On product representations of powers, I

The solvability of the equation a₁a₂ … a_k = x², a₁, a₂, …, a_k ε is studied for fixed k and ‘dense’ sets of positive integers. In particular, it is shown that if k is even and k 4, and is of positive upper density, then this equation can be solved.     

Moment integrals of powers of airy functions

Difference equations for moment integrals which contain powers of the Airy functionAi(z) and its derivativeAi'(z) i.e. $$\int_0^\infty {z^n [Ai(z)]^\alpha dz,} \int_0^\infty {z^n [Ai'(z)]^\alpha dz,} $$ are obtained. These equations are easily solved for the special case of integer α. The moments occur in recent analytical attempts at solving certain nonlinear problems in Physics and Chemistry.     

On some integrals over the product of three Legendre functions

The definite integrals \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2}[P_{\nu}(x)]^{3}\, \mathrm{d}x\) , \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2} [P_{\nu}(x)]^{2}P_{\nu}(-x)\, \mathrm{d}x\) , \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2}[P_{\nu+1}(x)]^{3}\,\mathrm{d}x\) , and \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2} [P_{\nu+1}(x)]^{2}P_{\nu +1}(-x)\,\mathrm{d}x \) are evaluated in closed form, where P _ν is the Legendre function of degree ν, and \(\operatorname{Re}\nu>-1\) . Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.     

On the minimum problem for nonconvex, multiple integrals of product type

We consider the problem of minimizing multiple integrals of product type, i.e. where is a bounded, open set in , is a possibly nonconvex, lower semicontinuous function with p-growth at infinity for some and the boundary datum is in (or simply in if ). Assuming that the convex envelope off is affine on each connected component of the set , we prove attainment for () for every continuous, positively bounded below function g such that (i) every point is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that the class of coefficents g that yield existence to () is dense in the space of continuous, positive functions on . We present examples which show that these conditions for attainment are essentially sharp. Received April 12, 2000 / Accepted May 9, 2000 / Published online November 9, 2000     

Numerical approximation of product integrals

A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, I_m(x, y) denotes

$\text{∫}\text{χ}\text{y}\text{∫}\text{χ}\text{v}{}_2\text{?}\text{∫}\text{χ}\text{v}{}_2\text{∏}\text{i=1}\text{m}\text{F(ν}{}_{\mathrm{i}}\text{)dν}{}_1\text{dν}{}_2\text{?dν}{}_{\mathrm{m}}$

, and A_m(x, y) denotes an approximation of I_m(x, y) of the form

$\text{(y?x)}{}^{\mathrm{m}}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{∏}\text{i=1}\text{m}\text{F(χ}{}_{\mathrm{ik}}\text{)}$

, where a_k and y_ik are fixed numbers for i = 1, 2,…, m and k = 1, 2,…, N and x_ik = x + (y ? x)y_ik. The following result is established. If p is a positive integer, F is a function from the real numbers to the set of w × w matrices with real elements and F⁽¹⁾ exists and is continuous on [a, b], then there exists a bounded interval function H such that, if n, r, and s are positive integers, $\text{(b ? a)}\text{n}\text{= h < 1, x}{}_{\mathrm{i}}\text{= a + hi}\text{for}\text{i = 0, 1,…, n}\text{and}\text{0 < r ? s ? n}$, then

$\text{∏}\text{χ}{}_{\mathrm{r?}}\text{χ}{}_{\mathrm{s}}\text{(I+F dχ)?}\text{∏}\text{i=r}\text{s}\text{I+}\text{∑}\text{j=1}\text{p}\text{I}{}_{\mathrm{j}}\text{(χ}{}_{\mathrm{i?1}}\text{,χ}{}_{\mathrm{i}}\text{)}$

$\text{=h}{}^{\mathrm{p}}\text{H(χ}{}_{\mathrm{r?1}}\text{,χ}{}_{\mathrm{s}}\text{)+O(h}{}^{\mathrm{p+1}}\text{)}$

Further, if F^(j) exists and is continuous on [a, b] for j = 1, 2,…, p + 1 and A is exact for polynomials of degree less than p + 1 ? j for j = 1, 2,…, p, then the preceding result remains valid when A_j is substituted for I_j.     

A constrained minimization problem with integrals on the entire space

In this paper we consider the question of minimizing functionals defined by improper integrals. Our approach is alternative to the method of concentration-compactness and it does not require the verification of strict subaddivity.Dedicated to the memory of Antonio Gilioli (1945–1989)     

On an interpolatory product rule for evaluating Cauchy principal value integrals

Convergence results for interpolatory product rules for evaluating Cauchy principal value integrals of the form f _?1 ¹ v(x)f(x)/x ? λ dx wherev is an admissible weight function have been extended to integrals of the form f _?1 ¹ k(x)f(x)/x ? λ dx wherek is an arbitrary integrable function subject to certain conditions. Further, whereas the above convergence results were shown when the interpolation points were the Gauss points with respect to some admissible weight functionw, they are now shown to hold when the interpolation points are Radau or Lobatto points with respect tow.     

The product of powers in a finitep-group

Archiv der Mathematik -     

Polynomial averages converge to the product of integrals

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges inL ² to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilmanifolds. Dedicated to Hillel Furstenberg upon his retirement The second author was partially supported by NSF grant DMS-0244994.     

Anisotropic singular integrals in product spaces

In this paper, the authors introduce a class of product anisotropic singular integral operators, whose kernels are adapted to the action of a pair A := (A1, A2) of expansive dilations on R n and R m , respectively. This class is a generalization of product singular integrals with convolution kernels introduced in the isotropic setting by Fefferman and Stein. The authors establish the boundedness of these operators in weighted Lebesgue and Hardy spaces with weights in product A∞ Muckenhoupt weights on R n × R m . These results are new even in the unweighted setting for product anisotropic Hardy spaces.     

On solutions in the form of product of powers for second-order differential equations of the Fuchs type

Solutions in the form of the product of powers are found for second-order differential equations of the Fuchs type. Necessary and sufficient conditions are formulated for the existence of such solutions.     

Fractional integrals of periodic functions of several variables

In this paper it has been systematically studied the imbedding properties of fractional integral operators of periodic functions of several variables, and isomorphic properties of fractional integral operators in the spaces of Lipschitz continuous functions. It has also been proved that the space of fractional integration, the space of Lipschitz continuous functions and the Sobolev space are identical in L²-norm. Results obtained here are not true for fractional integrals (or Riesz potentials) in ℝ ⁿ . Supported by NNSFC     

On the first-return integrals

Some pathological properties of the first-return integrals are explored. In particular it is proved that there exist Riemann improper integrable functions which are first-return recoverable almost everywhere, but not first-return integrable, with respect to each trajectory. It is also proved that the usual convergence theorems fail to be true for the first-return integrals.     

On sums of powers

Proceedings - Mathematical Sciences -