Sobolev exponents of Butterworth refinable functions

AbstractThe precise Sobolev exponent s∞(φn) of the Butterworth refinable function φn associated with the Butterworth filter of order n, bn(ξ)≔cos2n(ξ/2)cos2n(ξ/2)+sin2n(ξ/2), is shown to be s∞(φn)=nlog23+log2(1+3−n). This recovers the previously given asymptotic estimate of s∞(φn) of Fan and Sun, and gives more accurate regularity of Butterworth refinable function φn

Regularity of Dual Gabor Windows

We present a construction of dual windows associated with Gabor frames with compactly supported windows. The size of the support of the dual windows is comparable to that of the given window. Under certain conditions, we prove that there exist dual windows with higher regularity than the canonical dual window. On the other hand, there are cases where no differentiable dual window exists, even in the overcomplete case. As a special case of our results, we show that there exists a common smooth dual window for an interesting class of Gabor frames. In particular, for any value of K∈ℕ, there is a smooth function h which simultaneously is a dual window for all B-spline generated Gabor frames {EmbTnBN(x/2)}m,n∈ℕ for B-splines BN of order N=1,…,2K+1 with a fixed and sufficiently small value of b

Approximation of circular arcs by Bézier curves

AbstractFor the circular arc of angle 0<α<π we present the explicit form of the best GC3 quartic approximation and the best GC2 quartic approximations of various types, and give the explicit form of the Hausdorff distance between the circular arc and the approximate Bézier curves for each case. We also show the existence of the GC4 quintic approximations to the arc, and find the explicit form of the best GC3 quintic approximation in certain constraints and their distances from the arc. All approximations we construct in this paper have the optimal order of approximation, twice of the degree of approximate Bézier curves

Sensitization rates of airborne pollen and mold in children

PurposeAeroallergens are important causative factors of allergic diseases. Previous studies on aeroallergen sensitization rates investigated patients groups that had visited pediatric allergy clinics. In contrast, we investigated sensitization rates in a general population group of elementary school to teenage students in Incheon, Jeju, and Ulsan.MethodsAfter obtaining parental consent, skin-prick tests were performed on 5,094 students between March and June 2010. Elementary school students were tested for 18 common aeroallergens, whereas middle and high school students were tested for 25 allergens. The 25 allergens included Dermatophagoides pteronyssinus, Dermatophagoides farinae, pollen (birch, alder, oak, Japanese cedar, pine, willow, elm, maple, Bermuda grass, timothy grass, rye grass, orchard grass, meadow grass, vernal grass, mugwort, Japanese hop, fat hen, ragweed, and plantain), and mold (Penicillatum, Aspergillus, Cladosporium, and Alternaria).ResultsThe sensitization rates in descending order were 25.79% (D. pteronyssinus), 18.66% (D. farinae), 6.20% (mugwort), and 4.07% (willow) in Incheon; 33.35% (D. pteronyssinus), 24.78% (D. farinae), 15.36% (Japanese cedar), and 7.33% (Alternaria) in Jeju; and 32.79% (D. pteronyssinus), 30.27% (D. farinae), 10.13% (alder), and 8.68% (birch) in Ulsan. The dust mite allergen showed the highest sensitization rate among the 3 regions. The sensitization rate of tree pollen was the highest in Ulsan, whereas that of Alternaria was the highest in Jeju. The ragweed sensitization rates were 0.99% in Incheon, 1.07% in Jeju, and 0.81% in Ulsan.ConclusionThe differences in sensitization rates were because of different regional environmental conditions and distinct surrounding biological species. Hence, subsequent nationwide studies are required