Harmonic analysis on the line is the theory of Fourier transforms, more complicated than Fourier series, due to the line’s non-compactness. On R the exponential functions, while still eigenfunctions for d dx and still giving group homomorphisms, are no longer in L(R). Entangled with this point is the fact that Fourier inversion expresses functions as integrals of exponential functions, not as s...

Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation and best approximation by trigonometric functions, both direct and inverse theorems. One of the objective of this study is to demonstrate that Fourier series on spectral sets enjoy a rich structure ...

1. Introduction Recently [2,3] we found necessary and sufficient conditions for the convergence at a preassigned point of the spherical partial sums of the Fourier integral in a class of piece-wise smooth functions in Euclidean space. These yield elementary examples of divergent Fourier integrals in three dimensions and higher. Meanwhile, several years ago Gottlieb and Orsag[1] observed that in...

We introduce a method to estimate sums of oscillating functions on finite abelian groups over intervals or (generalized) arithmetic progressions, when the size of the interval is such that the completing techniques of Fourier analysis are barely insufficient to obtain non-trivial results. In particular, we prove various estimates for exponential sums over intervals in finite fields and related ...

We establish asymptotic normality of weighted sums of periodograms of a stationary linear process where weights depend on the sample size. Such sums appear in numerous statistical applications and can be regarded as a discretized versions of the quadratic forms involving integrals of weighted periodograms. Conditions for asymptotic normality of these weighted sums are simple and resemble Lindeb...

Approximation of positive polynomials by sums of squares.

It is now nearly a decade since the theories of perverse sheaves and of the /adic Fourier transform came into being and gave fundamental new insights into the behavior of additive character sums over finite fields. Most of these insights were spellings out of the basic fact that the Fourier transform of a perverse sheaf is itself a perverse sheaf, a statement that amounts to a succession of van...