We present a multidimensional analogue of an inequality by van der Corput-Visser concerning the coefficients of a real trigonometric polynomial. As an application, we obtain an improved estimate from below of the Bohr radius for the hypercone 𝓓₁ⁿ = {z ∈ ℂⁿ: |z₁|+. .. +|zₙ| < 1} when 3 ≤ n ≤ 10.

It is proved that the multi-dimensional maximal Fejér operator defined in a cone is bounded from the amalgam Hardy space $W(h_p,{\ell }_{\infty })$ to $W(L_p,{\ell }_{\infty })$. This implies the almost everywhere convergence of the Fejér means in a cone for all $f\in W(L₁,{\ell }_{\infty })$, which is larger than $L₁\left({ℝ}^d\right)$.

A famous theorem of Carleson says that, given any function $f\in L^p\left(𝕋\right)$, $p\in (1,+\infty )$, its Fourier series $\left(S_{n}f\left(x\right)\right)$ converges for almost every $x\in 𝕋$. Beside this property, the series may diverge at some point, without exceeding $O\left(n^{1/p}\right)$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f\left(x\right)=O\left(n^{\beta }\right)$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in 𝕋$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p\left(𝕋\right)$ have a multifractal behavior with respect to this definition....

We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class $Lip(\alpha ₁,...,{\alpha }_N)$ for some $0<\alpha ₁,...,{\alpha }_N<1$ and its marginal functions satisfy $f(·,x₂,...,x_N)\in Lip\beta ₁,...,f(x₁,...,x_{N-1},·)\in Lip{\beta }_N$ for some $0<\beta ₁,...,{\beta }_N<1$ uniformly in the indicated variables $x_l$, 1 ≤ l ≤ N, then $f{̃}^{(\eta ₁,...,{\eta }_N)}\in Lip(\alpha ₁,...,{\alpha }_N)$ for each choice of $(\eta ₁,...,{\eta }_N)$ with ${\eta }_l=0$ or 1 for 1 ≤ l ≤ N.

We present a new criterion for the weighted $L^p-L^q$ boundedness of multiplier operators for Laguerre and Hermite expansions that arise from a Laplace-Stieltjes transform. As a special case, we recover known results on weighted estimates for Laguerre and Hermite fractional integrals with a unified and simpler approach.

2000 Mathematics Subject Classification: 42A45.For a Hilbert space H ⊂ L1loc(R) of functions on R we obtain a representation theorem for the multipliers M commuting with the shift operator S. This generalizes the classical result for multipliers in L2(R) as well as our previous result for multipliers in weighted space L2ω(R). Moreover, we obtain a description of the spectrum of S.

We study the problem of construction of the smooth interpolation formula presented as the minimizer of suitable functionals subject to interpolation constraints. We present a procedure for determining the interpolation formula that in a natural way leads to a linear combination of polyharmonic splines complemented with lower order polynomial terms. In general, such formulae can be very useful e.g. in geographic information systems or computer aided geometric design. A simple computational example...