of the past 25 years, are a clear indication of his influence on the field during this period.