Generalized conjugate partial sums of Fourier series are used to find jumps of functions. The rate of convergence is studied and sharp results are obtained.

We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{\epsilon }\left(x\right):={\sum }_{j\in D_{\epsilon }\left(x\right)}e{*}_j\left(x\right)e_j$, where $D_{\epsilon }\left(x\right):=j:|e{*}_j\left(x\right)|\ge \epsilon$. We study a generalized version of $T_{\epsilon }$ that we call the weak thresholding approximation. We modify the $T_{\epsilon }\left(x\right)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t,\epsilon }\left(x\right):=j:t\epsilon \le |e{*}_j\left(x\right)|<\epsilon$ and consider the weak...

We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form $Gₘ\left(f\right):={\sum }_{k\in \Lambda }f̂\left(k\right)e^{i(k,x)}$, where $\Lambda \subset {\mathbb{Z}}^d$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in the case of...

We investigate some convergence and divergence properties of the logarithmic means of quadratic partial sums of double Fourier series of functions, in measure and in the L Lebesgue norm.

We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are $\left(j^{-\alpha }\right)$ for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.

We show that $L^2$-bounded singular integrals in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost everywhere existence of principal values.

Let X be a rearrangement-invariant space of Lebesgue-measurable functions on ${ℝ}^n$, such as the classical Lebesgue, Lorentz or Orlicz spaces. Given a nonnegative, measurable (weight) function on ${ℝ}^n$, define $X\left(w\right)=F:{ℝ}^n\to ℂ:\infty >\parallel F{\parallel }_{X\left(w\right)}:=\parallel Fw{\parallel }_X$. We investigate conditions on such a weight w that guarantee X(w) is an algebra under the convolution product F∗G defined at $x\in {ℝ}^n$ by $\left(F\ast G\right)\left(x\right)={ʃ}_{{ℝ}^n}F(x-y)G\left(y\right)dy$; more precisely, when $\parallel F\ast G{\parallel }_{X\left(w\right)}\le \parallel F{\parallel }_{X\left(w\right)}\parallel G{\parallel }_{X\left(w\right)}$ for all F,G ∈ X(w).

We reduce the convolution of radius functions to that of 1-variable functions. Then we present formulas for computing convolutions of an abstract radius function on ℝ³ with various integral kernels - given by elementary or discontinuous functions. We also prove a theorem on the asymptotic behaviour of a convolution at infinity. Lastly, we deduce some estimates which enable us to find the asymptotics of the velocity and pressure of a fluid (described by the Navier-Stokes equations) in the boundary...

We give sufficient conditions on the kernel K for the convolution operator Tf = K ∗ f to be bounded on Hardy spaces $H^p\left(G\right)$, where G is a homogeneous group.