By Javier Duoandikoetxea

Fourier research contains a number of views and strategies. This quantity provides the true variable equipment of Fourier research brought through Calderón and Zygmund. The textual content was once born from a graduate path taught on the Universidad Autónoma de Madrid and accommodates lecture notes from a direction taught by means of José Luis Rubio de Francia on the similar college. influenced via the examine of Fourier sequence and integrals, classical themes are brought, equivalent to the Hardy-Littlewood maximal functionality and the Hilbert remodel. the remainder parts of the textual content are dedicated to the examine of singular imperative operators and multipliers. either classical features of the speculation and newer advancements, reminiscent of weighted inequalities, $H^1$, $BMO$ areas, and the $T1$ theorem, are mentioned. bankruptcy 1 provides a assessment of Fourier sequence and integrals; Chapters 2 and three introduce operators which are uncomplicated to the sector: the Hardy-Littlewood maximal functionality and the Hilbert rework. Chapters four and five talk about singular integrals, together with smooth generalizations. bankruptcy 6 experiences the connection among $H^1$, $BMO$, and singular integrals; bankruptcy 7 provides the basic conception of weighted norm inequalities. bankruptcy eight discusses Littlewood-Paley idea, which had advancements that led to a few functions. the ultimate bankruptcy concludes with an immense outcome, the $T1$ theorem, which has been of the most important significance within the box. This quantity has been up-to-date and translated from the Spanish variation that was once released in 1995. Minor adjustments were made to the middle of the booklet; even though, the sections, "Notes and additional Results" were significantly extended and contain new issues, effects, and references. it really is aimed toward graduate scholars looking a concise creation to the most elements of the classical concept of singular operators and multipliers. necessities contain uncomplicated wisdom in Lebesgue integrals and useful research.

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**Example text**

It is named after J. Gibbs, who announced it in Nature 59 (1899), although it had already been discovered by H. Wilbraham in 1848. See Dym and McKean [4, Chapter 1] and the paper by E. Hewitt and R. E. Hewitt (The Gibbs- Wilbraham phenomenon: an episode in Fourier analysis, Arch. Hist. Exact Sci. 21 (1979/80), 129-160). Gibbs phenomenon is eliminated by replacing pointwise convergence by Cesaro summability. 8), m < ON f(x) ::; M. In fact, it can be shown that if m < f(x) < M on an interval (a, b), then for any E > 0, m - E < ON f(x) < M + E on (a + E, b - E) for N sufficiently large.

Mat. 25 (1961), 531-542). 1. 21). 6. Eigenfunctions for the Fourier transform in L2(JR). Since the Fourier transform has period 4, if j is a function such that j = Aj, we must have that A4 = 1. Hence, A = ±1, ±i are the only possible eigenvalues of the Fourier transform. 14 shows that exp( _TrX2) is an eigenfunction associated with the eigenvalue 1. The Hermite functions give the remaining eigenfunctions: for n > 0, ( l)n dn hn(x) = - , eXP(Trx 2)-d exp( _TrX2) n. xn satisfies hn = (-i)nh n . If we normalize these functions, en n )-n.

J0rsboe and L. Melbro (The Carleson-Hunt Theorem on Fourier Series, Lecture Notes in Math. 911, Springer-Verlag, Berlin, 1982) is devoted to the proof of this theorem. The original references for this are the articles by L. Carleson (On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157) and R. Hunt (On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. , 1967), pp. 235-255, Southern Illinois Univ. Press, Carbondale, 1968).