共查询到20条相似文献,搜索用时 31 毫秒
1.
Vladimir Drobot 《Aequationes Mathematicae》1970,5(1):120-122
It is well known ([1], [3]) that any measurable solution of the Cauchy functional equationf(x+y)=f(x)+f(y) must actually be continuous. The same is true of some other functional equations likef(x+y)=f(x)f(y),f(x+y)f(x–y)=f(x)
2
–f(y)
2, etc. (cf. [1]). In this note we prove a general result of this type for functional equations on groups. 相似文献
2.
Z. Ditzian 《Analysis Mathematica》1977,3(1):45-49
В РАБОтЕ ДОкАжАНО, ЧтО limk a *f(x)=f(x) пОЧтИ ВсУДУ, гДЕk a(t)=a?n k(a?1t), t?Rn, Для Дль ДОВОльНО шИРОкОг О клАссА ФУНкцИИk(t). ДАНы УслОВИь, пРИ кОтО Рых пОлУЧЕННыИ РЕжУл ьтАт РАспРОстРАНьЕтсь НА ФУНкцИУ $$k(x,y) = \gamma \frac{1}{{1 + |x|^\alpha }} \cdot \frac{1}{{1 + |y|^\beta }},$$ гДЕ α, β>1, А γ — НОРМИРУУЩ ИИ МНОжИтЕль тАкОИ, Чт О∫∫k(x, y) dx dy=1. 相似文献
3.
с. Б. кОжыРЕВ 《Analysis Mathematica》1985,11(4):311-329
A continuous real valued function defined on an intervalD is called crinkly iff the setf ?1(У) ∩I is uncountable for each interval \(I \subseteqq D\) and number \(y \in (\mathop {\inf }\limits_I f,\mathop {\sup }\limits_I f)\) . The main result of the paper consists in the following assertion. Let the closed segment [0, 1] be represented as a union of four measurable, mutually nonintersecting setsE 1,Е 2,E 3,E 4. Then, for each functionH(δ) such thatH(δ)→ + ∞ andδH(δ)→0 asδ→0, there exists a crinkly functionf possessing the following five properties:
- a.e. onE 1:D + f(x)=D-f(x)=+∞,D + f(x)=D?f(x)=?∞;
- a.e. onE 2:D + f(x)=+∞,D?f(x)=?∞,D +f(x)=D-f(x)=0;
- a.e. onE 3:D + f(x)=?∞,D ? f(x)=+∞,D + f(x)=D?f(x)=0;
- a.e. onE 4:Df(x)=0;
- the modulus of continuityΩ off on [0, 1] satisfies $$\omega (\delta ,f,[0,1]) \leqq \delta H(\delta ).$$
4.
A. B. Krygin 《Mathematical Notes》1978,23(6):479-485
Let f(x) be a smooth function on the circle S1, x mod 1, \(\smallint _{S^1 } f(x)dx = 0\) , α be an irrational number, and qn be the denominators of convergents of continued fractions. In this note a classification of ω-limit sets for the cylindrical cascade $$T:(x,y) \to (x + \alpha , y + f(x)),$$ x ε S1, y ε R, is obtained. Criteria for the solvability of the equation g(x +α) — g(x)=f (x) are found. Estimates for the speed of decrease of the function $$h_{q_n } (x) = \sum _{i = 0}^{q_n - 1} f(x + i\alpha )$$ as n → ∞ are obtained. 相似文献
5.
设a是大于1的正整数,f(a)是a的非负整系数多项式,f(1)=2rp+4,其中r是大于1的正整数,p=2~l-1是Mersenne素数.本文讨论了方程(a-1)x~2+f(a)=4a~n的正整数解(x,n)的有限性,并且证明了:当f(a)=91a+9时,该方程仅当a=5,7和25时分别有解(x,n)=(3,3),(11,3)和(3,4). 相似文献
6.
V. M. Veselinov 《Mathematical Notes》1974,16(2):695-700
We study the approximation of continuous real functionsf(x)=0((1+|x|)σ) (|x|→∞,σ≥0) by means of entire functions of exponential type in some metric of Hausdorff type. We generalize a theorem due to N. I. Akhiezer concerning uniform weighted approximations. 相似文献
7.
Roman Ger 《Results in Mathematics》1994,26(3-4):281-289
Let X, Y be two linear spaces over the field ? of rationals and let D ≠ ? be a (?—convex subset of X. We show that every function ?: D → Y satisfying the functional equation $${\mathop\sum^{n+1}\limits_{j=0}}(-1)^{n+1-j}\Bigg(^{n+1}_{j}\Bigg)f\Bigg((1-{j\over {n+1}})x+{j\over{n+1}}y\Bigg)=0,\ \ \ x,y\in\ D,$$ admits an extension to a function F: X → Y of the form $$F(x)=A^o+A^1(x)+\cdot\cdot\cdot+A^n(x),\ \ \ x\in\ X,$$ where A o ∈ Y, Ak(x) ? Ak(x,…,x), x ∈ X, and the maps A k: X k → Y are k—additive and symmetric, k ∈ {1,…, n}. Uniqueness of the extension is also discussed. 相似文献
8.
D. A. Mit'kin 《Mathematical Notes》1973,14(1):597-602
Let n≥4 be even, p > (n2?2n)/2 be simple odd, andf(x)=a 0+a 1+...+a nxn be a polynomial with integral coefficients that are not quadratic over the residue field modulo p, (a n, p)=1. The following inequality is proved: $$\left| {\sum\nolimits_{x = 1}^p {\left( {\frac{{f(x)}}{p}} \right)} } \right| \leqslant (n - 2)\sqrt {p + 1 - \frac{{n(n - 4)}}{4}} + 1.$$ 相似文献
9.
Guido Fubini 《Rendiconti del Circolo Matematico di Palermo》1916,42(1):135-154
Let Ω ? ? n be a convex bounded open set, of class\(C^2 ,Q_\tau = \Omega \times \left[ {\tau ,\tau + T} \right],\tau \in \mathbb{R},T > 0.\). LetB be a linear continuous operator ofL 2Ω ? ? N inL 2Ω ? ? N . It is shown that if\(f \in L^2 (Q_\tau ,\mathbb{R}^N )\) then there exists a unique solution of the problem:\(u \in W^{2,1} (Q_\tau ,\mathbb{R}^N ),\alpha (x,t,H(u)) - \frac{{\partial u}}{{\partial t}} = f(x,t)\), in\(Q_\tau \), such thatu(x,t)=B u(x, τ+T) in Ω, wherea(x, t, ζ) is misurable in(x,t), continuous in ζ,a(x,t, 0)=0, and verifies condition (A). IfB=Id this is the classical periodic problem. If moreovera(x,t,ζ)=a(x,t+T, ζ) anda(x,t, H (Bu))=B a(x,t,H (u)) ?t ∈ ?, the analogous problem in Ω × ? is studied. 相似文献
10.
V. E. Katznel'son 《Journal of Mathematical Sciences》1981,16(3):1095-1101
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?Cδ>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. 相似文献
11.
Janusz Brzdȩk 《Results in Mathematics》1996,30(1-2):25-38
Let E be a real inner product space with dimension at least 2, D ? E, f: E → R with f(x+y)?f(x)?f(y) ∈ Z for all orthogonal x,y ∈ E, and f(D) ? (?γ,γ)+Z witn some real γ > 0. We prove that, under some additional assumptions, there are a unique linear functional A: E → R and a unique constant d ∈ R with f(x)?d∥x∥2?A(x) ∈ Z for x ∈ E. We also show some applications of this result to the determination of solutions F: E → C of the conditional equation: F(x+y) = F(x)F(y) for all orthogonal x,y ∈ E. 相似文献
12.
Pitambar Das 《Proceedings Mathematical Sciences》1993,103(3):341-347
In this paper, we obtain the sufficient and necessary conditions for all solutions of the odd-order nonlinear delay differential equation.x (n)+Q(t)f(x(g(t)))=0 to be oscillatory. In particular, ifn=1, Q(t)>0, f(x)=x α, where α∈(0,1) and is a ratio of odd integers andg(t)=t?? for some ?>0, then every solution of (*) oscillates if and only if ∫∞Q(s)ds=∞. 相似文献
13.
The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F?X and every real number ε>0 one can choosen∈ω such that ∥f(x)?fn(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A?X, there exists a continuous mapf A:X→R ω such that A=f A ?1 (A). Splitting spaces will be studied systematically. 相似文献
14.
该文得到齐型空间中分数次积分交换子[b,I_α]的加权端点估计ω({x∈X:|[b,I_α]f(x)|t})≤Cψ(∫_xA(||b||_*(|f(x)|/t)■(ω(x))dμ(x))其中b∈BMO(X,d,μ),A(t)=tlog(e+t),ψ(t)=[tlog(e+t~α)]~(1/(1-α)),■(t)=t~(1-α)log(e+t~(-α)). 相似文献
15.
Elena Prestini 《Monatshefte für Mathematik》1990,109(2):135-143
Letf be a radial function and setT * f(x)=sup0<t<1 |T t f(x)|, x ∈ ?n, n≥2, where(Tt f)^ (ξ)=e it|ξ|a \(\hat f\) (ξ),a > 1. We show that, ifB is the ball centered at the origin, of radius 100, then \(\int\limits_B {|T^ * f(x)|} dx \leqslant c(\int {|\hat f(\xi )|^2 (l + |\xi |^s )ds} )^{1/2} \) if and only ifs≥1/4. 相似文献
16.
给出了三次Hyers—Ulam—Rassias型泛函方程的一种新表示方法af(x+ay)-f(ax+y)=(a(a~2-1))/2[f(x+y)+f(x-y)]+(a~4-1)f(y)-2a(a~2-1)f(x)其中a为整数且a≠0,土1.关于9个三次泛函方程给出等价性证明。对Banach空间三次方程的稳定性问题给予讨论。 相似文献
17.
P. Oswald 《Analysis Mathematica》1982,8(4):287-303
Обозначим через ? (L) кл асс всех ?-интегрируе мых 2π-периодических функ ций. При ограничении?(L)?L 1 устанавливаются необходимые и достат очные условия, которым долж на удовлетворять?(t), что бы для каждой функцииf(x)ε ?(L) а) тригонометрически й ряд Фурье сходился в метрике ?, в) (С, а) средние ряда Фур ье сходились в метрик е ? для некоторого α > 0, с) сопряженная функци я \(\tilde f(x)\) также принадлежал а классу?(L). 相似文献
18.
Diego Averna 《Rendiconti del Circolo Matematico di Palermo》1986,35(1):22-31
LetE be a Lebesgue measurable set in IR p+q andY a metric space. Iff:E→Y is such thatf(.,x) isL-measurable for almost allx andf(t,.) is continuous in each of theq variables separately for almost allt, thenf must beL-measurable (Theorem 1). By this result we deduce that a functionf:E→Y is almost-continuous iff it is almost-separately continuous. Finally, we give another characterization of the measurability of a functionf:IRp+q p+q→Y by means of properties of its sections (Theorem 2). 相似文献
19.
Jorge Ferreira 《Rendiconti del Circolo Matematico di Palermo》1995,44(1):135-146
In this paper we study the existence of weak solutions of the mixed problem for the nonlinear hyperbolic-parabolic equation $\begin{gathered} K_1 (x)u'' + K_2 (x)u' + A(t)u + H(u) = f in Q \\ u = 0 on \Sigma \\ u(0) = u_0 ,K_1 (x)u'(0) = \sqrt {K_1 (x)} u_1 (x) \\ \end{gathered} $ whereQ is a noncylindrical domain of ?n=1 with lateral boundary Σ, {A(t); t≥0} is a family of operators of ?(H 0 1 (ω),H -1(ω)) andH(s) is a continuous function which satisfies some appropriate conditions. 相似文献
20.
L. Vietoris 《Monatshefte für Mathematik》1986,102(1):85-89
I begin with a new short proof of: (I) LetP(t) inR d be a function oft havingn continuous derivatives fora≤t≤x. ThenP(x)∈ convK, where $$K = \left\{ {\sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}} P^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}P^{(n)} (t),a \leqslant t \leqslant x} \right\}.$$ for applying (I) let bef(t) a real function such that the point ((t?a) n+1,f(t)) fulfills the conditions of (I). Then (I) gives a sharper estimate of then th remainder term off(x) than the Lagrange remainder formula. Iff( n )(t) is also convex ina≤t≤x, thenf(x)∈[c,d], where $$\begin{gathered} c = \sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}f^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}f^{(n)} \left( {\frac{{na + x}}{{n + 1}}} \right)} , \hfill \\ d = \sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}f^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}} \frac{{nf^{(n)} (a) + f^{(n)} (x)}}{{n + 1}}. \hfill \\ \end{gathered} $$ 相似文献