There are two grounds the spline theory stems from - the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We use the general variational approach called smooth interpolation introduced by Talmi and Gilat and show that it covers not only the cubic spline and its 2D and 3D analogues but also the well known tension spline...

Let $\mu$ be a Fourier-Stieltjes transform, defined on the discrete real line and such that the corresponding measure on the dual group vanishes on the set of characters, continuous on $\mathbf{R}$. Then for every $ϵ\>0$, $\{x\in \mathbf{R}|\mathrm{Re}\left(\mu \right(x\left)\right)\>ϵ\}$ has a vanishing interior Lebesgue measure. If $ϵ=0$ the statement is not generally true. The result is applied to prove a theorem of Rosenthal.

We study the behaviour of the $n$-dimensional centered Hardy-Littlewood maximal operator associated to the family of cubes with sides parallel to the axes, improving the previously known lower bounds for the best constants $c_n$ that appear in the weak type $(1,1)$ inequalities.

The Hardy-Littlewood maximal function ℳ and the trigonometric function sin x are two central objects in harmonic analysis. We prove that ℳ characterizes sin x in the following way: Let $f\in C^{\alpha }(ℝ,ℝ)$ be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator $\left(A_{x}f\right)\left(r\right)=1/2r{\int }_{x-r}^{x+r}f\left(z\right)dz$ has a critical point at r = γ for every x ∈ ℝ, then f(x) = a + bsin(cx+d) for some a,b,c,d ∈ ℝ. This statement can be used to derive a characterization of trigonometric functions as those nonconstant...

We prove that $|{\sum }_{k=1}^{n}sin\left((2k-1)x\right)/k|<Si\left(\pi \right)=1.8519...$ for all integers n ≥ 1 and real numbers x. The upper bound Si(π) is best possible. This result refines inequalities due to Fejér (1910) and Lenz (1951).

Under certain conditions on a function space X, it is proved that for every $L^{\infty }$-function f with $\parallel f{\parallel }_{\infty }\le 1$ one can find a function φ, 0 ≤ φ ≤ 1, such that φf ∈ X, $mes\phi \ne 1\le \varepsilon \parallel f{\parallel }_1$ and $\parallel \phi f{\parallel }_X\le const(1+log{\varepsilon }^{-1})$. For X one can take, e.g., the space of functions with uniformly bounded Fourier sums, or the space of $L^{\infty }$-functions on ${ℝ}^n$ whose convolutions with a fixed finite collection of Calderón-Zygmund kernels are also bounded.

A general method is given for recovering a function $f:{\mathbb{R}}^N\to C$, $N\ge 1$, knowing only an approximation of its Fourier transform.

We consider a problem of intervals raised by I. Ya. Novikov in [Israel Math. Conf. Proc. 5 (1992), 290], which refines the well-known theorem of J. Marcinkiewicz concerning structure of closed sets [A. Zygmund, Trigonometric Series, Vol. I, Ch. IV, Theorem 2.1]. A positive solution to the problem for some specific cases is obtained. As a result, we strengthen the theorem of Marcinkiewicz for generalized Cantor sets.

Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space . We show that if W and its inverse $W^{-1}$ both satisfy a matrix reverse Hölder property introduced by Christ and Goldberg, then the weighted Hilbert transform $H:L{²}_W(ℝ,)\to L{²}_W(ℝ,)$ and also all weighted dyadic martingale transforms $T_{\sigma }:L{²}_W(ℝ,)\to L{²}_W(ℝ,)$ are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.