Bounds for modified bessel functions

Two-side inequalities for the modified Bessel functionI _v(x), K_v(x) of the first and third kind and of order v, are established. The chief tool is the monotonocity of the functionsI _v+1(x)/I _v(x),K _v+1(x)/K _v(x).     

On the power of the zeros of Bessel functions

For ν≥0 let c_νk be the k-th positive zero of the cylinder functionC _v(t)=J _v(t)cosα-Y _v(t)sinα, 0≤α>π whereJ _ν(t) andY _ν(t) denote the Bessel functions of the first and the second kind, respectively. We prove thatC _v,k ^1+H(x) is convex as a function of ν, ifc _νk≥x>0 and ν≥0, whereH(x) is specified in Theorem 1.1.     

On the zeros of derivatives of Bessel functions

In this paper we are interested in the behaviour respect tov of thekth positive zeroc′ _vk of the derivative of the general Bessel functionC _v(x)=J _v(x)cosα?Y _v(x)sinα, 0≤α<π, whereJ _v(x) andY _v(x) indicate the Bessel functions of first and second kind respectively. It is well known that forc′ _vk>∥v∥,c′ _vk increases asv increases. Here we prove several additional properties forc′ _vk. Our main result is thatc′ _vk is concave as a function ofv, whenc′ _vk>∥v∥>0. This implies the concavity ofc′ _vk for everyk=2,3, ?. In the case of the zerosJ′ _vk of _d ^dx J _v(x) we extend this property tok=1 for everyv≥0.     

A convexity property of zeros of Bessel functions

Fork=1, 2,... letj _vk andc _vk be thek-th positive zeros of the Bessel function $$C_v \left( x \right) = C_v \left( {\alpha ;x} \right) = J_v \left( x \right)\cos \alpha - Y_v \left( x \right)\sin \alpha , 0 \leqslant \alpha< \pi$$ whereY _v (X) is the Bessel function of the second kind. Using the notationj _vκ =C _vk withκ=k?α/π introduced in [3] we show that the functionj _vκ +f(v) is convex with respect toυ≥0 forκ≥0.7070..., wheref(υ) is defined in the theorem of section 2. As an application we find the inequality 0 >j _0κ +j _1κ ? 2κπ > log 8/9, where κ≥0.7070....     

Funktionalgleichungen und Charakterisierungen für die Riemannsche Funktion

Riemann's functionR=Σ _v=1 ^∞ v ^?2 sin(2πv ² x) satisfies the following infinite system of functional equations: (*) $$\sum\limits_{k = 0}^{n - 1} {R\left( {\frac{{x + k}}{n}} \right) = \frac{1}{q}R(qx)} $$      

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.     

An upper bound for the zeros of the derivative of Bessel functions

Letj _vk ^′ denotes thekth positive zero of the derivativeJ _v ^′ (x)=dJ _v (x)/dx of Bessel functionJ _v (x) fork=1, 2,…. We establish the upper bound

$$j'_{\nu k}< \nu + a_k \left( {\nu + \frac{{{\rm A}_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} + \frac{3}{{10}}a_k^2 \left( {\nu + \frac{{A_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} , \nu \geqslant 0, k = 1,2, \ldots $$     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Very slowly varying functions

A real-valued functionf of a real variable is said to be?-slowly varying (?-s.v.) if lim _x→∞ ?(x) [f(x+α)?f(x)]=0 for each α. It is said to be uniformly?-slowly varying (u.?-s.v.) if lim _x→∞ sup _{α ∈ I} ?(x) |f(x+α)?f(x)|=0 for every bounded intervalI. It is supposed throughout that ? is positive and increasing. It is proved that if? increases rapidly enough, then every?-s.v. functionf must be u.?-s.v. and must tend to a limit at ∞. Regardless of the rate of increase of?, a measurable functionf must be u.?-s.v. if it is?-s.v. Examples of pairs (?,f) are given that illustrate the necessity for the requirements on? andf in these results.     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.     

Characterization of measures by potentials

Let μ be a measure in a Banach spaceE, f be an even function onR. We consider the potentialg(a)=f _E f(‖x?a‖)dμ(x). The question is as follows: For whichf does the potentialg determine μ uniquely? In this article we give answers in the cases whereE=l _∞ ⁿ and wheref(t)=|t| ^p andE is a finite dimensional Banach space with symmetric analytic norm. Calculating the Fourier transform of the functionf(‖x‖ _∞) we give a new proof of the J. Misiewicz's result that the functionf(‖x‖ _∞) is positive definite only iff is a constant function.     

Method for Evaluation of Zeros of Bessel Functions

The paper presents a method for finding any of the positivezeros of Bessel function J_v(x),for v 0. The zeros are computedby the process of bisection as eigenvalues of the infinite matrixobtained from the recurrence relations among the Bessel functions.     

Elementary approximations for zeros of Bessel functions

Let j_vk, y_vk and c_vk denote the kth positive zeros of the Bessel functions J_v(x), Y_v(x) and of the general cylinder function C_v(x) = cos αJ_v(x)?sin αY_v(x), 0 ? α < π, respectively. In this paper we extend to c_vk, k = 2, 3,..., some linear inequalities presently known only for j_vk. In the case of the zeros y_vk we are able to extend these inequalities also to k = 1. Finally in the case of the first positive zero j_v1 we compare the linear enequalities given in [9] with some other known inequalities.     

Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains

LetD be a domain inR ² with the Green functionG(x, y) for the Laplace equation. We give a generalization of the Cranston-McConnell inequality concerning the integrability of positive harmonic functions onD. A typical new inequality is {fx137-01} whereu andv ₁,..., v_nare positive superharmonic functions onD andc _nis a constant depending only onn. The generalized Cranston-McConnell inequality is used for the determination of the Martin boundary of a certain unbounded domain.     

Spectral functions of zeros in the Besselq-functions

Zeta functions _v(z; q)= _n=1 [j_vn(q)]^–z and partition functions Z_v(t; q)=_n exp[–tj _vn ² (q)] related to the zeros j_vn(q) of the Bessel q-functions J_v(x; q) and J _v ⁽²⁾ (x; q) are studied and explicit formulas for _v(2n; q) at n=±1, ±2, ... are obtained. The poles of _v(z; q) in the complex plane and the corresponding residues are found. Asymptotics of the partition functions Z_v(t; q) at t 0 are investigated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 397–414, June, 1996.     

Symbolic and numerical computation on Bessel functions of complex argument and large magnitude

The Lanczos τ-method, with perturbations proportional to Faber polynomials, is employed to approximate the Bessel functions of the first kind J_v(z) and the second kind Y_v(z), the Hankel functions of the first kind H_v⁽¹⁾(z) and the second kind H_v⁽²⁾(z) of integer order v for specific outer regions of the complex plane, i.e. ¦z¦ ⩾ R for some R. The scaled symbolic representation of the Faber polynomials and the appropriate automated τ-method approximation are introduced. Both symbolic and numerical computation are discussed. In addition, numerical experiments are employed to test the proposed τ-method. Computed accuracy for J₀(z) and Y₀(z) for ¦z¦ ⩾ 8 are presented. The results are compared with those obtained from the truncated Chebyshev series approximations and with those of the τ-method approximations on the inner disk ¦z¦ ⩽ 8. Some concluding remarks and suggestions on future research are given.     

High-precision evaluation of the regular and irregular Coulomb wavefunctions

Steed's method for the calculation of both the regular and irregular Coulomb functions for positive energy, F_λ(η,x),G_λ(η,x), and their derivatives, is extended into the region of very high precision (～ 30S). Other methods in general use result in less than one half of this precision. The test-case results of Strecok and Gregory for G₀(η,x) and G′₀(η,x) to 22S over a restricted range of parameter values close to the transition line are almost completely verified. Limiting forms of the functions are given for x < x_TP (the turning point) and an heuristic estimate of the errors in the functions is obtained. The method is valid for real η, including η = 0 (i.e. the spherical Bessel functions), and for real λ > ?1; thus cylindrical Bessel functions, Airy functions, and even the real Gamma function can also be obtained to high accuracy. For each of these, in the oscillating region (where appropriate) results are available to within about 2S of the machine accuracy; in the monotonic region the loss of accuracy is quantitatively predictable.     

Multi-Set designs and numbers of common triples

A two-fold multi-set triple system is a collection of triples chosen, possibly with repetitions, from av-set in such a way that each pair (whether distinct or not) occurs precisely two times. For instance the triplex, x, y (which we shall write asxxy) contains the pairxx once and the pairxy twice. Such a design has v(v + l)/3 blocks, and a necessary and sufficient condition for existence is thatv = 0 or 2 (mod 3) and y≥5. The design is called simple if it contains no repeated blocks. Let I₂(v) denote the set of non-negative integerss such that there exist two simple two-fold multi-set triple systems of orderv with preciselys blocks in common. In this paper we determine I₂(v) for all admissiblev.     

New Effects of an Old Z. Zalcwasser's Theorem

It is well known that the strong (C, 1)-summability of an orthogonal series does not imply its very strong (C, 1)-summability, generally. For a given index-sequence {v _n }, first, Z. Zalcwasser gave an interesting condition implying the strong (C, 1)-summability of these partial sums s _{v n} (x). We show that Zalcwasser's condition on {v _n } holds if and only if the subsequence {v ₂ n} is quasi geometrically increasing. Utilizing this fact and known theorems several strong summability results are presented for given index-sequences {v _n }.