Continuous Functionf Research Articles

A note on an integral operator induced by Zygmund function

In this note, by means of a kernel function induced by a continuous function f on the unit circle, we show that corresponding integral operator on Banach space AP is bounded or compact precisely when f belongs to the big Zygmund class ?* or the little Zygmund class ?*, where AP consists of all holomorphic functions ? on ?C\S1 with the finite corresponding norm. This generalizes the result in Hu, Song, Wei and Shen (2013) [5] and meanwhile may be considered as the infinitesimal version of main result obtained in Tang and Wu (2019) [8].

Interval universal approximation for neural networks

To verify safety and robustness of neural networks, researchers have successfully applied abstract interpretation , primarily using the interval abstract domain. In this paper, we study the theoretical power and limits of the interval domain for neural-network verification. First, we introduce the interval universal approximation (IUA) theorem. IUA shows that neural networks not only can approximate any continuous function f (universal approximation) as we have known for decades, but we can find a neural network, using any well-behaved activation function, whose interval bounds are an arbitrarily close approximation of the set semantics of f (the result of applying f to a set of inputs). We call this notion of approximation interval approximation . Our theorem generalizes the recent result of Baader et al. from ReLUs to a rich class of activation functions that we call squashable functions . Additionally, the IUA theorem implies that we can always construct provably robust neural networks under ℓ ∞ -norm using almost any practical activation function. Second, we study the computational complexity of constructing neural networks that are amenable to precise interval analysis. This is a crucial question, as our constructive proof of IUA is exponential in the size of the approximation domain. We boil this question down to the problem of approximating the range of a neural network with squashable activation functions. We show that the range approximation problem (RA) is a Δ 2 -intermediate problem, which is strictly harder than NP -complete problems, assuming coNP ⊄ NP . As a result, IUA is an inherently hard problem : No matter what abstract domain or computational tools we consider to achieve interval approximation, there is no efficient construction of such a universal approximator. This implies that it is hard to construct a provably robust network, even if we have a robust network to start with.

Cardinal invariants of Haar null and Haar meager sets

A subsetXof a Polish groupGisHaar nullif there exists a Borel probability measure μ and a Borel setBcontainingXsuch that μ(gBh) = 0 for everyg,h∈G. A setXisHaar meagerif there exists a compact metric spaceK, a continuous functionf:K→Gand a Borel setBcontainingXsuch thatf−1(gBh) is meager inKfor everyg,h∈G. We calculate (inZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case$G = \mathbb {Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.

Images of Bernstein sets via continuous functions

Abstract We examine images of Bernstein sets via continuous mappings. Among other results, we prove that there exists a continuous function f : ℝ → ℝ {f\colon\mathbb{R}\to\mathbb{R}} that maps every Bernstein subset of ℝ {\mathbb{R}} onto the whole real line. This gives the positive answer to a question of Osipov.

On the PropertyN-1

We construct a continuous functionf:[0,1]→Rsuch thatfpossessesN-1-property, butfdoes not have approximate derivative on a set of full Lebesgue measure. This shows that Banach’s Theorem concerning differentiability of continuous functions with Lusin’s property(N)does not hold forN-1-property. Some relevant properties are presented.

Robust Satisfiability of Systems of Equations

We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f : K → Rn on a finite simplicial complex K and α>0, it holds that each function g : K → Rn such that ║ g−f ║∞ ≤ α , has a root in K . Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n , assuming dim K ≤ 2 n −3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis. Via a reverse reduction, we prove that the problem is undecidable when dim K ≥ 2 n −2, where the threshold comes from the stable range in homotopy theory. For the lucidity of our exposition, we focus on the setting when f is simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.

Positive Solution for the Nonlinear Hadamard Type Fractional Differential Equation withp-Laplacian

We study the following nonlinear fractional differential equation involving thep-Laplacian operatorDβφpDαut=ft,ut,1<t<e,u1=u′1=u′e=0,Dαu1=Dαue=0, where the continuous functionf:1,e×0,+∞→[0,+∞),2<α≤3,1<β≤2.Dαdenotes the standard Hadamard fractional derivative of the orderα, the constantp>1, and thep-Laplacian operatorφps=sp-2s. We show some results about the existence and the uniqueness of the positive solution by using fixed point theorems and the properties of Green's function and thep-Laplacian operator.

A Note on Functional Averages over Gaussian Ensembles

We find a new formula for matrix averages over the Gaussian ensemble. LetHbe ann×nGaussian random matrix with complex, independent, and identically distributed entries of zero mean and unit variance. Given ann×npositive definite matrixAand a continuous functionf:ℝ+→ℝsuch that∫0∞‍e-αt|f(t)|2dt<∞for everyα>0, we find a new formula for the expectation𝔼[Tr(f(HAH*))]. Takingf(x)=log(1+x)gives another formula for the capacity of the MIMO communication channel, and takingf(x)=(1+x)-1gives the MMSE achieved by a linear receiver.

Continuous extensions of functions defined on subsets of products

A subset Y of a space X is Gδ-dense if it intersects every nonempty Gδ-set. The Gδ-closure of Y in X is the largest subspace of X in which Y is Gδ-dense.The space X has a regularGδ-diagonal if the diagonal of X is the intersection of countably many regular-closed subsets of X×X.Consider now these results: (a) (N. Noble, 1972 [18]) every Gδ-dense subspace in a product of separable metric spaces is C-embedded; (b) (M. Ulmer, 1970 [22], 1973 [23]) every Σ-product in a product of first-countable spaces is C-embedded; (c) (R. Pol and E. Pol, 1976 [20], also A.V. Arhangelʼskiĭ, 2000 [3]; as corollaries of more general theorems), every dense subset of a product of completely regular, first-countable spaces is C-embedded in its Gδ-closure.The present authorsʼ Theorem 3.10 concerns the continuous extension of functions defined on subsets of product spaces with the κ-box topology. Here is the case κ=ω of Theorem 3.10, which simultaneously generalizes the above-mentioned results. TheoremLet{Xi:i∈I}be a set ofT1-spaces, and let Y be dense in an open subspace ofXI:=∏i∈IXi. Ifχ(qi,Xi)⩽ωfor everyi∈Iand every q in theGδ-closure of Y inXI, then for every regular space Z with a regularGδ-diagonal, every continuous functionf:Y→Zextends continuously over theGδ-closure of Y inXI.Some examples are cited to show that the hypothesis χ(qi,Xi)⩽ω cannot be replaced by the weaker hypothesis ψ(qi,Xi)⩽ω.

The number of records within a random interval of the current record value

LetX 1,X 2, … be a sequence of i.i.d. random variables with some continuous distribution functionF. LetX(n) be then-th record value associated with this sequence and μ n − , μ n + be the variables that count the number of record values belonging to the random intervals(f−(X(n)), X(n)), (X(n), f+(X(n))), wheref−, f+ are two continuous functions satisfyingf−(x) x. Properties of μ n − , μ n + are studied in the present paper. Some statistical applications connected with these variables are also provided.

Lebesgue Measurability of Separately Continuous Functions and Separability

A connection between the separability and the countable chain condition of spaces withL-property (a topological spaceXhasL-property if for every topological spaceY, separately continuous functionf:X×Y→ℝand open setI⊆ℝ,the setf−1(I)is anFσ-set) is studied. We show that every completely regular Baire space with theL-property and the countable chain condition is separable and constructs a nonseparable completely regular space with theL-property and the countable chain condition. This gives a negative answer to a question of M. Burke.

Convergence of mock Fourier series

For certain Cantor measures μ on ℝn, it was shown by Jorgensen and Pedersen that there exists an orthonormal basis of exponentialse 2πiγ·x for λeΛ. a discrete subset of ℝn called aspectrum for μ. For anyL 1 functionf, we define coefficientsc γ(f)=∝f(y)e −2πiγiy dμ(y) and form the Mock Fourier series ∑λ∈Λcλ(f)e 2πiλ·x . There is a natural sequence of finite subsets Λn increasing to Λ asn→∞, and we define the partial sums of the Mock Fourier series by $$s_n (f)(x) = \sum\limits_{\lambda \in \Lambda _n } {c_n (f)e^{2\pi i\lambda \cdot x} } .$$ We prove, under mild technical assumptions on μ and Λ, thats n(f) converges uniformly tof for any continuous functionf and obtain the rate of convergence in terms of the modulus of continuity off. We also show, under somewhat stronger hypotheses, almost everywhere convergence forfeL 1.

Separate continuity, joint continuity and the Lindelöf property

In this paper we prove a theorem more general than the following. Suppose that X X is Lindelöf and α \alpha -favourable and Y Y is Lindelöf and Čech-complete. Then for each separately continuous function f : X × Y → R f:X\times Y \rightarrow \mathbb {R} there exists a residual set R R in X X such that f f is jointly continuous at each point of R × Y R\times Y .

Strongly meager sets can be quite big

AssumeCH. There exists a strongly meager setX⊆2ω and a continuous functionF: 2ω → 2ω such thatF″ (X)=2ω. The analogous statement for the strong measure zero, the notion dual to strongly meager, is false.

The argument principle and holomorphic extendibility

LetD be a bounded domain in the complex plane whose boundary consists of finitely many pairwise disjoint simple closed curves. GivebD the standard orientation, and letA(D) be the algebra of all continuous functions on $$\bar D$$ which are holomorphic onD. We prove that a continuous functionf onbD extends to a function inA(D) if and only if for eachg∈A(D) such thatf+g≠0 onbD, the change of argument off+g alongbD is nonnegative.

Dirichlet solutions on bordered Riemann surfaces and quasiconformal mappings

In this paper, we consider quasiconformal homeomorphisms ϕn ;S 0→S n (n = 1, 2, ...) of a bordered Riemann surfaceS 0 and discuss how the Dirichlet solutions $$H_{fo\varphi n^{ - 1} }^{S_n } $$ for a continuous functionf on ϖS 0 vary when the maximal dilatations of ϕn converge to one. Furthermore, we consider the smoothness of Dirichlet solutions for parameters of the quasiconformal deformation.

Continuity and differentiability of Nemytskii operators on the Hardy space $\mathcal{H}^{1,1} (T^1 )$

Let\(\mathcal{H}^{1,1} (T^1 )\) denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space\(\mathcal{H}^1 (T^1 )\). It is shown that when the weak derivative of a locally Lipschitz continuous functionf has bounded variation on compact sets the Nemytskii operatorF, defined byF(u)=f·u, maps\(\mathcal{H}^{1,1} (T^1 )\) continuously into itself. A further condition sufficient for the continuous Frechet differentiability ofF is then added.

Stability of the topological pressure for piecewise monotonic maps underC 1-perturbations

Assume thatX is a finite union of closed intervals and consider aC 1-mapX→ℝ for which {c∈X: T′c=0} is finite. Set $$T\left( T \right) = \cap _{j = 1}^\infty \overline {T^{ - j} X} $$ . Fix ann ∈ ℕ. For e>0, theC 1-map $$\tilde T:X \to \mathbb{R}$$ is called an e-perturbation ofT if $$\tilde T$$ is a piecewise monotonic map with at mostn intervals of monotonicity and $$\tilde T$$ is e-close toT in theC 1-topology. The influence of small perturbations ofT on the dynamical system (R(T),T) is investigated. Under a certain condition on the continuous functionf:X → ℝ, the topological pressure is lower semi-continuous. Furthermore, the topological pressure is upper semi-continuous for every continuous functionf:X → ℝ. If (R(T),T) has positive topological entropy and a unique measure μ of maximal entropy, then every sufficiently small perturbation $$\tilde T$$ ofT has a unique measure $$\tilde \mu $$ of maximal entropy, and the map $$\tilde T \mapsto \tilde \mu $$ is continuous atT in the weak star-topology.

Locally Analytic Functions over Completions of [formula omitted]r[U

It is well known that a continuous functionf:Zp→Qpcan be expanded by Mahler's basis: [formula] withan→0. Amice (Bull. Soc. Math. France92(1964), 117–180) has established conditions on the coefficientsanfor the functionfto be locally analytic, as well as more general results whenZpis replaced by some compact subset of a local field. We study the function field case in this paper. The function field analogue of Mahler's basis is the Carlitz polynomials, and the corre- sponding result for continuous functions has already been established by Wagner (Acta Arith.17(1971), 389–406). We show that the conditions for a continuous function to be locally analytic in the function field case are completely similar to theQpcase. An application to using integral calculus to analytically continue characteristicp-valuedL-series is briefly mentioned at the end of the paper.

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, Σ, μ) we choose an additive measureν: Σ → Z (Z is a Banach space) with the following property: for alle ∈Σ, we have\(\int_e {x d\nu = 0 \Rightarrow x \mathop \sim \limits^\mu 0}\); this measure defines an indefinite integral over the measureν onL2(T, Σ,μ). We prove that if {τn(t)}n=1/∞ is an orthonormal basis inL2 andθn(e)=e τn(t) dν, then any additive measureϕ: Σ → Z whose Radon-Nikodým derivativedϕ/dν belongs toL2 is uniquely expandable in a seriesϕ(e)=Σn=1/∞αnθn(e) that converges toϕ(e) uniformly with respect toe ∈Σ can be differentiated term-by-term, and satisfies Σn=1/∞ αn/2<∞. In the caseL2[0,2π],Z=l, the Fourier series of a 2π-periodic absolutely continuous functionF(t) such thatF'(t) ∈L2[0, 2π] is superuniformly convergent toF(t).