Consider the Euclidean space \\(\\mathbb {R}^n\\), \\(n\\ge 1\\), with \\(x=(x_1,\\ldots , x_n)\\in \\mathbb {R}^n\\) and with \\(|x|=\\sqrt{x_1^2+\\cdots +x_n^2}\\) and scalar product \\((x, y)=\\sum _{j=1}^n x_j y_j\\)....
Spherical transformSchwartz functionsInvariantsPrimary 13A5043A32Secondary 43A8543A90The spectrum of a Gelfand pair of the form ${(Kltimes N,K)}$ , where N is a nilpotent group, can be embedded in a Euclidean space ${{mathbb R}^d}$ . The identification of the spherical transforms of K...
In this case, the Gelfand spaceΔ:(K, Hn) is equipped with the Godement–Plancherel measure, and the spherical transform∧:L2K(Hn)→L2(Δ(K, Hn)) is an isometry. The main result in this paper provides a complete characterization of the set SK(Hn)∧={f | f∈SK(Hn)} of spherical...
after testing for room size as described herein, the space doesn't match up to "normal" definitions, or can not be computed at all, the computer enters another program from EPROM called "Obstructs" that tests to determine if the anomaly is due to forward obstruction of the speaker by a ...
The spectrum of a Gelfand pair of the form (K\ltimes N,K){(K\ltimes N,K)}, where N is a nilpotent group, can be embedded in a Euclidean space \mathbb Rd{{\mathbb R}^d}. The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions ...
Fourier transformdouble Fourier transformwavelet transformPrimary: 44A1542C15Secondary: 46F12In the present paper, we discuss about extension of the wavelet transform on distribution space of compact support and develop the Paley–Wiener–Schwartz type theorem for the wavelet transform on the same. ...
It is also possible to coat certain portions of an instrument's interior with a sound-absorbing material, such as felt, or fill some of the space inside the instrument with a suitable material, to accomplish this. The material may be permanently attached to the instrument, or may simply be...
It was L. Schwartz who in [108] and [109] pointed out that the spaceS(n), nowadays called the Schwartz space, is best suited to handle the Fourier integral and the Fourier transform. In this chapter we want to introduce this space鈥...
The Paley-Wiener-Schwartz theorem characterizes the Fourier transforms of distributions with bounded (compact) support as being exactly the entire functions of exponential type which are slowly increasing (cf.[4], [18], [20], [2l]). Nachbin and Dineen [9] defined the Frechet space Nbc (E;...
We demonstrate that the Fourier transform (FT) acts as an isomorphism on the GS space of test functions and, by duality, on TSE distributions. Crucially, the GS space proves to be minimal in the sense that its dual, encompassing TSE distributions, represents the largest possible linear space ...