Approximation theorems for double orthogonal series

AbstractLet {φik(x): i, k = 1, 2,…} be a double orthonormal system on a positive measure space (X, ƒ, μ) and {aik} a double sequence of real numbers for which ∑i = 1∞ ∑k = 1∞ aik2 < ∞. Then the sum f(x) of the double orthogonal series ∑i = 1∞ ∑k = 1∞ aikφik(x) exists in the sense of L2-metric. If, in addition, ∑i = 1∞ ∑k = 1∞ aik2κ2(i, k) < ∞ with an appropriate double sequence {κ(i,k)} of positive numbers, then a rate of approximation to f(x) can be concluded by the rectangular partial sums smn(x) = ∑i = 1m ∑k = 1n aikφik(x), by the first arithmetic means of the rectangular partial sums σmn(x) = (1mn) ∑i = 1m ∑k = 1n sik(x), by the first arithmetic means of the square partial sums σr(x) = (1r) ∑k = 1r skk(x), etc. The so-called strong approximation to f(x) by smn(x) is also studied

Boundary non-crossings of Brownian pillow

Let B_0(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]^2\to R be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability \psi(u;h):=P{B_0(s,t)+h(s,t) \le u(s,t), \forall s,t\in [0,1]}. Further we investigate the asymptotic behaviour of $\psi(u;\gamma h)$ with $\gamma$ tending to infinity, and solve a related minimisation problem.Comment: 14 page

On the order of summability of the Fourier inversion formula

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems

Evaluation of Hungarian Wines for Resveratrol by Overpressured Layer Chromatography

A method, including solid phase extraction sample preparation, overpressured layer chromatographic separation and subsequent densitometric evaluation, was developed for measurement of total resveratrol (cis- and trans-isomers) content of wine. The amount of resveratrol was determined in wine samples from different winemaking regions of Hungary. The total resveratrol was high in Hungarian red wines (3.6–11 mg/L), and much lower in white ones (0.04–1.5 mg/L)

Four-dimensional generalized difference matrix and some double sequence spaces

In this study, I introduce some new double sequence spaces B(Mu), B(Cp), B(Cbp), B(Cr) and B(Lq) as the domain of four-dimensional generalized difference matrix B(r,s,t,u) in the spaces Mu, Cp, Cbp, Cr and Lq, respectively. I show that the double sequence spaces B(Mu), B(Cbp) and B(Cr) are the Banach spaces under some certain conditions. I give some inclusion relations with some topological properties. Moreover, I determine the α-dual of the spaces B(Mu) and B(Cbp), the β(ϑ)-duals of the spaces B(Mu), B(Cp), B(Cbp), B(Cr) and B(Lq), where ϑ∈{p,bp,r}, and the γ-dual of the spaces B(Mu), B(Cbp) and B(Lq). Finally, I characterize the classes of four-dimensional matrix mappings defined on the spaces B(Mu), B(Cp), B(Cbp), B(Cr) and B(Lq) of double sequences