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Register a new userConvolution, Fourier analysis, and distributions generated by Riesz bases
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In this note we discuss notions of convolutions generated by biorthogonal systems of elements of a Hilbert space. We develop the associated biorthogonal Fourier analysis and the theory of distributions, discuss properties of convolutions and give a number of examples.
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This work is devoted to the study of Bessel and Riesz systems of the type $big{L_{gamma}mathsf{f}big}_{gammain Gamma}$ obtained from the action of the left regular representation $L_{gamma}$ of a discrete non abelian group $Gamma$ which is a semidirect product, on a function $mathsf{f}in ell^2(Gamma)$. The main features about these systems can be conveniently studied by means of a simple matrix-valued function $mathbf{F}(xi)$. These systems allow to derive sampling results in principal $Gamma$-invariant spaces, i.e., spaces obtained from the action of the group $Gamma$ on a element of a Hilbert space. Since the systems $big{L_{gamma}mathsf{f}big}_{gammain Gamma}$ are closely related to convolution operators, a connection with $C^*$-algebras is also established.
We introduce hybrid fractals as a class of fractals constructed by gluing several fractal pieces in a specific manner and study energy forms and Laplacians on them. We consider in particular a hybrid based on the $3$-level Sierpinski gasket, for which we construct explicitly an energy form with the property that it does not capture the $3$-level Sierpinski gasket structure. This characteristic type of energy forms that miss parts of the structure of the underlying space are investigated in the more general framework of finitely ramified cell structures. The spectrum of the associated Laplacian and its asymptotic behavior in two different hybrids is analyzed theoretically and numerically. A website with further numerical data analysis is available at http://www.math.cornell.edu/~harry970804/.
We provide explicit sequence space representations for the test function and distribution spaces occurring in the Valdivia-Vogt structure tables by making use of Wilson bases generated by compactly supported smooth windows. Furthermore, we show that these kind of bases are common unconditional Schauder bases for all separable spaces occurring in these tables. Our work implies that the Valdivia-Vogt structure tables for test functions and distributions may be interpreted as one large commutative diagram.
We use a noncommutative generalization of Fourier analysis to define a broad class of pseudo-probability representations, which includes the known bosonic and discrete Wigner functions. We characterize the groups of quantum unitary operations which correspond to phase-space transformations, generalizing Gaussian and Clifford operations. As examples, we find Wigner representations for fermions, hard-core bosons, and angle-number systems.
We find necessary and sufficient conditions on a family $mathcal{R} = (r_i)_{i in I}$ in a Boolean algebra $mathcal{B}$ under which there exists a unique positive probability measure $mu$ on $mathcal{B}$ such that $mu ( bigcap_{k=1}^n theta_k r_{i_k} ) = 2^{-n}$ for all finite collections of distinct indices $i_1, ldots, i_n in I$ and all collections of signs $theta_1, ldots, theta_n in {-1,1}$, where the product $theta x$ of a sign $theta$ by an element $x in mathcal{B}$ is defined by setting $1 x = x$ and $-1 x = - x = mathbf{1} setminus x$. Such a family we call a complete Rademacher family. We prove that Dedekind $sigma$-complete Boolean algebras admitting complete Rademacher systems of the same cardinality are isomorphic. As a consequence, we obtain that a Dedekind $sigma$-complete Boolean algebra is homogeneous measurable if and only if it admits a complete Rademacher family. This new way to define a measure on a Boolean algebra allows us to define classical systems on an arbitrary Riesz space, such as Rademacher and Haar. We define a complete Rademacher system of any cardinality and a countable complete Haar system on an element $e > 0$ of a vector lattice $E$ in such a way that if $e$ is an order unit of $E$ then the corresponding systems become complete for the entire $E$. We prove that if $E$ is Dedekind complete then any complete Haar system on $e$ is an order Schauder basis for the ideal $A_e$ generated by $e$. Finally, we develop a theory of integration in a Riesz space of elements of the band $B_e$ generated by a fixed $e > 0$ with respect to the measure on the Boolean algebra $mathfrak{F}_e$ of fragments of $e$ generated by a complete Rademacher family on $mathfrak{F}_e$. Much space is devoted to examples showing that our way of thinking is sharp (e.g., we show the essentiality of each of the condition in the definition of a Rademacher family).
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