Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

On a linear combination of some expressions in the theory of the univalent functions

LetH(α) denote the class of regular functionsf(z) normalized so thatf(0)=0 andf′(0)=1 and satisfying in the unit discE the condition $$\operatorname{Re} \left\{ {(1 - \alpha )f'(z) + \alpha (1 + zf''(z)/f'(z))} \right\} > 0$$ for fixed α. It is known thatH(0) is a particular class NW of close-to-convex univalent functions. The authors show the following results:Theorem 1. Letf(z)∈H(α). Thenf(z)∈NW if α≤0 andz∈E.Theorem 2. Letf(z)∈NW. Thenf(z)∈H(α) in |z|=r<r _α where i) \(r_\alpha = (1 + \sqrt {2\alpha } )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}\) , α≥0 and ii) \(r_\alpha = \sqrt {\frac{{1 - \alpha - \sqrt {\alpha (\alpha - 1)} }}{{1 - \alpha }}}\) , α<0. All results are sharp.Theorem 3. Iff(z)=z+a ₂ z ²+a ₃ z ³+... is inH(α) and if μ is an arbitrary complex number, then $$\left| {1 + \alpha } \right|\left| {a_3 - \mu a_2^2 } \right| \leqslant ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})\max \left[ {1,\left| {1 + 2\alpha - {3 \mathord{\left/ {\vphantom {3 {2\mu }}} \right. \kern-\nulldelimiterspace} {2\mu }}(1 + \alpha )} \right|} \right].$$ .     

Regularity property of Donsker's delta function

LetL be the space of rapidly decreasing smooth functions on ? andL ^* its dual space. Let (L ²)⁺ and (L ²)^? be the spaces of test Brownian functionals and generalized Brownian functionals, respectively, on the white noise spaceL ^* with standard Gaussian measure. The Donsker delta functionδ(B(t)?x) is in (L ²)^? and admits the series representation $$\delta (B(t) - x) = (2\pi t)^{ - 1/2} \exp ( - x^2 /2t)\sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} H_n (x/\sqrt {2t} )} \times H_n (B(t)/\sqrt {2t} )$$ , whereH _n is the Hermite polynomial of degreen. It is shown that forφ in (L ²)⁺,g _t,φ(x)≡〈δ(B(t)?x), φ〉 is inL and the linear map takingφ intog _t,φ is continuous from (L ²)⁺ intoL. This implies that forf inL ^* is a generalized Brownian functional and admits the series representation $$f(B(t)) = (2\pi t)^{ - 1/2} \sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} \langle f,\xi _{n, t} \rangle } H_n (B(t)/\sqrt {2t} )$$ , whereξ _n,t is the Hermite function of degreen with parametert. This series representation is used to prove the Ito lemma forf inL ^*, $$f(B(t)) = f(B(u)) + \int_u^t {\partial _s^ * } f'(B(s)) ds + (1/2)\int_u^t {f''} (B(s)) ds$$ , where? _s ^* is the adjoint of \(\dot B(s)\) -differentiation operator? _s .     

On the gaps of the Markoff spectrum

Letf be a reduced real indefinite binary quadratic form, having an arithmetic minimum that is the inverse of a left endpoint of a gap of the Markoff spectrum. Then the roots off are eventually periodic continued fractions, provided that $$\min \{ |f(p,q)|:p^2 + q^2 > 0,p,q \operatorname{int} egers\} $$ exists.     

Estimates of maximal functions measuring local smoothness

Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(I ⁿ ) (I≡[0,1]. Set $${\mathcal{N}}_\eta f(x) = \sup \frac{1}{{\left| Q \right|\eta (\left| Q \right|^{1/n} )}} \smallint _Q \left| {f(t) - f(x)} \right|dt,$$ , where the supremum is taken over all cubes containing the pointx. Forη=t ^α (0<α≤1) this definition was given by A.Calderón. In the paper we prove estimates of the maximal functions ${\mathcal{N}}_\eta f$ , along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if $$1 \leqslant p< q< \infty , \theta \equiv n(1/p - 1/q)< 1, and \eta (t) \leqslant t^\theta \sigma (t),$$ , then $$\parallel {\mathcal{N}}_\sigma {f} {\parallel_{q,p}} \leqslant c \parallel {\mathcal{N}}_\eta {f} {\parallel_p} .$$ . Furthermore, we obtain estimates of ${\mathcal{N}}_\eta f$ in terms of theL ^p -modulus of continuity off. We find sharp conditions for ${\mathcal{N}}_\eta f$ to belong toL ^p (I ⁿ ) and the Orlicz class?(L), too.     

Characterization of unbounded spectral operators with spectrum in a half-line

LetT be a possibly unbounded linear operator in the Banach spaceX such thatR(t)=(t+T)^?1 is defined onR ⁺. LetS=TR(I?TR) and letB(.,.) denote the Beta function. Theorem 1.1.T is a scalar-type spectral operator with spectrum in [0, ∞) if and only if $$sup\left\{ {B\left( {k,k} \right)^{ - 1} \int_0^\infty {\left| {x*S^k \left( t \right)x} \right|{{dt} \mathord{\left/ {\vphantom {{dt} t}} \right. \kern-\nulldelimiterspace} t};\left\| x \right\| \leqslant 1,} \left\| {x*} \right\| \leqslant 1,k \geqslant 1} \right\}< \infty .$$ A “local” version of this result is formulated in Theorem 2.2.     

Rows and diagonals of the Walsh array for entire functions with smooth Maclaurin series coefficients

Let \(f(z): = \sum\nolimits_{j = 0}^\infty {a_j z^J } \) be entire, witha _j≠0,j large enough, \(\lim _{J \to \infty } a_{j + 1} /a_J = 0\) , and, for someq∈C, \(q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q\) asj→∞. LetE _mn(f; r) denote the error in best rational approximation off in the uniform norm on |z‖≤r, by rational functions of type (m, n). We study the behavior ofE _mn(f; r) asm and/orn→∞. For example, whenq above is not a root of unity, or whenq is a root of unity, butq _m has a certain asymptotic expansion asm→∞, then we show that, for each fixed positive integern, ,m→∞. In particular, this applies to the Mittag-Leffler functions \(f(z): = \sum\nolimits_{j = 0}^\infty {z^j /\Gamma (1 + j/\lambda )} \) and to \(f(z): = \sum\nolimits_{j = 0}^\infty {z^j /(j!)^{I/\lambda } } \) , λ>0. When |q‖<1, we also handle the diagonal case, showing, for example, that ,n→∞. Under mild additional conditions, we show that we can replace 1+0(1) ⁿ by 1+0(1). In all cases we show that the poles of the best approximants approach ∞ asm→∞.     

Distribution of points for convergent interpolatory integration rules on (−∞, ∞)

Letw be a “nice” positive weight function on (?∞, ∞), such asw(x)=exp(??x?^α) α>1. Suppose that, forn≥1, $$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} } f(x_{jn} )$$ is aninterpolatory integration rule for the weightw: that is for polynomialsP of degree ≤n-1, $$I_n [P]: = \int\limits_{ - \infty }^\infty {P(x)w(x)dx.} $$ Moreover, suppose that the sequence of rules {I _n} _n=1 ^t8 isconvergent: $$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - \infty }^\infty {f(x)w(x)dx} $$ for all continuousf:R→R satisfying suitable integrability conditions. What then can we say about thedistribution of the points {x _jn} _j=1 ⁿ ,n≥1? Roughly speaking, the conclusion of this paper is thathalf the points are distributed like zeros of orthogonal polynomials forw, and half may bearbitrarily distributed. Thus half the points haveNevai-Ullmann distribution of order α, and the rest are arbitrarily distributed. We also describe the possible distributions of the integration points, when the ruleI _n has precision other thann-1.     

Weighted approximation of entire functions of exponential type of several variables by trigonometric polynomials

Letf be an entire function (in Cⁿ) of exponential type for whichf(x)=0(?(x)) on the real subspace \(\mathbb{R}^w (\phi \geqslant 1,{\mathbf{ }}\mathop {\lim }\limits_{\left| x \right| \to \infty } \phi (x) = \infty )\) and ?δ>0?C_δ>0 $$\left| {f(z)} \right| \leqslant C_\delta \exp \left\{ {h_s (y) + S\left| z \right|} \right\},z = x + iy$$ where h, (x)=sup〈3, x〉, S being a convex set in ?ⁿ. Then for any ?, ?>0, the functionf can be approximated with any degree of accuracy in the form p→ \(\mathop {\sup }\limits_{x \in \mathbb{R}^w } \frac{{\left| {P(x)} \right|}}{{\varphi (x)}}\) by linear combinations of functions x→expi〈λx〉 with frequenciesX belonging to an ?-neighborhood of the set S.     

On the order of the error function of the square-full integers,II

LetL(x) denote the number of square-full integers not exceedingx. It is well-known that $$L\left( x \right) \sim \frac{{\zeta \left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}{{\zeta \left( 3 \right)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta \left( {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \right)}}{{\zeta \left( 2 \right)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$ whereζ(s) denotes the Riemann Zeta function, LetΔ(x) denote the error function in the asymptotic formula forL(x). On the assumption of the Riemann hypothesis (R.H.), it is known that $$\Delta x = O\left( {x^{13/81 + 8} } \right)$$ for everyε > 0. In this paper, we prove on the assumption of R.H. that $$\frac{1}{x}\int\limits_x^1 {\left| {\Delta \left( t \right)} \right|dt = O\left( {x^{1/10 + ^8 } } \right)} .$$ In fact, we prove a more general result. We conjecture that $$\Delta x = O\left( {x^{1/10 + ^8 } } \right)$$ under the assumption of the R.H.     

Distribution of integral functionals of a Brownian motion process

In this paper one considers methods which enable one to determine the distribution of certain functionals of a Brownian motion process. Among such functionals we have: the positive continuous additive functional of a Brownian motion, defined by the formula $$A\left( t \right) = \int\limits_{ - \infty }^\infty {\hat t\left( {t, y} \right)dF\left( y \right),} $$ where \(\hat t\left( {t, y} \right)\) is the Brownian local time process while F(y) is a monotonically increasing right continuous function; the functional $$B\left( t \right) = \mathop {\mathop \smallint \limits_{ - \infty } }\nolimits^\infty f\left( {y,\hat t\left( {t, y} \right)} \right)dy,$$ where f(y, x) is a continuous function; and the functional $$C\left( t \right) = \mathop {\mathop \smallint \limits_0 }\nolimits^t f\left( {w\left( s \right),\hat t\left( {sr} \right)} \right)ds$$ As an application of these methods one considers some concrete functionals such that \(\hat t^{ - 1} \left( z \right) = \min \left\{ {s:\hat t\left( {s, o} \right) = z} \right\},\mathop {\mathop \smallint \limits_{ - \infty } }\nolimits^\infty \hat t^2 \left( {t, y} \right)dy,\mathop {\sup }\limits_{y \in R^1 } \hat t\left( {T, y} \right)\) , where T is an exponential random time, independent of \(\hat t\left( {t, y} \right)\) .     

On integral graphs which belong to the class\overline {\alpha K_a \cup \beta K_b }

LetG be a simple graph and let $\bar G$ denotes its complement. We say thatG is integral if its spectrum consists entirely of integers. If $\overline {\alpha K_a \cup \beta K_b } $ is integral we show that it belongs to the class of integral graphs $$\overline {[\frac{{kt}}{\tau }x_o + \frac{{mt}}{\tau }z]K(t + \ell n)k + \ell m \cup [\frac{{kt}}{\tau }y_o + \frac{{(t + \ell n)k + \ell m}}{\tau }z]nK\ell m,} $$ where (i) t, k, l, m, n ∈ ? such that (m, n) =1, (n, t) =1 and (l, t)=1; (ii) τ=((t+ln)k+lm, mt) such that τ| kt; (iii) (x₀, y₀) is aparticular solution of the linear Diophantine equation ((t+ln)k+lm)x-(mt)y=τ and (iv) z≥z₀ where z₀ is the least integer such that $(\frac{{kt}}{\tau }x_0 + \frac{{mt}}{\tau }z_0 ) \geqslant 1$ and $(\frac{{kt}}{\tau }y_0 + \frac{{(t + \ell n)k + \ell m}}{\tau }z_0 ) \geqslant 1$ .     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

A reciprocity formula for quadratic forms

LetQ(x) denote a quadratic form over the rational integers in four variables (x=(x₁,...,x₄)). ThenQ is representable as a symmetric matrix. Assume this matrix to be non-singular modp(p≠2 prime); then the “inverse” quadratic formQ ^?1 modp can be defined. Letf:?⁴→? be defined such that the Fourier transformf exists and the sum $$\sum\limits_{x \in \mathbb{Z}^4 } {f(c x), c \in \mathbb{R}, c \ne 0} $$ is convergent. Furthermore, letm=p ₁...p _k be the product ofk distinct primes withm>1, 2×m; let $$\varepsilon = \prod\limits_{i = 1}^k {\left( {\frac{{\det Q}}{{p_i }}} \right)} \ne 0$$ for the Legendre symbol $$\left( {\frac{ \cdot }{p}} \right)$$ ; define $$B_i (Q,x) = \left\{ {\begin{array}{*{20}c} {1 for Q(x) \equiv 0\bmod p_i } \\ , \\ {0 for Q(x)\not \equiv 0\bmod p_i } \\ \end{array} } \right.$$ and forr∈?,r>0, $$F(Q,f,r) = \sum\limits_{x \in \mathbb{Z}^4 } {\left( {\prod\limits_{i = 1}^k {\left( {B_i (Q,x) - \frac{1}{{p_i }}} \right)} } \right)f(r^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} x)} $$ Then we have $$F(Q,f,m) = \varepsilon F(Q^{ - 1} ,\hat f,m)$$      

Über arithmetische Eigenschaften von Lückenreihen mit algebraischen Koeffizienten und algebraischem Argument

Let \(f(z) = \sum\limits_{h = 0}^\infty {f_h z^h } \) be a power series with positive radius of convergenceR _f ≤1,f _h algebraic and lacunary in the following sense: Let {r _n }, {s _n } be two infinite sequences of integers, satisfying $$0 = s_0 \leqslant r_1< s_1 \leqslant r_2< s_2 \leqslant r_3< s_3 \leqslant ..., \mathop {lim}\limits_{n \to \infty } (s_n /F(n)) = \infty $$ such that $$f_h = 0 if r_n< h< s_n ,f_{r_n } \ne 0,f_{s_n } \ne 0 for n = 1,2,3,...;$$ F(n) denotes a certain function ofn, dependent onr _n and \(f_0 ,f_1 ,f_2 , \ldots f_{r_n } \) . Using ideas from a note ofK. Mahler, among other results the following main theorem is proved: The function valuef(α) (with α algebraic, 0<|α|<R _f ) is algebraic if and only if there exists a positive integerN=N(α) such that $$P_n (\alpha ): = \sum\limits_{h = s_n }^{r_{n + 1} } {f_h \alpha ^h = 0 for all n \geqslant N.} $$      

Some new estimates of the Fourier transform in \mathbb{L}_2 (ℝ)

Given a function $\mathbb{L}_2 $ (?), its Fourier transform $g(x) = \hat f(x) = F[f](x) = \frac{1} {{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {f(x)e^{ - ixt} dt} ,f(t) = F^{ - 1} [g](t) = \frac{1} {{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {g(x)e^{ - ixt} dx} $ and the inverse Fourier transform are considered in the space f ε $\mathbb{L}_2 $ (?). New estimates are presented for the integral $\int\limits_{|t| \geqslant N} {|g(t)|^2 dt} = \int\limits_{|t| \geqslant N} {|\hat f(t)|^2 dt} ,N \geqslant 1,$ in the vase of f ε $\mathbb{L}_2 $ (?) characterized by the generalized modulus of continuity of the kth order constructed with the help of the Steklov function. Some other estimates associated with this integral are proved.     

On a result of Hardy and Littlewood concerning Fourier series

Let f ∈ L ¹( $ \mathbb{T} $ ) and assume that $$ f\left( t \right) \sim \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos kt + b_k \sin kt} \right)} $$ Hardy and Littlewood [1] proved that the series $ \sum\limits_{k = 1}^\infty {\frac{{a_k }} {k}} $ converges if and only if the improper Riemann integral $$ \mathop {\lim }\limits_{\delta \to 0^ + } \int_\delta ^\pi {\frac{1} {x}} \left\{ {\int_{ - x}^x {f(t)dt} } \right\}dx $$ exists. In this paper we prove a refinement of this result.     

On the approximation by de la Vallée Poussin sums and interpolatory polynomials in Lipschitz norms

ПустьM ^m,α - множество 2π-п ериодических функци йf с конечной нормой $$||f||_{p,m,\alpha } = \sum\limits_{k = 1}^m {||f^{(k)} ||_{_p } + \mathop {\sup }\limits_{h \ne 0} |h|^{ - \alpha } ||} f^{(m)} (o + h) - f^{(m)} (o)||_{p,} $$ где1 ≦ p ≦ ∞, 0≦α≦1. Рассмотр им средние Bалле Пуссе на $$(\sigma _{n,1} f)(x) = \frac{1}{\pi }\int\limits_0^{2x} {f(u)K_{n,1} (x - u)du} $$ и $$(L_{n,1} f)(x) = \frac{2}{{2n + 1}}\sum\limits_{k = 1}^{2n} {f(x_k )K_{n,1} } (x - x_k ),$$ де0≦l≦n и x _k=2kπ/(2n+1). В работе по лучены оценки для вел ичин \(||f - \sigma _{n,1} f||_{p,r,\beta } \) и $$||f - L_{n,1} f||_{p,r,\beta } (r + \beta \leqq m + \alpha ).$$      

Some variations on Kac's formula for Brownian motion

We show that under suitable conditions $$\begin{gathered} E_x f\left\{ {a + \int_0^t \beta \left[ {b + \int_0^s {\alpha \left( {X_r } \right)dr, c + s, X_s } } \right]ds, b + \int_0^t {\alpha \left( {X_s } \right)ds, c + t, X_t } } \right\} \hfill \\ = e^{tG} f\left[ {a, b, c, x} \right] \hfill \\ \end{gathered} $$ whereX _t is a Brownian motion andG is the generator of a (C ₀) contraction semigroupe ^tG.