Mathematics

We consider the asymptotic expansion of the functional series S μ,γ (a;λ)= ??n=1 ??n γ e ?��?n 2 / a 2 ( n 2 + a 2 ) μ for real values of the parameters γ , λ>0 and μ?? as |a|?��? in the sector |arga|<?/4 . For general values of γ the expansion is of algebraic type with terms involving the Riemann zeta function and a terminating confluent hypergeometric function. Of principal interest in this study is the case corresponding to even integer values of γ , where the algebraic-type expansion consists of a finite number of terms together with a contribution comprising an infinite sequence of increasingly subdominant exponentially small expansions. This situation is analogous to the well-known Poisson-Jacobi formula corresponding to the case μ=γ=0 . Numerical examples are provided to illustrate the accuracy of these expansions.

For a parameter λ>0 , we investigate greedy λ -energy sequences ( a n ) ∞ n=0 on the unit sphere S d ⊂ R d+1 , d≥1 , satisfying the defining property that each a n , n≥1 , is a point where the potential ∑ n−1 k=0 |x− a k | λ attains its maximum value on S d . We show that these sequences satisfy the symmetry property a 2k+1 =− a 2k for every k≥0 . The asymptotic distribution of the sequence undergoes a sharp transition at the value λ=2 , from uniform distribution ( λ<2 ) to concentration on two antipodal points ( λ>2 ). We investigate first-order and second-order asymptotics of the λ -energy of the first N points of the sequence, as well as the asymptotic behavior of the extremal values ∑ n−1 k=0 | a n − a k | λ . The second-order asymptotics is analyzed on the unit circle. It is shown that this asymptotic behavior differs significantly from that of N equally spaced points on the unit circle, and a transition in the behavior takes place at λ=1 .

We investigate the asymptotic expansion of integrals analogous to Ball's integral ????0 ( ?(1+ν)| J ν (x)| (x/2 ) ν ) n dx for large n in which the Bessel function J ν (x) is replaced by the modified Bessel functions I ν (x) and K ν (x) together with appropriate exponential factors e ?�x , respectively. The above integral with J ν (x) replaced by a hyper-Bessel function of the type recently discussed in Aktas {\it et al.} [The Ramanujan J., 2019] and taken over a finite interval determined by the first positive zero of the function is also considered for n?��? . We give the leading asymptotic behaviour of the hyper-Bessel function for x????in an appendix. Numerical examples are given to illustrate the accuracy of the various expansions obtained.

In this paper asymptotic formulas are given for the Lebesgue constants generated by three special approximation processes related to the ℓ 1 -partial sums of Fourier series. In particular, we consider the Lagrange interpolation polynomials based on the Lissajous-Chebyshev node points, the partial sums of the Fourier series generated by the anisotropically dilated rhombus, and the corresponding discrete partial sums.

We study averages along the integers using the divisor function d(n) , and defined as K N f(x)= 1 D(N) ??n?�N d(n)f(x+n), where D(N)= ??N n=1 d(n) . We shall show that these averages satisfy a uniform, scale free ??p -improving estimate for p??1,2) , that is ( 1 N ?�| K N f | p ??) 1/ p ????( 1 N ?�|f | p ) 1/p as long as f is supported on [0,N] . We will also show that the associated maximal function K ??f= sup N | K N f| satisfies (p,p) sparse founds for p??1,2) , which implies that K ??is bounded on ??p (w) for p??1,?? , for all weights w in the Muckenhoupt A p class.

We prove the continuity of the map f↦ M ˜ f from BV(R) to itself, where M ˜ is the uncentered Hardy--Littlewood maximal operator. This answers a question of Carneiro, Madrid and Pierce.

Given n polynomials p 1 ,…, p n of degree at most n with ∥ p i ∥ ∞ ≤1 for i∈[n] , we show there exist signs x 1 ,…, x n ∈{−1,1} so that ∥ ∥ ∑ i=1 n x i p i ∥ ∥ ∞ <30 n − − √ , where ∥p ∥ ∞ := sup |x|≤1 |p(x)| . This result extends the Rudin-Shapiro sequence, which gives an upper bound of O( n − − √ ) for the Chebyshev polynomials T 1 ,…, T n , and can be seen as a polynomial analogue of Spencer's "six standard deviations" theorem.

In this paper, we present a new type of α− Bernstein-Păltănea operators having a better order of approximation than itself. We establish some approximation results concerning the rate of convergence, error estimation and asymptotic formulas for the new modifications. Also, the theoretical results are verified by using MAPLE algorithms.

Let p∈(0,1) , α:=1/p−1 and, for any τ∈[0,∞) , Φ p (τ):=τ/(1+ τ 1−p ) . Let H p ( R n ) , h p ( R n ) and Λ nα ( R n ) be, respectively, the Hardy space, the local Hardy space and the inhomogeneous Lipschitz space on R n . In this article, applying the inhomogeneous renormalization of wavelets, the authors establish a bilinear decomposition for multiplications of elements in h p ( R n ) [or H p ( R n ) ] and Λ nα ( R n ) , and prove that these bilinear decompositions are sharp in some sense. As applications, the authors also obtain some estimates of the product of elements in the local Hardy space h p ( R n ) with p∈(0,1] and its dual space, respectively, with zero ⌊nα⌋ -inhomogeneous curl and zero divergence, where ⌊nα⌋ denotes the largest integer not greater than nα . Moreover, the authors find new structures of h Φ p ( R n ) and H Φ p ( R n ) by showing that h Φ p ( R n )= h 1 ( R n )+ h p ( R n ) and H Φ p ( R n )= H 1 ( R n )+ H p ( R n ) with equivalent quasi-norms, and also prove that the dual spaces of both h Φ p ( R n ) and h p ( R n ) coincide. These results give a complete picture on the multiplication between the local Hardy space and its dual space.

In this paper, we determine the L p (R)× L q (R)→ L r (R) boundedness of the bilinear Hilbert transform H γ (f,g) along a convex curve γ H γ (f,g)(x):=p.v. ∫ ∞ −∞ f(x−t)g(x−γ(t)) dt t , where p , q , and r satisfy 1 p + 1 q = 1 r , and r> 1 2 , p>1 , and q>1 . Moreover, the same L p (R)× L q (R)→ L r (R) boundedness property holds for the corresponding (sub)bilinear maximal function M γ (f,g) along a convex curve γ M γ (f,g)(x):= sup ε>0 1 2ε ∫ ε −ε |f(x−t)g(x−γ(t))|dt.