We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear (GL) transformation behavior of the QFT with matrices, Clifford geometric algebra ...

In [4], D. Osipov and A. Parshin developed an approach to harmonic analysis on higher dimensional local fields through categories of filtered vector spaces as in [3]. In the present paper we give a variant of this approach that behaves nicely for inductive arguments. We establish Pontryagin duality, the Fourier inversion formula, and a Plancherel formula for all dimensions. ∗This paper was writ...

1. S. Bernstein, Sur la borne supérieure du module de la dérivée d'une fonction de degré fini, Dokl. Akad. Nauk SSSR 51 (1946), 567-568; Collected Works, Vol. 2, Izdatelstvo Akademii Nauk SSSR, Moscow, 1954, pp. 396-398. 2. R. P. Boas, Jr., Entire functions, Academic Press, New York, 1954. 3. S. Bochner, Vorlesungen über Fouriersche Integrale, Akademische Verlagsgesellschaft, Leipzig, 1932. 4. ...

We study the action of standard intertwining operators on H-fixed generalized vectors in the minimal principal series of a reductive symmetric space G/H of Harish-Chandra's class. The main result is that-after an appropriate normalization-this action is unitary for the unitary principal series. This is an extension of previous work under more restrictive hypotheses on G and H. The present resul...

We obtain Plancherel-Rotach type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form e−NV (x) on the real line, assuming that V has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application we extend the universality class for bulk and edge asymptotics o...

We consider the Plancherel measure on irreducible components of tensor powers spinor representation so(2n+1). The representations correspond to generalized Young diagrams. With respect this probability an is product its multiplicity and dimension, divided by total dimension product. study limit shape diagram when power N rank n algebra tend infinity with N/n fixed. derive explicit formula for p...

We establish the inverse Lebedev expansion with respect to parameters and arguments of the modified Bessel functions for an arbitrary function from Hardy’s space H2,A, A> 0. This gives another version of the Fourier-integral-type theorem for the Lebedev transform. The result is generalized for a weighted Hardy space Ĥ2,A ≡ Ĥ2((−A,A);|Γ(1+Rez+iτ)|2dτ), 0 < A < 1, of analytic functions f(z),z = R...

Let G be a real semisimple Lie group. Harish-Chandra has defined the Schwartz space, V[G), on G. A tempered distribution on G is a continuous linear functional on R G ) . If the real rank of G equals one, Harish-Chandra has published a version of the Plancherel formula for I^(G) [3(k), 5241. We restrict the Fourier transform map to %(G), and we compute the image of the space V(G) [Theorem 31. T...

provided that {X„} is an increasing sequence of real numbers such that |X„ — n\ 0. This is the L2-analogue of the theorem, proved in different ways by Duffin and Schaeffer [6] and by B. Levin [8], that f(x) is bounded if {/(X„)} is bounded ; the cases of the two theorems when \n = n are due respectively to Plancherel and Pólya [il] and Cartwright [3]. The object of this pap...

We initiate a Stein’s method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov’s central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein’s method arises from the theory of harmonic function...