Journal of Fourier Analysis and Applications最新文献

In this paper, we enhance a recent algorithm for approximate spectral factorization of matrix functions, extending its capabilities to precisely factorize rational matrices when an exact lower-upper triangular factorization is available. This novel approach leverages a fundamental component of the improved algorithm for the precise design of rational paraunitary filter banks, allowing for the predetermined placement of zeros and poles. The introduced algorithm not only advances the state-of-the-art in spectral factorization but also opens new avenues for the tailored design of paraunitary filters with specific spectral properties, offering significant potential for applications in signal processing and beyond.

Let (rho ) be a rearrangement-invariant (r.i.) norm on the set (M({mathbb {R}}^n)) of Lebesgue-measurable functions on ({mathbb {R}}^n) such that the space (L_{rho }({mathbb {R}}^n) = left{ f in M({mathbb {R}}^n): rho (f) < infty right} ) is an interpolation space between (L_{2}({mathbb {R}}^n)) and (L_{{infty }}({mathbb {R}}^n).) The principal result of this paper asserts that given such a (rho ,) the inequality

holds for any r.i. norm (sigma ) on ( M({mathbb {R}}^n)) if and only if

Here, ({bar{rho }}) is the unique r.i. norm on (M({mathbb {R}}_+)), ({mathbb {R}}_+ = (0, infty )), satisfying ({bar{rho }}(f^{*})=rho (f)) and (U f^{*} (t) = int _{0}^{1/t} f^{*}), in which (f^{*}) is the nonincreasing rearrangement of f on (mathbb {R_+}). Further, in this case the smallest r.i. norm (sigma ) for which (rho ( {hat{f}}) le C sigma (f)) holds is given by

where, necessarily, ({bar{rho }} left( int _{0}^{1/t} chi _{(0, a)} right) = {bar{rho }} left( min {1/t, , a} right) < infty ), for all (a>0). We further specialize and expand these results in the contexts of Orlicz and Lorentz Gamma spaces.

In this paper we consider the semiclassical version of pseudo-differential operators on the lattice space (hbar {{mathbb {Z}}^{n}}). The current work is an extension of the previous work (Botchway et al. in J Funct Anal 278(11):108473, 33, 2020) and agrees with it in the limit of the parameter (hbar rightarrow 1). The various representations of the operators will be studied as well as the composition, transpose, adjoint and the link between ellipticity and parametrix of operators. We also give the conditions for the (ell ^p), weighted (ell ^2) boundedness and (ell ^p) compactness of operators. We investigate the relation between the classical and semi-classical quantization in the spirit of Ruzhansky and Turunen (Pseudo-differential operators and symmetries. Pseudo-differential operators, vol 2. Theory and Applications, Birkhäuser, Basel, 2010; J Fourier Anal Appl 16(6):943–982, 2010) RTspsJFAA and employ its applications to Schatten–von Neumann classes on (ell ^2( hbar mathbb {Z}^n)). We establish Gårding and sharp Gårding inequalities, with an application to the well-posedness of parabolic equations on the lattice (hbar mathbb {Z}^n). Finally we verify that in the limiting case where (hbar rightarrow 0) the semi-classical calculus of pseudo-differential operators recovers the classical Euclidean calculus, but with a twist.

We investigate a class of Fourier extension operators on fractional surfaces ((xi ,|xi |^alpha )) with (alpha ge 2). For the corresponding (alpha )-Strichartz inequalities, we characterize the precompactness of extremal sequences by applying the missing mass method and bilinear restriction theory. Our result is valid in any dimension. In particular for dimension two, our result implies the existence of extremals for (alpha in [2,alpha _0)) with some (alpha _0>5).

We introduce the Hadamard–Bergman convolution on the half-plane. We show that it exists in terms of the Hadamard product and it is commutative on the Bergman space (more appropriately called the Bergman–Jerbashian space) in the half-plane. Further, we explore mapping properties of the generalized Bergman-type operators with exponential weights in weighted Bergman spaces in the half-plane. Finally, we deduce sharp inclusions for weighted Bergman spaces, from corresponding Sobolev-type inequalities.

For the density of Galton-Watson processes in the Schröder case, we derive a complete left tail asymptotic series consisting of power terms multiplied by periodic factors.

We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets B. As a corollary we obtain the corresponding pointwise convergence result on (L^1). This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on (L^1) of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund along B on (L^p), (p>1), which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.

We introduce and prove small cap square function estimates for the unit parabola and the truncated light cone. More precisely, we study inequalities of the form

where (Gamma _alpha (R^{-1})) is the set of small caps of width (R^{-alpha }). We find sharp upper and lower bounds of the constant (C_{alpha ,p}(R)).

We prove the following group analogue of the well-known Heyde theorem on a characterization of the Gaussian distribution on the real line. Let X be a second countable locally compact Abelian group containing no subgroups topologically isomorphic to the 2-dimensional torus. Let G be the subgroup of X generated by all elements of X of order 2 and let (alpha ) be a topological automorphism of the group X such that (textrm{Ker}(I+alpha )={0}). Let (xi _1) and (xi _2) be independent random variables with values in X and distributions (mu _1) and (mu _2) with nonvanishing characteristic functions. If the conditional distribution of the linear form (L_2 = xi _1 + alpha xi _2) given (L_1 = xi _1 + xi _2) is symmetric, then (mu _j) are convolutions of Gaussian distributions on X and distributions supported in G. We also prove that this theorem is false if X is the 2-dimensional torus.