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Section 8.3 Fourier transform of tempered distributions 435 8.3 Fourier transform of tempered distributions Definition 8.3.1. Let T 2 S 0.Rn/ be a tempered distribution. Then its Fourier trans- form F T 2 S 0.Rn/ and its co-transform NF T 2 S 0.Rn/ are tempered distributions defined by: hF T; i D hT;F i 8 2 S.Rn/; (8.3.1) h NF T; i D hT; NF i 8 2 S.Rn/ (8.3.2) ((8.3.1)–(8.3.2) were, in fact, suggested in (8.1.5)), with .F /.x/ D Z Rn ./ei2h;xid; . NF /.x/ D Z Rn ./ei2h;xid: (8.3.3) Justification By Theorem 7.7.2, F W S.Rn/ ! S.Rn/ is an isomorphism from S

Chapter 2 Fourier Transformation and Tempered Distributions Summary In this chapter the Fourier transformation is introduced and their most important properties for the study of pseudodifferential operators are proved. This includes Plancherel’s Theorem, the Fourier Inversion Theorem and Fourier Transformation of tempered distributions. Moreover, the relation between smoothness properties of a function and decay properties of its Fourier transform are discussed. This will be es- sential for the study of regularity questions for partial differential equations and

Analysis 18, 2 5 7 - 2 6 7 (1998) Analysis O R. Oldenbourg Verlag München 1998 AMALGAMS OF KUCERA TYPE A N D TEMPERED DISTRIBUTIONS Jorge J. Betancor a n d Benito J . Gonzalez Received: October 17. 1995; revised: July 1. 1997 Abstract: In this paper we introduce a new dass of amalgams of LP and V on the real line. We establish topological properties and we investigate the behaviour of the Fourier transform on these spaces. Also we characterize the Schwartz space S and its dual space in terms of the new amalgams. Key words. Amalgams, Fourier

general. In fact,we have the following result: If 2 D.Rn/; ¤ 0; then F … D.Rn/ [7]: (8.1.6) Thus, the Fourier transform F T of arbitrary distribution T 2 D 0.Rn/ does not exist. (8.1.7) But the right-hand side of (8.1.5) becomes meaningful for 2 S.Rn/, since, by Theorem 7.6.1, 8 2 S.Rn/;F 2 S.Rn/ with S.Rn/ ,! L1.Rn/. Hence, we are going to identify a particular subclass of distributions containing L1.Rn/ as a subspace. This particular subclass of distributions, described by Laurent Schwartz, are called tempered distributions, or distributions tempérés in French. 8

.5.23)). 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives of tempered distributions Theorem 8.6.1. Let T 2 S 0.Rn/ be a tempered distribution on Rn. Then the follow- ing relations hold: I. 8 multi-index ˛, F Œ@˛x T D .i2/˛F T I @˛ .F T /./ D F Œ.i2x/˛T I F Œx˛T D .1/j˛j 1 .i2/j˛j @˛ .F T /I (8.6.1) II. 8a 2 Rn, F ŒaT D ei2h;aiF T I a.F T / D F Œei2hx;aiT ; (8.6.2) where @˛x D @ j˛j @x ˛1 1 @x ˛2 2 :::@x ˛n n , @˛ D @j˛j @ ˛1 1 @ ˛2 2 :::@ ˛n n . Section 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives 459 Proof

COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.6(2006), No.3, pp.336–344 c© 2006 Institute of Mathematics of the National Academy of Sciences of Belarus SPLITTING OF THE S′+ SPACE INTO A UNION OF BANACH SPACES THROUGH LAGUERRE EXPANSIONS M.STOJANOVIĆ1 Abstract — We split the space S′+ of tempered distributions into a union of Banach spaces with respect to the scale of spaces LG′0s, s 0, which allows us to give the approximation of the generalization of function therein. 2000 Mathematics Subject Classification: 45D05, 41A10, 46F10, 44A15. Keywords: tempered

Adv. Pure Appl. Math. 3 (2012), 113–122 DOI 10.1515/APAM.2011.015 © de Gruyter 2012 A characterisation of the Weyl transform R. Lakshmi Lavanya and S. Thangavelu Abstract. A theorem of Alesker et al. says that the Fourier transform on Rn is essentially the only transform on the space of tempered distributions which interchanges convolutions and products. In this note we obtain a similar characterisation for the Weyl transform. Keywords. Schwartz class, tempered distributions, Weyl transform, Fourier–Weyl transform, noncommutative derivations. 2010 Mathematics

Abstract

For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ℝn. Due to the conditions on M each function of this space admits a holomorphic extension in ℂn. In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions satisfying some weighted L 2-bounds in a domain of holomorphy in ℂn are obtained with the aid of L. Hörmander’s method of L 2-bounds for the $$\bar \partial$$ operator. Also, some new facts on the Fourier-Laplace transform of tempered distributions complementing some well-known results of V.S. Vladimirov are employed.

Analysis 32, 271–279 (2012) / DOI 10.1524/anly.2012.1146 c Oldenbourg Wissenschaftsverlag, München 2012 Weierstrass and Picard summability of more-dimensional Fourier transforms Ferenc Weisz Received: September 20, 2011 Summary: It is proved that the maximal operator of the Weierstrass and Picard summability means of a tempered distribution is bounded from Hp.Rd / to Lp.Rd / for all 0 < p 1 and, consequently, is of weak type (1,1). As a consequence we obtain that the summability means of a function f 2 L1.Rd / converge a.e. to f . Similar results are shown for

not equal norm) to the complexification of the Banach space 𝓑( X , Y ). (For the problem of the many possible equivalent complexifications of real Banach spaces, see [ 25 ].) If A ∈ 𝓑( X ℂ , Y ℂ ) we can define A ∈ 𝓑( X ℂ , Y ℂ ) by A ¯ x := A x ¯ ¯ , $$\begin{array}{} \displaystyle \overline A x:=\overline{A\overline x}, \end{array}$$ where in the right hand side we use the natural conjugations of X ℂ and Y ℂ . With this definition Ax = A x . An X -valued tempered distribution is a continuous linear map from the Schwartz class 𝓢(ℝ) to X . We