Search

It is well known that Heron’s equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having $$n>4$$ n > 4 edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron’s and Brahmagupta’s equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron’s and Brahmagupta’s results, which consequently represent special cases of the provided general equality.

PurposeThe authors propose a rather elementary method to compute a family of integrals on the half line, involving positive powers of sin x and negative powers of x, depending on the integer parameters n≥q≥1.Design/methodology/approachCombinatorics, sine and cosine integral functions.FindingsThe authors prove an explicit formula to evaluate sinc-type integrals.Originality/valueThe proof is not present in the current literature, and it could be of interest for a large audience.

We study the asymptotic decay as t → + ∞ of integral transforms ∫ 0 + ∞ g ( x ) Φ ( t x ) d x . Examples are the cosine and sine Fourier transforms, the Hankel transforms, and the Laplace transforms. Under appropriate assumptions on the kernels and on the functions involved, we prove that the integral transforms can be controlled by the support, or by the first oscillation of the kernels. We also prove that if f ( x ) and g ( x ) are asymptotic at the origin, then the associated integral transforms are asymptotic at infinity. Finally, we give an asymptotic estimate for the integral transform of h ( x ) ⋅ g ( x ) when h ( x ) is suitably slowly varying.

An elementary proof of a quantitative version of the Riemann-Lebesgue lemma for functions supported on the half line is given. Applications to differential models with memory are discussed.

We present a convergence–divergence test for series of nonnegative terms. Our proof is elementary, and yet we show examples of application to some apparently difficult cases.

The Hilbert transform H can be extended to an isometry of L 2 . We prove this fact working directly on the principal value integral, completely avoiding the use of the Fourier transform and the use of orthogonal systems of functions. Our approach here is a byproduct of our attempts to understand the rearrangement properties of H.

We give an alternative proof of a theorem of Stein and Weiss: The distribution function of the Hilbert transform of a characteristic function of a set E only depends on the Lebesgue measure | E | of such a set. We exploit a rational change of variable of the type used by George Boole in his paper "On the comparison of transcendents, with certain applications to the theory of definite integrals" together with the observation that if two functions f and g have the same Lp norm in a range of exponents p1 < p < p2 then their distribution functions coincide. © 2009 Elsevier Inc. All rights reserved.

We split the classical Hilbert Transform H into the sum of two convolution integrals H (δ) +R (δ) , where the kernel of H (δ) is supported away from the origin in {|t| δ }, while the kernel of R (δ) is supported near the origin in {|t| δ }. We prove that the L p -norm of H (δ) , known in the literature as the Truncated Hilbert Transform, is equal to the norm of H. Namely ||H (δ) ||p,p = cot(π/2p) for 2 p < +∞ ,a nd||H (δ) ||p,p = tan(π/2p) for 1 < p 2. We then prove that the L p -norm of R (δ) is stricly larger. In particular ||R (δ) ||2,2 = 1.17897..., a constant related to the Gibbs phenomenon.

We prove that when a function on the real line is symmetrically rearranged, the distribution function of its uncentered Hardy-Littlewood maximal function increases pointwise, while it remains unchanged only when the function is already symmetric. Equivalently, if ℳ is the maximal operator and script S sign the symmetrization, then script S signℳf(x)≤ℳscript S signf(x) for every x, and equality holds for all x if and only if, up to translations, f(x) = script S sign f(x) almost everywhere. Using these results, we then compute the exact norms of the maximal operator acting on Lorentz and Marcinkiewicz spaces, and we determine extremal functions that realize these norms. © 2008 London Mathematical Society.

Let m be a bounded and monotonic function on R with λ=infm and Λ=supm. We study the best constant in the inequality ‖Tmf‖p≤c‖f‖p for 1

In the early 1930s, Wiener proved that if f ( x ) is a strictly positive periodic function whose Fourier series is absolutely convergent, then the Fourier series of g ( x )=1/ f ( x ) is also absolutely convergent [ 8 , pp. 10–14]. This phenomenon can be easily understood nowadays using Banach algebra techniques (see, for example, [ 4 , pp. 202–203]). In fact, these techniques allow us to study the absolute convergence of g ( x )= F ( f ( x )), where F is holomorphic in an open subset of [Copf ] that contains the range of f ( x ) (for x ∈ℝ). In this context, Wiener's original problem corresponds to the choice F ( z )=1/ z . In this work we want to analyse the constraints on the simultaneous rate of vanishing of the Fourier coefficients fˆ ( n ) and ĝ ( n ) as n →∞. We shall focus on g =1/ f , but we shall also study the general case g = F ( f ). In either case, there are obviously no constraints when f is a constant function. Although this problem does not seem to be directly related to uncertainty inequalities for the Fourier Transform, we observe that there are some analogies, both in the nature of the results and in the proof techniques. The general fact with which we are dealing is that fˆ ( n ) and ĝ ( n ) cannot vanish too quickly at the same time as n →∞, unless f ( x ) is constant. The general fact that underlies uncertainty inequalities is that a non-periodic function ϕ( x ) and its Fourier Transform ϕcirc;( u ) cannot vanish too quickly at the same time as x →∞ and u →∞, unless ϕ( x ) is zero (almost everywhere). For a simple introduction to some aspects of uncertainty inequalities, see [ 5 ]; for a thorough and recent introduction to this vast subject, see [ 3 ].

We study the maximal function Mf(x) = sup |f(x + y, t)| when Ω is a region in the (y,t) Ω upper half space R N+1+ and f(x, t) is the harmonic extension to R+N+1 of a distribution in the Besov space Bαp,q(RN) or in the Triebel‐Lizorkin space Fαp,q(RN). In particular, we prove that when Ω= {|y|N/ (N‐αp) < t < 1} the operator M is bounded from F αp,∞ (RN) into Lp (RN). The admissible regions for the spaces B αp,q (RN) with p < q are more complicated