A new proof of some polynomial inequalities related to pseudo-splines

Pseudo-splines of type I were introduced in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003) 1–46] and [Selenick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2000) 163–181] and type II were introduced in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104]. Both types of pseudo-splines provide a rich family of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. In [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104], Dong and Shen gave a regularity analysis of pseudo-splines of both types. The key to regularity analysis is Proposition 3.2 in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104], which also appeared in [A. Cohen, J.P. Conze, Régularité des bases d'ondelettes et mesures ergodiques, Rev. Mat. Iberoamericana 8 (1992) 351–365] and [I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, 1992] for the case l=N−1. In this note, we will give a new insight into this proposition.     

Construction of biorthogonal wavelets from pseudo-splines

Pseudo-splines constitute a new class of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. Pseudo-splines were first introduced by Daubechies, Han, Ron and Shen in [Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14(1) (2003), 1–46] and Selenick in [Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10(2) (2001) 163–181], and their properties were extensively studied by Dong and Shen in [Pseudo-splines, wavelets and framelets, 2004, preprint]. It was further shown by Dong and Shen in [Linear independence of pseudo-splines, Proc. Amer. Math. Soc., to appear] that the shifts of an arbitrarily given pseudo-spline are linearly independent. This implies the existence of biorthogonal dual refinable functions (of pseudo-splines) with an arbitrarily prescribed regularity. However, except for B-splines, there is no explicit construction of biorthogonal dual refinable functions with any given regularity. This paper focuses on an implementable scheme to derive a dual refinable function with a prescribed regularity. This automatically gives a construction of smooth biorthogonal Riesz wavelets with one of them being a pseudo-spline. As an example, an explicit formula of biorthogonal dual refinable functions of the interpolatory refinable function is given.     

Determination of jumps via advanced concentration factors

If a periodic function f with period 2π has a discontinuity at $\xi \in [?\pi ,\pi )$, then the partial sums of its Fourier conjugate series diverge to + or ? infinity at ξ. It was shown in [F. Lukács, Über die Bestimmung des Sprunges einer Function aus iher Fourierreihe, J. Reine Angew. Math. 150 (1920) 107–122] that, if we divide these partial sums at n by $\ln n$, however, we obtain essentially the length of the jump in the limit, as n tends to infinity. But the convergence rate in this way is very slow. This result has been improved if these partial sums are replaced by other sequences, similar to appropriate summability sequences (see [N.S. Banerjee, J. Geer, Exponentially accurate approximations to piecewise smooth periodic functions, ICASE report 95-17, NASA Langley Research Center, 1995; B.I. Golubov, Determination of jump of a function of bounded p-variation by its Fourier series, Math. Notes 12 (1972) 444–449; G. Kvernadge, Determination of jumps of a bounded function by its Fourier series, J. Approx. Theory 92 (1998) 107–122]). In the present paper we obtain still stronger results along these lines. We will prove that there are concentration factors $\{{\mu }_{n,k}\}$ such that the condition of Dini-type introduced in [A. Gelb, E. Tadmor, Detection of edges in spectral data, Appl. Comput. Harmon. Anal. 7 (1999) 101–135] can be canceled and for piecewise smooth functions the convergence speed keeps the same rate as A. Gelb and E. Tadmor's method shown in [A. Gelb, E. Tadmor, Detection of edges in spectral data, Appl. Comput. Harmon. Anal. 7 (1999) 101–135].     

Symmetric and antisymmetric tight wavelet frames

For a given set of wavelets Ψ, we provide a general, and yet simple, method to derive a new set of wavelets ${\Psi }^{\prime }$ such that each wavelet in ${\Psi }^{\prime }$ is either symmetric or antisymmetric. The affine system generated by ${\Psi }^{\prime }$ is a tight frame for the space $L_2({\mathbb{R}}^d)$ whenever the affine system generated by Ψ is so. Further, when Ψ is constructed via a multiresolution analysis, ${\Psi }^{\prime }$ can also be derived from a, but possibly different, multiresolution analysis. If moreover the multiresolution analysis for constructing Ψ is generated by a symmetric refinable function, then ${\Psi }^{\prime }$ is obtained from the same multiresolution analysis.     

Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.     

Boolean functions with a simple certificate for CNF complexity

In this paper we study relationships between CNF representations of a given Boolean function $f$ and essential sets of implicates of $f$. It is known that every CNF representation and every essential set must intersect. Therefore the maximum number of pairwise disjoint essential sets of $f$ provides a lower bound on the size of any CNF representation of $f$. We are interested in functions, for which this lower bound is tight, and call such functions coverable. We prove that for every coverable function there exists a polynomially verifiable certificate (witness) for its minimum CNF size. On the other hand, we show that not all functions are coverable, and construct examples of non-coverable functions. Moreover, we prove that computing the lower bound, i.e. the maximum number of pairwise disjoint essential sets, is $NP$-hard under various restrictions on the function and on its input representation.     

Some expansions of the exponential integral in series of the incomplete Gamma function

In a recent paper in this journal, Gautschi et al. [W. Gautschi, F.E. Harris, N.M. Temme, Expansions of the exponential integral in incomplete Gamma functions, Appl. Math. Lett. 16 (2003) 1095–1099] presented an interesting expansion formula for the exponential integral $E_1\left(z\right)$ in a series of the incomplete Gamma function $\gamma (\alpha ,z)$. Their investigation was motivated by a search for better methods of evaluating the exponential integral $E_1\left(z\right)$ which occurs widely in applications, most notably in quantum-mechanical electronic structure calculations. The object of the present sequel to the work by Gautschi et al. [Expansions of the exponential integral in incomplete Gamma functions, Appl. Math. Lett. 16 (2003) 1095–1099] is to give a rather elementary demonstration of the aforementioned expansion formula and to show how easily it can be put in a much more general setting. Some analogous expansion formulas in series of the complementary incomplete Gamma function $\Gamma (\alpha ,z)$ are also considered.     

Bifurcation theory for finitely smooth planar autonomous differential systems

In this paper we establish bifurcation theory of limit cycles for planar $C^k$ smooth autonomous differential systems, with $k\in \mathbb{N}$. The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and $C^{\infty }$ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case.     

Identifying defective sets using queries of small size

We examine the following version of a classic combinatorial search problem introduced by Rényi: Given a finite set $X$ of $n$ elements we want to identify an unknown subset $Y$ of $X$, which is known to have exactly $d$ elements, by means of testing, for as few as possible subsets $A$ of $X$, whether $A$ intersects $Y$ or not. We are primarily concerned with the non-adaptive model, where the family of test sets is specified in advance, in the case where each test set is of size at most some given natural number $k$. Our main results are nearly tight bounds on the minimum number of tests necessary when $d$ and $k$ are fixed and $n$ is large enough.