Almost Sure Convergence of the Numerical Discretization of Stochastic Jump Diffusions

Based on the shuffle product expansion of exponential Lie series in terms of a Philip Hall basis for the stochastic differential equations of jump-diffusion type, we can establish Stratonovich–Taylor–Hall (STH) schemes. However, the STHr scheme converges only at order r in the mean-square sense. In order to have the almost sure Stratonovich–Taylor–Hall (ASTH) schemes, we have to include all the terms related to multiple Poissonian integrals as the moments of multiple Poissonian integrals always have lower orders of magnitudes as compared with those of multiple Brownian integrals.     

Chaos decomposition of multiple fractional integrals and applications

Chaos decomposition of multiple integrals with respect to fractional Brownian motion (with H > 1/2) is given. Conversely the chaos components are expressed in terms of the multiple fractional integrals. Tensor product integrals are introduced and series expansions in those are considered. Strong laws for fractional Brownian motion are proved as an application of multiple fractional integrals. Received: 22 September 1998 / Revised version: 20 April 1999     

The Feynman-Stratonovich semigroup and stratonovich integral expansions in nonlinear filtering

Representations for the solution of the Zakai equation in terms of multiple Stratonovich integrals are derived. A new semigroup (the Feynman-Stratonovich semigroup) associated with the Zakai equation is introduced and using the relationship between multiple Stratonovich integrals and iterated Stratonovich integrals, a representation for the unnormalized conditional density,u(t,x), solely in terms of the initial density and the semigroup, is obtained. In addition, a Fourier seriestype representation foru(t,x) is given, where the coefficients in this representation uniquely solve an infinite system of partial differential equations. This representation is then used to obtain approximations foru(t,x). An explicit error bound for this approximation, which is of the same order as for the case of multiple Wiener integral representations, is obtained. Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL03-92-G0008.     

<Emphasis Type="Italic">p</Emphasis>-Adic limit of the Fourier coefficients of weakly holomorphic modular forms of half integral weight

Serre obtained the p-adic limit of the integral Fourier coefficients of modular forms on SL ₂(ℤ) for p = 2, 3, 5, 7. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on Γ₀(4N) for N = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications to our main result, we obtain congruences on various modular objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space.     

Multiple fourier series and integrals

One gives a brief survey of the investigations on the theory of multiple Fourier series and integrals, reviewed in Referativnyi Zhurnal Matematika in the period 1953–1980. Principal attention is given to the following questions: localization principles, uniform convergence and summability, convergence and summability at a point, in the L_p metric, and almost everywhere, absolute convergence, uniqueness theorems, conjugate Fourier series and integrals, equiconvergence and equisummability of Fourier series and integrals, properties of the kernel and of the Lebesgue constant of summation methods of Fourier series, Fourier coefficients and Fourier transforms.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 19, pp. 3–54, 1982.     

On the absolute convergence of multiple fourier series

Let f: R ^N → C be a periodic function with period 2π in each variable. We prove suffcient conditions for the absolute convergence of the multiple Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to multiple Fourier series. This research was started while the first author was a visiting professor at the Department of Mathematics, Texas A&M University, College Station during the fall semester in 2005; and it was also supported by the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

Fast direct solvers for Poisson equation on 2D polar and spherical geometries

A simple and efficient class of FFT‐based fast direct solvers for Poisson equation on 2D polar and spherical geometries is presented. These solvers rely on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by the second‐ and fourth‐order finite difference discretizations. Using a grid by shifting half mesh away from the origin/poles, and incorporating with the symmetry constraint of Fourier coefficients, the coordinate singularities can be easily handled without pole condition. By manipulating the radial mesh width, three different boundary conditions for polar geometry including Dirichlet, Neumann, and Robin conditions can be treated equally well. The new method only needs O(MN log₂ N) arithmetic operations for M × N grid points. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 56–68, 2002     

On Taylor coefficients of Cauchy-type integrals of finite complex measures

Boundary values of Cauchy-type integrals of finite complex measures given on a unit circle, generally speaking, are not Lebesgue integrable, and therefore at expansion of Cauchy-type integrals in Taylor series, the expansion coefficients cannot be expressed by boundary values using the Lebesgue integral. In this paper, using the notion of A-integration and N-integration, we get a formula for calculating the Taylor expansion coefficients of Cauchy-type integrals of finite complex measures.     

A fast transform for spherical harmonics

Spherical harmonics arise on the sphere S² in the same way that the (Fourier) exponential functions {e^ik}k arise on the circle. Spherical harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform (FFT). Without a fast transform, evaluating (or expanding in) spherical harmonic series on the computer is slow—for large computations probibitively slow. This paper provides a fast transform.For a grid ofO(N²) points on the sphere, a direct calculation has computational complexityO(N⁴), but a simple separation of variables and FFT reduce it toO(N³) time. Here we present algorithms with timesO(N^5/2 log N) andO(N²(log N)²). The problem quickly reduces to the fast application of matrices of associated Legendre functions of certain orders. The essential insight is that although these matrices are dense and oscillatory, locally they can be represented efficiently in trigonometric series.     

On Sets of Unbounded Divergence at Each Point of Multiple Fourier Series of a Function Equal Zero on a Closed Set

The problem investigated is to characterize sets E, the sets of unbounded divergence (at each point) of single and multiple Fourier series under condition of convergence of these series to zero at each point of the complement of E.For any nonempty open set B T ^N = [0, 2] ^N , N 1, a Lebesgue integrable function f ₀ is constructed which equals zero on the set U = T ^N \ B whose multiple trigonometric Fourier series diverges unboundedly (in the case of summation over squares) at each point of the set

Near-best multivariate approximation by Fourier series, Chebyshev series and Chebyshev interpolation

A set of results concerning goodness of approximation and convergence in norm is given for L_∞ and L₁ approximation of multivariate functions on hypercubes. Firstly the trigonometric polynomial formed by taking a partial sum of a multivariate Fourier series and the algebraic polynomials formed either by taking a partial sum of a multivariate Chebyshev series of the first kind or by interpolating at a tensor product of Chebyshev polynomial zeros are all shown to be near-best L_∞ approximations. Secondly the trigonometric and algebraic polynomials formed by taking, respectively, a partial sum of a multivariate Fourier series and a partial sum of a multivariate Chebyshev series of the second kind are both shown to be hear-best L₁ approximations. In all the cases considered, the relative distance of a near-best approximation from a corresponding best approximation is shown to be at most of the order of Π log n_j, where n_j (j = 1, 2,…, N) are the respective degrees of approximation in the N individual variables. Moreover, convergence in the relevant norm is established for all the sequences of near-best approximations under consideration, subject to appropriate restrictions on the function space.     

On the Riemann Summability of Fourier Integrals and Real Hardy Spaces

We consider the Riemann means of single and multiple Fourier integrals of functions belonging to L¹ or the real Hardy spaces defined on ℝⁿ, where n ≥ 1 is an integer. We prove that the maximal Riemann operator is bounded both from H¹(ℝ) into L¹(ℝ) and from L¹(ℝ) into weak –L¹(ℝ). We also prove that the double maximal Riemann operator is bounded from the hybrid Hardy spaces H^(1,0)(ℝIsup2), H^(0,1)(ℝ²⁾ into weak –L¹(ℝ²). Hence pointwise Riemann summability of Fourier integrals of functions in H^(1,0) ∪ H^(0,1)(ℝ²) follows almost everywhere.The maximal conjugate Riemann operators as well as the pointwise convergence of the conjugate Riemann means are also dealt with.     

A weak generalized localization of multiple Fourier series of continuous functions with a certain module of continuity

Let E be an arbitrary measurable set, E ⊂ T ^N = [−π, π)^N, N ≥ 1, μE > 0, and let μ be a measure. In this paper, a weak generalized almost everywhere localization is studied, i.e., for given subsets E ₁ ⊂ E, μE ₁ > 0 we study the almost everywhere convergence of multiple trigonometric Fourier series of functions that are zero on E. We obtain sufficient conditions for the almost everywhere convergence of multiple Fourier series (summable over rectangles) of functions from {ie031-01}, as δ → 0 on E ₁. These conditions are given in terms of the structure and geometry of the sets E ₁ and E and are related to certain orthogonal projections of the sets; they are called the {ie031-02} property of the set E. Previously, one of the authors had introduced the {ie031-03}, k = 1, 2, properties of the set E, which are related to one-dimensional and two-dimensional projections of the sets E and E ₁ respectively, as sufficient conditions for the almost everywhere convergence of Fourier series of functions from L ₁(T ^N ) and L _p (T ^N ), p > 1. The results presented generalize these ideas. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

On the order of summability of the Fourier inversion formula

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.     

The generalized localization for multiple Fourier integrals

In this paper we investigate almost-everywhere convergence properties of the Bochner-Riesz means of N-fold Fourier integrals under summation over domains bounded by the level surfaces of the elliptic polynomials. It is proved that if the order of the Bochner-Riesz means s?(N−1)(1/p−1/2), then the Bochner-Riesz means of a function f∈Lp(RN), 1?p?2 converge to zero almost-everywhere on RN?supp(f).     

Spherical Convergence of Double Fourier Integrals of Functions in Waterman Classes

The aim of the present paper is to obtain new results on the spherical convergence of double Fourier integrals of functions belonging to certain Waterman classes. A two-dimensional Waterman class of functions in L(R ²) is introduced in which the partial spherical Fourier integrals are uniformly bounded and converge at each point of continuity of the function in question, and which class is as large as possible. In addition, a one-dimensional Waterman class is established such that if the mean of a function belongs to this class, than its Fourier integral converges spherically at a given point, and this class is the largest possible in a certain sense.     

On the Pinsky Phenomenon

When n>2 it is well known that the spherical partial sums of n-fold Fourier integrals of a characteristic function of the ball D={x:|x|²<1} do not converge at the origin. In the mathematical literature this result is called “the Pinsky phenomenon”. In 1993 Pinsky established necessary and sufficient conditions for a piecewise smooth function, supported on D, which guarantee the convergence at the origin its spherical partial sums. We prove this result for nonspherical partial sums, i.e. for Fourier integrals under summation over domains bounded by level surfaces of elliptic polynomials.     

Approximation of spherical means of multiple Fourier integrals and Fourier series on set of full measure

In this paper we consider the approximation for functions in some subspaces of L² by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained. Supported by NNSFC.     

On an Integral Equation in the Problem of Diffraction of a Plane Wave on a Transparent Circular Cone

The problem of diffraction on a transparent convex cone is studied. A uniqueness theorem is proved for the case where the cone is illuminated by a compact source. For a circular cone, the solution is obtained in the form of Kontorovich-Lebedev integrals and Fourier series expansions. A singular integral equation is deduced for the Fourier coefficients, and its regularization is carried out. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 101–123.     

The Finite Product Theorem for Certain Epireflections

We discuss a number of topics relating to multiple stochastic integration, where notions and ideas from point process theory seem particularly useful. Thus we give conditions for summability of certain multiple random series in terms of associated Poisson integrals, prove a decoupling result for divergence in probability to infinity, and give conditions for the existence of certain multiple integrals with respect to compensated POISSON and asymmetric LÉVY processes. In particular, the existence criteria for multiple p-stable integrals are shown to be independent of the skewness parameter.