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We study the compactness of commutator of a locally integrable function and an oscillatory singular integral operator defined by a real-valued polynomial and a kernel function satisfying the Hölder condition. We show that such a commutator is compact from L p {L^{p}} to L q {L^{q}} if and only if the locally integrable function is the sum of a constant and a function in certain Lebesgue space. [ABSTRACT FROM AUTHOR]

Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version. The method demonstrated here has applications to the subconvexity of some of the twisted zeta functions introduced by F. Sato. The argument demonstrates that the symmetric space condition used by Sato is not necessary to estimate the zeta function in the critical strip. [ABSTRACT FROM AUTHOR]

For an integer d ≥ 2 , t ∈ R , and a 0-homogeneous function Φ ∈ C ∞ (R d \ { 0 } , R) , we consider the family of Fourier multiplier operators T Φ t associated with symbols ξ ↦ exp (i t Φ (ξ)) and prove that for a generic phase function Φ , one has the estimate ‖ T Φ t ‖ L p → L p ≳ d , p , Φ ⟨ t ⟩ d | 1 p - 1 2 | . That is the maximal possible order of growth in t → ± ∞ , according to the previous work by V. Kovač and the author and the result shows that the two special examples of functions Φ that induce the maximal growth, given by V. Kovač and the author and independently by D. Stolyarov, to disprove a conjecture of Maz'ya actually exhibit the same general phenomenon. [ABSTRACT FROM AUTHOR]

This paper focuses on constructing an O(1, 3)-invariant oscillatory integral on Minkowski space. We then proceed to calculate the wave front set of this operator. Additionally, we provide a detailed computation of the cut-off functions compactly supported in Minkowski space. [ABSTRACT FROM AUTHOR]

In this paper, we consider uniform estimates for the oscillatory integrals with stationary phase, which were previously studied by Farah–Rousset–Tzvetkov and Alazard–Burq–Zuily. We significantly reduce the order of required regularity on the phase and amplitude functions for the uniform estimate. We also study estimates for the oscillatory integrals of which phase and amplitude functions depend on the oscillation parameter. The novelty of this article lies in the use of the wave packet decomposition, which has been widely utilized in the study of the Fourier restriction problems. This allows one to obtain the decay estimate for the oscillatory integral from disjointness of the spatial supports of wave packets. The latter is geometric in its nature and less sensitive to the smoothness of the phase and amplitude functions. [ABSTRACT FROM AUTHOR]

In this article, we studied the convolution operators M k with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator M k is associated to the characteristic hypersurfaces Σ ⊂ R 3 of a hyperbolic equation and smooth amplitude function, which is homogeneous of the order - k for large values of the argument. We investigated the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point v ∈ Σ at which, exactly one of the principal curvatures of the surface Σ does not vanish. Such surfaces exhibit singularities of the type A in the sense of Arnold’s classification. Denoting by k p the minimal number such that M k is L p ↦ L p ′ -bounded for k > k p , we showed that the number k p depends on some discrete characteristics of the surface Σ . [ABSTRACT FROM AUTHOR]

This paper proposes an original approach to calculating integrals of rapidly oscillating functions, based on Levin's algorithm, which reduces the search for an anti-derivative function to solve an ODE with a complex coefficient. The direct solution of the differential equation is based on the method of integrating factors. The reduction in the original integration problem to a two-stage method for solving ODEs made it possible to overcome the instability that arises in the standard (in the form of solving a system of linear algebraic equations) approach to the solution. And due to the active use of Chebyshev interpolation when using the collocation method on Gauss–Lobatto grids, it is possible to achieve high speed and stability when taking into account a large number of collocation points. The presented spectral method of integrating factors is both flexible and reliable and allows for avoiding the ambiguities that arise when applying the classical method of collocation for the ODE solution (Levin) in the physical space. The new method can serve as a basis for solving ordinary differential equations of the first and second orders when creating high-efficiency software, which is demonstrated by solving several model problems. [ABSTRACT FROM AUTHOR]

Since Varchenko’s seminal paper, the asymptotics of oscillatory integrals and related problems have been elucidated through the Newton polyhedra associated with the phase P. The supports of those integrals are concentrated on sufficiently small neighborhoods. The aim of this paper is to investigate the estimates of sublevel sets and oscillatory integrals whose supports are global domains D. A basic model of D is R d . For this purpose, we define the Newton polyhedra associated with (P, D) and establish analogues of Varchenko’s theorem in global domains D, under nondegeneracy conditions of P. [ABSTRACT FROM AUTHOR]

Our aim is to answer the following question: "Among the Filon-type methods for computing oscillatory integrals, which one is the most efficient in practice?". We first discuss why we should seek the answer among the family of Filon-Clenshaw-Curtis rules. A theoretical analysis accompanied by a set of numerical experiments reveals that the plain Filon-Clenshaw-Curtis rules reach a given accuracy faster than the (adaptive) extended Filon-Clenshaw-Curtis rules. The comparison is based on the CPU run-time for certain wave numbers (medium and large).

In this article, we study convolution operators Mk with oscillatory kernel, which are related to solutions to the Cauchy problem for strictly hyperbolic equations. The operator Mk is associated to the characteristic hypersurface Σ ⊂ R³ of the equation and smooth amplitude function, which is homogeneous of order −k for large values of the argument. We study the convolution operators assuming that the support of the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point v ∈ Σ. Such surfaces is given as the graph of a smooth function 1 + Φ. Denoting by kp the minimal exponent such that Mk is Lp 7→ Lp′ -bounded for any k > kp, we show that the number kp depends on some discrete characteristics of the surface Σ defined by the Newton polyhedron associated to the function Φ. Moreover, we found the sharp value of kp for smooth convex surfaces. [ABSTRACT FROM AUTHOR]

The unscaled small-time asymptotics of the density is approached by means of the Brownian bridge, regarding planar analogues of the Langevin diffusion, which are strictly hypoelliptic. While the non-quadratic case remains open, in the quadratic case a new precision is derived. [ABSTRACT FROM AUTHOR]

This paper proposes an original approach to calculating integrals of rapidly oscillating functions, based on Levin’s algorithm, which reduces the search for an anti-derivative function to solve an ODE with a complex coefficient. The direct solution of the differential equation is based on the method of integrating factors. The reduction in the original integration problem to a two-stage method for solving ODEs made it possible to overcome the instability that arises in the standard (in the form of solving a system of linear algebraic equations) approach to the solution. And due to the active use of Chebyshev interpolation when using the collocation method on Gauss–Lobatto grids, it is possible to achieve high speed and stability when taking into account a large number of collocation points. The presented spectral method of integrating factors is both flexible and reliable and allows for avoiding the ambiguities that arise when applying the classical method of collocation for the ODE solution (Levin) in the physical space. The new method can serve as a basis for solving ordinary differential equations of the first and second orders when creating high-efficiency software, which is demonstrated by solving several model problems.

In this paper, the $L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness of the bilinear oscillatory integral along parabola \begin{equation*} T_\beta(f, g)(x)=p.v.\int_{{\mathbb R}} f(x-t)g(x-t^{2})\,{\rm e}^{i |t|^{\beta}}\,\frac{{\rm d}t}{t} \end{equation*} is set up, where β  > 1 or β  < 0, $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $\frac{1}{2}\lt r\lt\infty$ , p  > 1 and q  > 1. The result for the case β  < 0 extends the $L^\infty\times L^2\to L^2$ boundedness obtained by Fan and Li (D. Fan and X. Li, A bilinear oscillatory integral along parabolas, Positivity 13(2) (2009), 339–366) by confirming an open question raised in it. [ABSTRACT FROM AUTHOR]

Numerical stability of the Filon–Clenshaw–Curtis rules is considered, when applied to oscillatory integrals with the linear oscillator. The following results are proved: (1) the coefficients of the ( N + 1 )-point rule, for any N > 2 , never lie in a right sector of the complex plane; (2) the coefficients of the 2-point rule lie in a right sector only when k ∈ [ d π - 3 π / 4 , d π - π / 4) , for any integer d > 0 large enough; and (3) the coefficients of the 3-point rule lie in a right sector only when k ∈ (d π - π / 2 , d π - π / 4) , for any integer d > 0 large enough. These results imply that the condition numbers associated with the 2-point and the 3-point rules are bounded by π / 2 when k satisfies the aforementioned conditions. Then, we extend the stability intervals for k and show that in the following cases, the FCC rules can be applied in a stable manner: (1) the 2-point rule with k far enough from d π for any integer d > 0 ; (2) the 3-point rule with k ∈ [ d π - π / 2 , d π) far enough from d π ; and (3) the 4-point rule with k ∈ [ d π - π / 2 , d π) far enough from both d π - π / 2 and d π . [ABSTRACT FROM AUTHOR]

In this paper we consider the uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4 in two variables. The obtained estimate is sharp and the result is an analogue of the more general theorem of Karpushkin (Proc I.G.Petrovsky Seminar 9:3–39, 1983) for sufficiently smooth functions, thus, in particular, removing the analyticity assumption. [ABSTRACT FROM AUTHOR]

In this paper, we present some explicit exponents in the estimates for the volumes of sub-level sets of polynomials on bounded sets and applications to the decay of oscillatory integrals and the convergence of singular integrals.

We study the boundedness of the oscillatory integral Tα,βf(x,y)=∫Q2f(x−γ1(t),y−γ2(s))e−2πit−β1s−β2t−α1−1s−α2−1dtds{T}_{\alpha ,\beta }f\left(x,y)=\mathop{\int }\limits_{{Q}^{2}}f\left(x-{\gamma }_{1}\left(t),y-{\gamma }_{2}\left(s)){e}^{-2\pi i{t}^{-{\beta }_{1}}{s}^{-{\beta }_{2}}}{t}^{{-\alpha }_{1}-1}{s}^{-{\alpha }_{2}-1}{\rm{d}}t{\rm{d}}s on Wiener amalgam spaces, where Q2=[0,1]×[0,1]{Q}^{2}=\left[0,1]\times \left[0,1] is the unit square in two dimensions, (x,y)∈Rn×Rm,γ1(t)=(tp1,tp2,…,tpn),γ2(s)=(sq1,sq2,…,sqm)\left(x,y)\in {{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{m},{\gamma }_{1}\left(t)=\left({t}^{{p}_{1}},{t}^{{p}_{2}},\ldots ,{t}^{{p}_{n}}),{\gamma }_{2}\left(s)=\left({s}^{{q}_{1}},{s}^{{q}_{2}},\ldots ,{s}^{{q}_{m}}) are homogeneous curves on Rn{{\mathbb{R}}}^{n} and Rm{{\mathbb{R}}}^{m}.

Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a subconvex estimate for the Shintani \zeta function enumerating class numbers of binary cubic forms. [ABSTRACT FROM AUTHOR]

The modified Filon–Clenshaw–Curtis rules, proposed earlier by the author, are combined with the (double) exponential transformations in such a way that (1) the necessity of computing the inverse of the oscillator function is released, (2) possible inaccuracy due to rounding error, when the amplitude function has endpoint singularities, is treated, (3) difficulties raised by the stationary points of the oscillator function are treated, and (4) all the benefits of the original Filon–Clenshaw–Curtis rules are preserved. We also carry out some numerical experiments, which illustrate the efficiency of the proposed algorithms while earlier methods based on Filon–Clenshaw–Curtis rules result in poor approximations or even fail. [ABSTRACT FROM AUTHOR]

This paper may be viewed as a companion paper to Greenblatt (arXiv:1910.04547). In that paper, L 2 Sobolev estimates derived from a Newton polyhedron-based resolution of singularities method are combined with interpolation arguments to prove L p to L s q estimates, some sharp up to endpoints, for translation invariant Radon transforms over hypersurfaces and related operators. Here q ≥ p and s can be positive, negative, or zero. In this paper, we instead use L 2 Sobolev estimates derived from the resolution of singularities methods of Greenblatt (arXiv:1810.10507) and combine with analogous interpolation arguments, again resulting in L p to L s q estimates for translation invariant Radon transforms for q ≥ p which can be sharp up to endpoints. It will turn out that sometimes the results of this paper are stronger, and sometimes the results of Greenblatt (arXiv:1910.04547) are stronger. As in Greenblatt (submitted), some of the sharp estimates of this paper occur when s = 0 , thereby giving some new sharp L p to L q estimates for such operators, again up to endpoints. Our results lead to natural global analogues whose statements can be recast in terms of a hyperplane integrability condition analogous to that of Iosevich and Sawyer in their work (Adv Math 132(1):46–119, 1997) on the L p boundedness of maximal averages over hypersurfaces. [ABSTRACT FROM AUTHOR]

In this paper, we present an oscillatory version of the celebrated Breuer–Major theorem that is motivated by the random corrector problem. As an application, we are able to prove new results concerning the Gaussian fluctuation of the random corrector. We also provide a variant of this theorem involving homogeneous measures. [ABSTRACT FROM AUTHOR]

We consider an irreducible finite range random walk on the d-dimensional integer lattice and study asymptotic behavior of its transition function p(n; x) close to the boundary of Cramér's zone. [ABSTRACT FROM AUTHOR]

In this paper, the authors investigate the boundedness of the oscillatory singular integrals with variable Calderón-Zygmund kernel on generalized Morrey spaces Mp,φ(Rn) and the vanishing generalized Morrey spaces VMp,φ(Rn). When 1

Let ψ be a positive function defined near the origin such that limt→0+ ψ(t) = 0. We consider the operator T θ f (x) = lim ε → 0 0 ∫ ε 1 e i γ (t) f (x − t) d t t θ ψ (t) 1 − θ , where γ is a real function with limt→0+ |γ(t)| = ∞ and 0 ≤ θ ≤ 1. Assuming certain regularity and growth conditions on ψ and γ, we show that T1 is of weak type (1, 1). [ABSTRACT FROM AUTHOR]

We develop further the notion of a formal oscillatory integral interpreted as a functional on the amplitudes supported near a fixed non-degenerate critical point x 0 of the phase function with zero critical value which depends on a formal parameter. We relate to a formal oscillatory integral two objects, its formal oscillatory integral kernel and the full formal asymptotic expansion at x 0 . This formal asymptotic expansion is a formal distribution supported at x 0 which is applied to the amplitude. We establish a correspondence between the formal oscillatory integral kernels and formal distributions based upon a number of axiomatic properties of a formal integral expressed in terms of its integral kernel. Then we consider a family of multidifferential operators related to a star product with separation of variables on a pseudo-Kähler manifold. We show that these operators evaluated at a point are given by formal oscillatory integrals. We completely identify their formal oscillatory integral kernels. [ABSTRACT FROM AUTHOR]

Abstract This paper studies the existence and scattering of global small amplitude solutions to the nonlinear Boussinesq paradigm equation in R n. Firstly, with a dispersive estimate obtained by oscillatory integral and some properties of Bessel function, the existence of global small amplitude solutions is established by the method of contractive mapping principle. Then, with the help of the decay of solutions with respect to time, we construct the suitable scattering state corresponding to the small amplitude solutions. [ABSTRACT FROM AUTHOR]

In the paper (Greenblatt in J Funct Anal, 10.1016/j.jfa.2018.05.014, 2018) the author proved L p Sobolev regularity results for averaging operators over hypersurfaces and connected them to associated Newton polyhedra. In this paper, we use rather different resolution of singularities techniques along with oscillatory integral methods applied to surface measure Fourier transforms to prove L p Sobolev regularity results for a class of averaging operators over surfaces which can be of any codimension. [ABSTRACT FROM AUTHOR]

In this article, we continue the study of the problem of Lp-boundedness of the maximal operator M associated to averages along isotropic dilates of a given, smooth hypersurface S of finite type in 3-dimensional Euclidean space. An essentially complete answer to this problem was given about eight years ago by the third and fourth authors in joint work with M. Kempe [Acta Math 204 (2010), pp. 151-271] for the case where the height h of the given surface is at least two. In the present article, we turn to the case h < 2. More precisely, in this Part I, we study the case where h < 2, assuming that S is contained in a sufficiently small neighborhood of a given point x0 ∈ S at which both principal curvatures of S vanish. Under these assumptions and a natural transversality assumption, we show that, as in the case h ≥ 2, the critical Lebesgue exponent for the boundedness of M remains to be pc = h, even though the proof of this result turns out to require new methods, some of which are inspired by the more recent work by the third and fourth authors on Fourier restriction to S. Results on the case where h < 2 and exactly one principal curvature of S does not vanish at x0 will appear elsewhere. [ABSTRACT FROM AUTHOR]

In this paper we consider the summation problem for two-dimensional trigonometrical integrals with a third-order polynomial phase. We find the sharp convergence exponent using the estimates for trigonometric integrals. [ABSTRACT FROM AUTHOR]

A novel bidirectional ray-tracing method for solving antenna coupling problems is presented where the transfer function between the transmit and receive antennas is obtained by evaluating surface interaction integrals according to the reciprocity theorem. In order to eliminate the limitations of the reception spheres in the traditional unidirectional shooting and bouncing rays (SBR) method, both the transmitter and receiver are used for ray launching. The resulting rays are captured on an interaction surface, which is much more flexible in terms of size and shape than a reception sphere. The transfer function between two antennas is obtained by evaluating the reciprocity integral on this interaction surface by a specifically designed numerical integration procedure. It is shown that the new method has two advantages compared to the traditional unidirectional ray tracing. First, flexible interaction surfaces help to retain the accuracy in large geometries where the traditional unidirectional SBR fails due to reception spheres being limited in size and being missed by the rays. Second, explicit computation of single and multiple diffraction effects by the uniform theory of diffraction can be avoided and replaced by a physical optics like integration over the reciprocity interaction surface. Numerical results demonstrate the capabilities and advantages of the new approach. [ABSTRACT FROM AUTHOR]

In this work we used radial kernels for computing more generalized fast oscillatory integral equations. The proposed method is based on radial kernels. The present method is efficient for computing oscillatory integral equations with large oscillation parameters. The proposed method is very robust and capable of handling fast oscillatory integral equations. [ABSTRACT FROM AUTHOR]

Invariant estimates of oscillatory integrals with polynomial phase are studied. The main result is a theorem on uniform invariant estimates of trigonometric integrals. The obtained estimates improve Popov’s well-known results on invariant estimates of trigonometric integrals in the case where the phase function is a third-degree polynomial. [ABSTRACT FROM AUTHOR]

Estimates of the Fourier transform of measures concentrated on analytic hypersurfaces containing a damping factor are considered. The solution of the Sogge and Stein problem on the optimal decrease of the Fourier transform of measures with a damping factor for the particular class of analytic surfaces of three-dimensional space is given. [ABSTRACT FROM AUTHOR]

We show that, for odd d, the Ld+22 bounds of Sogge (J Am Math Soc 12:1-31, 1999) and Xi (Trans Am Math Soc 369:6351-6372, 2017) for the Nikodym maximal function over manifolds of constant sectional curvature are unstable with respect to metric perturbation, in the spirit of the work of Minicozzi and Sogge (Math Res Lett 4:221-237, 1997). A direct consequence is the instability of the bounds for the corresponding oscillatory integral operator. Furthermore, we extend our construction to show that the same phenomenon appears for any d-dimensional Riemannian manifold with a local totally geodesic submanifold of dimension ⌈d+12⌉ if d≥3. In contrast, Sogge’s L73 bound for the Nikodym maximal function on 3-dimensional variably curved manifolds is stable with respect to metric perturbation. [ABSTRACT FROM AUTHOR]

We study oscillatory integrals in several variables with analytic, smooth, or Ck phases satisfying a nondegeneracy condition attributed to Varchenko. With only real analytic methods, Varchenko’s sharp estimates are rediscovered and generalized. The same methods are pushed further to obtain full asymptotic expansions of such integrals with analytic and smooth phases, and finite expansions with error assuming the phase is only Ck. The Newton polyhedron appears naturally in the estimates; in particular, we show precisely how the exponents appearing in the asymptotic expansions depend only on the geometry of the Newton polyhedron of the phase. All estimates proven hold for oscillatory parameter real and nonzero, not just asymptotically. [ABSTRACT FROM AUTHOR]

Abstract: With the help of Van der Corput lemmas, decay estimates are proven for Fourier transforms of mixed homogeneous hypersurface measures with densities that can be quite irregular. The primary results are local in nature, but can be extended to global theorems in an appropriate sense. The estimates are sharp for a certain range of indices in the theorems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study time‐asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of phases which may have a stationary point of real order and amplitudes allowed to have an integrable singular point. The resulting estimates provide optimal decay rates which show explicitly the influence of these two particular points. Then we apply these abstract results to solution formulas of a class of dispersive equations on the line defined by Fourier multipliers. Under the hypothesis that the Fourier transform of the initial data has a compact support or an integrable singular point, we derive uniform estimates of the solutions in space‐time cones, describing their motions when the time tends to infinity. The method permits also to show that symbols having a restricted growth at infinity may influence the dispersion of the solutions: we prove the existence of a cone, depending only on the symbol, in which the solution is time‐asymptotically localized. This corresponds to an asymptotic version of the notion of causality for initial data without compact support. [ABSTRACT FROM AUTHOR]

In this paper we study singular oscillatory integrals with a nonlinear phase function. We prove estimates of $$L^2 \rightarrow L^2$$ and $$L^p\rightarrow L^p$$ type. [ABSTRACT FROM AUTHOR]

It is proved that the local smoothing conjecture for wave equations implies certain improvements on Stein's analytic family of maximal spherical means. Some related problems are also discussed. [ABSTRACT FROM AUTHOR]

In this note we study the relation between oscillatory integral operators and convolution operators, and also the sharpness of Lp-estimates for singular oscillatory integral operators. [ABSTRACT FROM AUTHOR]

In this paper, we propose an adaptive Filon-type method to approximate oscillatory integrals. By using some S-shaped functions connecting Gauss-Legendre nodes and Filon-type method, the adaptive schemes behave well for small as well as for large frequencies. Adding frequency-dependent nodes can improve the asymptotic order at the same time. Moreover, there exist some complex nodes leading to a higher asymptotic order method approximating to the integral, and the special nodes work in other methods as well. The error can be further decreased by taking extra Chebyshev nodes. The efficiency and accuracy are tested by some experiments. [ABSTRACT FROM AUTHOR]

Green's function of the problem describing steady forward motion of bodies in an open ocean in the framework of the linear surface wave theory (the function is often referred to as Kelvin's wave source potential) is considered. Methods for numerical evaluation of the so-called ‘single integral’ (or, in other words, ‘wavelike’) term, dominating in the representation of Green's function in the far field, are developed. The difficulty in the numerical evaluation is due to integration over infinite interval of the function containing two differently oscillating factors and the presence of stationary points. This work suggests two methods to approximate the integral. First of them is based on the idea put forward by D. Levin in 1982 — evaluation of the integral is converted to finding a particular slowly oscillating solution of an ordinary differential equation. To overcome well-known numerical instability of Levin's collocation method, an alternative type of collocation is used; it is based on a barycentric Lagrange interpolation with a clustered set of nodes. The second method for evaluation of the wavelike term involves application of the steepest descent method and Clenshaw–Curtis quadrature. The methods are numerically tested and compared. [ABSTRACT FROM AUTHOR]

For indefinite integrals Q ( f ; x , ω ) = ∫ − 1 x f ( t ) e i ω t d t ( x ∈ [ − 1 , 1 ] ) Torii and the first author (Hasegawa and Torii, 1987) developed a quadrature method of Clenshaw–Curtis (C–C) type. Its improvement was made and combined with Sidi’s m W -transformation by Sidi and the first author (Hasegawa and Sidi, 1996) to compute infinite oscillatory integrals. The improved method per se, however, has not been elucidated in its attractive features, which here we reveal with new results and its detailed algorithm. A comparison with a method of C–C type for definite integrals Q ( f ; 1 , ω ) due to Domínguez et al. (2011) suggests that a smaller number of computations is required in our method. This is achieved by exploiting recurrence and normalization relations and their associated linear system. We show their convergence and stability properties and give a verified truncation error bound for a result computed from the linear system with finite dimension. For f ( z ) analytic on and inside an ellipse in the complex plane z the error of the approximation to Q ( f ; x , ω ) of the improved method is shown to be bounded uniformly. Numerical examples illustrate the stability and performance of the method. [ABSTRACT FROM AUTHOR]

We consider the oscillatory hyper Hilbert transform H f( x) = ∫ f( x - Γ( t))etd t; where Γ( t) = ( t, γ(t)) in ℝ is a general curve. When γ is convex, we give a simple condition on γ such that H is bounded on L when β > 3 α, β > 0: As a corollary, under this condition, we obtain the L -boundedness of H when 2 β/(2 β - 3 α) < p < 2 β/(3 α). When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that H is bounded on L : As an application, we construct a class of strictly convex curves along which H is bounded on L only if β > 2 α > 0. [ABSTRACT FROM AUTHOR]

Adaptive extended Filon–Clenshaw–Curtis rules are introduced for computing oscillatory integrals, and efficient algorithms are developed for their construction. Also, an error estimate has been obtained that is explicit in the wave number and the rule parameters. In addition, we rigorously prove that an adaptive extended Filon–Clenshaw–Curtis rule preserves the asymptotic order of the corresponding extended Filon–Clenshaw–Curtis rule. Meanwhile, the connection between these rules and the Filon–Clenshaw–Curtis rules is declared. The connection enables one to construct an adaptive extended Filon–Clenshaw–Curtis rule from the corresponding Filon–Clenshaw–Curtis rule naturally. Also, we estimate complexity of the proposed construction algorithms asymptotically. In some cases of the proposed construction algorithms, one encounters bad-conditioned dense linear systems. We carry out some numerical experiments, which suggest efficient tools for achieving reliable solutions of the systems. In our numerical experiments, the adaptive extended Filon–Clenshaw–Curtis rules reveal themselves more efficient than the corresponding extended Filon–Clenshaw–Curtis rules. [ABSTRACT FROM AUTHOR]

An efficient algorithm for constructing extended Filon-Clenshaw-Curtis rules is developed. One important benefit of the proposed algorithm is that it reveals an explicit relation of the extended Filon-Clenshaw-Curtis rules and the corresponding plain Filon-Clenshaw-Curtis rules. This relation enables one to extend a Filon-Clenshaw-Curtis rule to the corresponding extended Filon-Clenshaw-Curtis rules naturally. [ABSTRACT FROM AUTHOR]