Integrable Complex-valued Functions Research Articles

Approximation of Complex-Valued Functions by Fractal Functions

Fractal approximants developed through iterated function systems (IFS) prove more versatile than classical approximants. In this paper, we introduce a new class of fractal approximants using the suitable bounded linear operators defined on the space C(I) of continuous functions and concept of $$$\alpha$$$-fractal functions. The convergence of the proposed fractal approximants towards the original continuous function does not need any condition on the scaling factors. The fractal approximants proposed in this paper possess fractality and convergence simultaneously. Without imposing any condition on the scaling vector, we establish the constrained approximation by the proposed fractal approximants. Existence of Schauder basis of fractal polynomials for the space of continuous functions C(I) is investigated. Using the proposed class of fractal approximants, we develop complex fractal approximants for representation of the square integrable complex-valued functions defined on a real compact interval.

Spectral parameter power series representation for solutions of linear system of two first order differential equations

A representation in the form of spectral parameter power series (SPPS) is given for a general solution of a one dimension Dirac system containing arbitrary matrix coefficient at the spectral parameter,BdYdx+P(x)Y=λR(x)Y,(*)where Y=(y1,y2)T is the unknown vector-function, λ is the spectral parameter, B=(01−10), and P is a symmetric 2 × 2 matrix, R is an arbitrary 2 × 2 matrix whose entries are integrable complex-valued functions. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a non-vanishing solution for one particular λ=λ0. The existence of such solution is shown.For a general linear system of two first order differential equationsP(x)dYdx+Q(x)Y=λR(x)Y,x∈[a,b],where P, Q, R are 2 × 2 matrices whose entries are integrable complex-valued functions, P being invertible for every x, a transformation reducing it to a system (*) is shown.The general scheme of application of the SPPS representation to the solution of initial value and spectral problems as well as numerical illustrations are provided.

English

For a Lebesgue integrable complex-valued function $f$ defined on $\mathbb R^+:=[0,\infty )$ let $\hat f$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\hat f(y)\to 0$ as $y\to \infty $. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^1(\mathbb R^+)$ there is a definite rate at which the Walsh-Fourier transform tends to zero. We determine this rate for functions of bounded variation on $\mathbb R^+$. We also determine such rate of Walsh-Fourier transform for functions of bounded variation in the sense of Vitali defined on $(\mathbb R^+)^N$, $N\in \mathbb N$.

Classical Equiconvergence Problem for the Sturm-Liouville Operator with a Singular Potential

We study the classical problem of equiconvergence of spectral expansions for the Sturm-Liouville operator with a singular potential. We present various conditions on the potential guaranteeing the equiconvergence for the expansions of an arbitrary integrable complex-valued function.

Convergence and Approximation of the Sturm–Liouville Operators with Potentials-Distributions

We study the operators L n y = −(p n y′)′+q n y, n ∈ ℤ+ , given on a finite interval with various boundary conditions. It is assumed that the function q n is a derivative (in a sense of distributions) of Q n and 1/p n , Q n /p n , and $$ {Q}_n^2/{p}_n $$ are integrable complex-valued functions. The sufficient conditions for the uniform convergence of Green functions G n of the operators L n on the square as n → ∞ to G 0 are established. It is proved that every G 0 is the limit of Green functions of the operators L n with smooth coefficients. If p 0 > 0 and Q 0(t) ∈ ℝ, then they can be chosen so that p n > 0 and q n are real-valued and have compact supports.

Determination of order of magnitude of multiple Fourier coefficients of functions of bounded ϕ-variation having lacunary Fourier series using Jensen's inequality

For a Lebesgue integrable complex-valued function f defined over the m -dimensional torus Tm := [0,2π)m , let f (n) denote the Fourier coefficient of f , where n = (n(1), . . . ,n(m)) ∈ Zm . Recently, in one of our papers [to appear in Mathematical Inequalities & Applications], we have defined the notion of bounded φ -variation for a complex-valued function on a rectangle [a1,b1 ]× . . .× [am,bm ] and studied the order of magnitude of Fourier coefficients of such functions on [0,2π]m . In this paper, the order of magnitude of Fourier coefficients of a function of bounded φ -variation from [0,2π]m to C and having lacunary Fourier series with certain gaps is studied and a generalization of our earlier result (Theorem in [Acta Sci. Math. (Szeged), 78, (2012), 97–109]) is proved. Interestingly, the Jensen’s inequality for integrals is used to prove the main result. Mathematics subject classification (2010): 42B05, 26B30, 26D15.

An application of Jensen's inequality in determining the order of magnitude of multiple Fourier coefficients of functions of bounded φ-variation

For a Lebesgue integrable complex-valued function f defined over the n -dimensional torus Tn := [0,2π)n , n∈N , let f (k) denote the Fourier coefficient of f , where k = (k1, . . . ,kn) ∈ Zn . The Riemann-Lebesgue lemma shows that f (k) = o(1) as |k| → 0 for any f ∈ L1(Tn) . However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. The definitive results are due to V. Fulop and F. Moricz for functions of bounded variation, and due to B. L. Ghodadra for functions of bounded p -variation. In this paper, defining the notion of bounded φ -variation for a function from [0,2π]n to C in two different ways, we prove that this is the case for Fourier coefficients of such functions also. Interestingly, in proving our main results we use the famous Jensen’s inequality for integrals. Our new results with φ(x) = xp (p 1) gives our earlier results [Acta Math. Hungar, 128 (4) (2010), 328–343]. Mathematics subject classification (2010): 42B05, 26D15, 26B30.

Order of magnitude of multiple Fourier coefficients of functions of bounded p-variation

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus \( \mathbb{T}^n \):= [0, 2π)n, let \( \hat f \)(k) denote the Fourier coefficient of f, where k = (k1, … kn) ∈ ℤn. In this paper, defining the notion of bounded p-variation (p ≧ 1) for a function from [0, 2π]n to ℜ in two diffierent ways, the order of magnitude of Fourier coefficients of such functions is studied. As far as the order of magnitude is concerned, our results with p = 1 give the results of Moricz [5] and Fulop and Moricz [3].

A Central Limit Theorem for Some Special Complex-Valued Probability Densities

We describe the behavior of the n-fold convolutions[formula] under a suitable scaling as n→∞, where f is an integrable complex-valued function on R. We consider only the unstable case and only two typical examples in that case. In the first example f is defined by f̂(t)=exp{i|t|p−|t|q} and in the second one by f̂(t)=exp{i|t|psgn(t)−|t|q}, where p and q are real numbers such that 2≤p<q.

Minimal Operators for Schrodinger-Type Differential Expressions with Discontinuous Principal Coefficients

The objective of this paper is to extend the recent results [7, 8, 9] concerning the self-adjointness of Schrödinger-type operators with singular potentials to a more general setting. We shall be concerned here with formally symmetric elliptic differential expressions of the form1.1where x = (x1, …, xm) ∈ Rm (and m ≧ 1), i = (–1)1/2, ∂j = ∂/∂xj, and the coefficients ajk, bj and q are real-valued and measurable on Rm.The basic problem that we consider is that of deciding whether or not the formal operator defined by (1.1) determines a unique self-adjoint operator in the space L2(Rm) of (equivalence classes of) square integrable complex-valued functions on Rm.

Local Λ sets for profinite groups

Preliminaries. Let G be a compact group with normalized Haar measure λβ. For p e [ l , [ the Banach space of pth power integrable complex-valued functions on G is denoted (L(G), || | |p). The dual object G of G is taken to be a maximal set of pairwise inequivalent continuous irreducible unitary representations of G. For each σeG let da be the degree or dimension of the representation space of σ and let Xσ denote its trace. The Fourier transform of / 6 L\G) is the matrix-valued function f on G defined by

Operational Calculus for Functions of Two Variables

Let L be the space of locally integrable complex-valued functions of two variables defined on the first quadrant. We can inject L into a commutative algebra of operators on a space of testing functions. Since this injection maps convolution into multiplication it serves as a generalization (there are no growth restrictions) of the two-dimensional Laplace transform. Some useful operational formulas are developed.

Positive Contractions on L 1 -Spaces

Let T be a positive linear operator on L1 of a probability space such that II T1j i: 1. In this note we consider the following question: Under what condition is T multiplicative on Let (X, F, m) be a probability space and L1=L1(X, F, m) the Banach space of equivalence classes of integrable complex-valued functions on X. Let T be a positive linear operator on L1 such that II TIIl< 1, and define G(T)={f E L1; If I = I Tf I = 1). The main purpose of this note is to prove the following: THEOREM. If G(T) is total in L1, i.e., the linear manifold generated by G(T) is dense in L1 in the norm topology, then for any bounded functions f and g we have T(fg)= (Tf)(Tg). As a corollary of the Theorem we have the following result; essentially the same idea has been used by Halmos [2, p. 45] to find a necessary and sufficient condition for a unitary operator on L2=L2(X, Y, m) to be induced by an automorphism of the measure algebra (F(m), m) associated with (X, F, m). COROLLARY. A necessary and sufficient condition that T be induced by a homomorphism of the measure algebra (f(m), m) into itself is that G(T) be total in L1. For the proof of the theorem we need some lemmas. LEMMA 1. Let K be a compact Hausdorff space and C(K) the Banach algebra of all complex-valued continuous functions f on K wvith norm lIf IIU=sup{If(s)I; s e K}. Let U be a positive linear operator on C(K) such that U1=1. Let f,geC(K), IfI=lgi=1, and IUfI=IUgi=1. Then U(fg)=(Uf)(Ug)Received by the editors February 28, 1973 and, in revised form, May 10, 1973. AMS (MOS) subject classifications (1970). Primary 47B55, 47A35; Secondary 28A65, 46J10.

Spaces of functions with values in a Banach algebra

Introduction. In the following pages certain spaces of abstract-valued functions are examined. Throughout the paper A will denote a commutative Banach algebra. In analogy to the group algebra L'(G) of a locally compact Abelian group G, that is, the space of absolutely integrable complex-valued functions on G, we form the set B1 =B'(G, A) of Bochner integrable functions defined to A from G. Bl is first of all a Banach space and it becomes a commutative Banach algebra if multiplication of two elementsf, gcBl is defined by the convolution formula, (f*g) (x) =f Gf(xy)g(y-1)dy. For the theory of the Bochner integral we shall rely mainly oni the presentatioIn in Hille's book [2, pp. 35-49]. Although the development there uses the Lebesgue measure for finite dimensional Euclidean spaces, the theorems which we shall need hold as well for more general measure spaces, in particular for a locally compact group with Haar measure. The calculus for the generalized convolution carries over directly from that for numerical functions and will be assumed. We note that the convolution of a function in B' and a funiction in L'(G) is well defined and in Bl. To the greatest possible extent the notation and definitions are those of Loomis [3]. Maximial means regular maximal ideal throughout. Special conventions are as follows: f =-f(x), g = g(x), * * denote elemenlts of B1. For the most part complex-valued functions are assigned Greek letters regardless of the set on which they are defined. X is always used for the characteristic function of a set and A is the Haar measure on G. 5TB denotes the maximal ideal space of B1, ORA that of A, and G, the character group of G, is the maximal ideal space of Ll. Typical elements are MB, M, and a respectively. Subscripts distinguish the various norms. For example, |f|I IB =_ff(X)llAdx, fEB'. For any complex or A-valued function on G and xCG, the subscript x applied to the function denotes its translate by x. The symbol used with