Frames and Bessel sequences in Fréchet spaces and their duals are defined and studied. Their relation with Schauder frames and representing systems is analyzed. The abstract results presented here, when applied to concrete spaces of analytic functions, give examples and consequences about sampling sets and Dirichlet series expansions.

In this article we study harmonic analysis on the proper velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. This PV addition is the relativistic addition of proper velocities in special relativity, and it is related with the hyperboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter $z$ and on the radius $t$ of the hyperboloid, and it comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace–Beltrami operator and its eigenfunctions, the generalized Poisson transform and its inverse, the generalized Helgason–Fourier transform and its inverse, and Plancherel’s theorem. In the limit of large $t$, $t\to +\infty$, the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on ${\mathbb{R}}^n$, thus unifying hyperbolic and Euclidean harmonic analysis.

This article presents a systematic study for abstract harmonic analysis aspects of wave-packet transforms over locally compact abelian (LCA) groups. Let $H$ be a locally compact group, let $K$ be an LCA group, and let $\theta :H\to Aut\left(K\right)$ be a continuous homomorphism. We introduce the abstract notion of the wave-packet group generated by $\theta$, and we study basic properties of wave-packet groups. Then we study theoretical aspects of wave-packet transforms. Finally, we will illustrate application of these techniques in the case of some well-known examples.

In this work we derive a convex dual representation for increasing convex functionals on a space of real-valued Borel measurable functions defined on a countable product of metric spaces. Our main assumption is that the functionals fulfill marginal constraints satisfying a certain tightness condition. In the special case where the marginal constraints are given by expectations or maxima of expectations, we obtain linear and sublinear versions of Kantorovich’s transport duality and the recently discovered martingale transport duality on products of countably many metric spaces.

We study the existence of infinite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations $P$ of the set $\{1,\dots ,n\}$, there exists a closed infinite-dimensional Banach subspace of the space of $n$-linear forms on ${\ell }_1$ such that, for all nonzero elements $B$ of such a subspace, the Arens extension associated to the permutation $\sigma$ of $B$ is norm-attaining if and only if $\sigma$ is an element of $P$. We also study the structure of the set of norm-attaining $n$-linear forms on $c_0$.

We develop a systematic approach to the study of ideals of Lipschitz maps from a metric space to a Banach space, inspired by classical theory on using Lipschitz tensor products to relate ideals of operator/tensor norms for Banach spaces. We study spaces of Lipschitz maps from a metric space to a dual Banach space that can be represented canonically as the dual of a Lipschitz tensor product endowed with a Lipschitz cross-norm, and we show that several known examples of ideals of Lipschitz maps (Lipschitz maps, Lipschitz $p$-summing maps, maps admitting Lipschitz factorization through subsets of $L_p$-space) admit such a representation. Generally, we characterize when the space of a Lipschitz map from a metric space to a dual Banach space is in canonical duality with a Lipschitz cross-norm. Finally, we introduce a concept of operators which are approximable with respect to one of these ideals of Lipschitz maps, and we identify them in terms of tensor-product notions.

Let $(X,+)$ be a topological abelian group. We discuss regularity of solutions $f:X\to \mathbb{R}$ of Hlawka’s functional inequality

$$f(x+y)+f(y+z)+f(x+z)\le f(x+y+z)+f\left(x\right)+f\left(y\right)+f\left(z\right),$$ postulated for all $x,y,z\in X$. We study the lower and upper hull of $f$. Moreover, we provide conditions which imply continuity of $f$. We prove, in particular, that if $X$ is generated by any neighborhood of zero, $f$ is continuous at zero, and $f\left(0\right)=0$, then $f$ is continuous on $X$.

In this article, we give the criteria for approximative compactness of every proximinal convex subset of Musielak–Orlicz–Bochner function spaces equipped with the Orlicz norm. As a corollary, we give the criteria for approximative compactness of Musielak–Orlicz–Bochner function spaces equipped with the Orlicz norm.

This article deals with unbounded composition operators with infinite matrix symbols acting in $L^2$-spaces with respect to the Gaussian measure on ${\mathbb{R}}^{\infty }$. We introduce weak cohyponormality classes ${\mathcal{S}}_{n,r}^{\ast }$ of unbounded operators and provide criteria for the aforementioned composition operators to belong to ${\mathcal{S}}_{n,r}^{\ast }$. Our approach is based on inductive limits of operators.

By using Blunck’s operator-valued Fourier multiplier theorem, we completely characterize the existence and uniqueness of solutions in Lebesgue sequence spaces for a discrete version of the Cauchy problem with fractional order $1<\alpha \le 2$. This characterization is given solely in spectral terms on the data of the problem, whenever the underlying Banach space belongs to the UMD-class.

We continue the study begun by the third author of $C^{\ast }$-Segal algebra-valued function algebras with an emphasis on the order structure. Our main result is a characterization theorem for $C^{\ast }$-Segal algebra-valued function algebras with an order unitization. As an intermediate step, we establish a function algebraic description of the multiplier module of arbitrary Segal algebra-valued function algebras. We also consider the Gelfand representation of these algebras in the commutative case.

We consider the triangular $\theta$-summability of $2$-dimensional Fourier transforms. Under some conditions on $\theta$, we show that the triangular $\theta$-means of a function $f$ belonging to the Wiener amalgam space $W(L_1,{\ell }_{\infty })\left({\mathbb{R}}^2\right)$ converge to $f$ at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points for the so-called modified Lebesgue points of $f\in W(L_p,{\ell }_{\infty })\left({\mathbb{R}}^2\right)$ whenever $1<p<\infty$. Some special cases of the $\theta$-summation are considered, such as the Weierstrass, Abel, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations.