Related papers: Minimal Fourier majorants in $L^p$

Denote the coefficients in the complex form of the Fourier series of a function $f$ on the interval $[-\pi, \pi)$ by $\hat f(n)$. It is known that if $p = 2j/(2j-1)$ for some integer $j>0$, then for each function $f$ in $L^p$ there exists…

We prove that if $f\in L^p(\mathbb{R}^k)$ with $p<(k^2+k+2)/2$ satisfies that $\widehat{f}$ is supported on a small perturbation of the moment curve in $\mathbb{R}^k$, then $f$ is identically zero. This improves the more general result of…

In this work we define a Fourier transform for each $f\in L^{p(\cdot)}(\mathbb{R})$, for a large class of exponent functions $p(\cdot)$, as the distributional derivative of a H\"older continuous function. A norm is defined in the space of…

It is proved that there does not exist any non zero function in $L^p(\R^n)$ with $1\leq p\leq 2n/\alpha$ if its Fourier transform is supported by a set of finite packing $\alpha$-measure where $0<\alpha<n$. It is shown that the assertion…

We consider several problems at or beyond endpoint in harmonic analysis. The solutions of these problems are related to the estimates of some classes of sublinear operators. To do this, we introduce some new functions spaces…

We give characterizations of radial Fourier multipliers as acting on radial L^p-functions, 1<p<2d/(d+1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding…

Let p be a prime, and let f : Z/pZ -> R be a function with average value 0 and ||f||_A <= 1, where ||f||_A denotes the algebra norm (L^1 norm of the Fourier transform). Then f(x) is small for some x, specifically min_x |f(x)| is no more…

The Hardy-Littlewood majorant problem asks whether L^p norms of functions on the circle grow if one replaces their Fourier coefficients with their absolute values. This is clear if p is an even integer, but false if p is any other number.…

Suppose that f : F_p^n -> [0,1] has expected value t in [p^(-n/9),1] (so, the density t can be quite low!). Furthermore, suppose that support(f) has no three-term arithmetic progressions. Then, we develop non-trivial lower bounds for f_j,…

In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: $$T(f)(x)=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^{2n}}e^{i\varphi(x,\xi,\eta)}\cdot…

Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K \leq -1$. In \cite{biswas6}, a Fourier transform was defined for functions on $X$, and a Fourier inversion formula and Plancherel theorem…

For each $f\in L^p({\mathbb R)}$ ($1\leq p<\infty$) it is shown that the Fourier transform is the distributional derivative of a H\"older continuous function. For each $p$ a norm is defined so that the space Fourier transforms is…

Let $\mathbb{F}_p$ be the finite field of prime order $p$. For any function $f \colon \mathbb{F}_p{}^n \to \mathbb{F}_p$, there exists a unique polynomial over $\mathbb{F}_p$ having degree at most $p-1$ with respect to each variable which…

Consider a trigonometric polynomial f of degree N, and associate to it the polynomial F in which each coefficient of f is replaced by its absolute value. F is called the majorant of f. We show that the L^3 norm of f can be larger than that…

A famous theorem of Carleson says that, given any function $f\in L^p(\TT)$, $p\in(1,+\infty)$, its Fourier series $(S_nf(x))$ converges for almost every $x\in \mathbb T$. Beside this property, the series may diverge at some point, without…

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing…

We give a simple necessary and sufficient condition for maximal operators associated with radial Fourier multipliers to be bounded on $L^p_{rad}$ and $L^p$ for certain $p$ greater than $2$. The range of exponents obtained for the…

The minimal Kolmogorov complexity of a total computable function that exceeds everywhere all total computable functions of complexity at most $n$, is $2^{n+O(1)}$. If we replace "everywhere" by "for all sufficiently large inputs", the…

We study the effect of a splitting operator S_t on the L^p norm of the Fourier transform of a function f and on the operator norm of a Fourier multiplier m. Most of our results assume p is an even integer, and are often stronger when f or m…

Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained in the paper. The results are given in terms of $L^p$ integrability of the function and its…