If $F$ is a continuous function on the real line and $f=F'$ is its distributional derivative then the continuous primitive integral of distribution $f$ is $\int_a^bf=F(b)-F(a)$. This integral contains the Lebesgue, Henstock--Kurzweil and wide Denjoy integrals. Under the Alexiewicz norm the space of integrable distributions is a Banach space. We define the convolution $f\ast g(x)=\intinf f(x-y)g(y) dy$ for $f$ an integrable distribution and $g$ a function of bounded variation or an $L^1$ function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For $g$ of bounded variation, $f\ast g$ is uniformly continuous and we have the estimate $\|f\ast g\|_\infty\leq \|f\|\|g\|_\bv$ where $\|f\|=\sup_I|\int_If|$ is the Alexiewicz norm. This supremum is taken over all intervals $I\subset\R$. When $g\in L^1$ the estimate is $\|f\ast g\|\leq \|f\|\|g\|_1$. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.
Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz's norm, the space…
In this paper we will study integrability of distributions whose primitives are left regulated functions and locally or globally integrable in the Henstock--Kurzweil, Lebesgue or Riemann sense. Corresponding spaces of distributions and their primitives are defined and their properties are studied. Basic properties of primitive integrals are derived and applications to systems of first order nonlinear distributional differential equations and to an $m$th order distributional differential equation are presented. The domain of solutions can be unbounded, as shown by concrete examples.
A function on the real line is called regulated if it has a left limit and a right limit at each point. If $f$ is a Schwartz distribution on the real line such that $f=F'$ (distributional or weak derivative) for a regulated function $F$ then the regulated primitive integral of $f$ is $\int_{(a,b)}f=F(b-)-F(a+)$, with similar definitions for other types of intervals. The space of integrable distributions is a Banach space and Banach lattice under the Alexiewicz norm. It contains the spaces of Lebesgue and Henstock--Kurzweil integrable functions as continuous embeddings. It is the completion of …
An integral is defined on the plane that includes the Henstock--Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions $F(x,y)$ defined on the extended real plane $[-\infty,\infty]^2$ that vanish when $x$ or $y$ is $-\infty$. With usual pointwise operations this is a Banach space under the uniform norm. The integrable functions and distributions (generalised functions) are those that are the distributional derivative $\partial^2/(\partial x\partial y)$ of this space of primitives. If $f=\partial^2/(\part…
For each $1\leq p<\infty$ a space of integrable Schwartz distributions, $L^'^{\,p}$, is defined by taking the distributional derivative of all functions in $L^p$. Here, $L^p$ is with respect to Lebesgue measure on the real line. If $f\in L^'^{\,p}$ such that $f$ is the distributional derivative of $F\in L^p$ then the integral is defined as $\int^\infty_{-\infty} fG=-\int^\infty_{-\infty} F(x)g(x)\,dx$, where $g\in L^q$, $G(x)= \int_0^x g(t)\,dt$ and $1/p+1/q=1$. A norm is $\lVert f\rVert'_p=\lVert F\rVert_p$. The spaces $L^'^{\,p}$ and $L^p$ are isometrically isomorphic. Distributions in $L^'^…
We consider mapping properties of the iterated Stieltjes transform, establishing its new relations with the iterated Hilbert transform (a singular integral) on the half-axis and proving the corresponding convolution and Titchmarsh's type theorems. Moreover, the obtained convolution method is applied to solve a new class of singular integral equations.
We show that the evaluation of an integral considered by Boros and Moll is a special case of a convolution result about Student t-densities obtained by the authors in 2008.
This paper is devoted to $L^2$ estimates for trilinear oscillatory integrals of convolution type on $\mathbb{R}^2$. The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp $L^2$ decay estimates of trilinear oscillatory integrals with smooth phases, and then give $L^2$ uniform estimates for these integrals with polynomial phases.
In this paper convolution type integral equations in the conservative case are studied. The conservative case of convolution type of equations relates to the case of non normal type of equations and is that of the corresponding symbols degenerate at some points of the real line, and the classical Furier transformation method meets difficulties with its application to the studying equations. To the study in the conservative case of convolution type equations in this paper we use the special factorization method.
In this paper, we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators in two-dimensional setting in the following form: \begin{equation*} L_{λ}\left( f;x,y\right) =\underset{D}{\iint }f\left( t,s\right) K_{λ}\left( t-x,s-y\right) dsdt,\text{ }\left( x,y\right) \in D, \end{equation*} where $D=\left \langle a,b\right \rangle \times \left \langle c,d\right \rangle $ is an arbitrary closed, semi-closed or open rectangle in $\mathbb{R}^{2}$ and $% λ\in Λ,$ $Λ$ is a set of non-negative indices with accumulation point $λ_{0}$. Also, we pro…
We prove Fatou type theorem on almost everywhere convergence of convolution integrals in spaces $L^p\,(1<p<\infty)$ for general kernels, forming an approximate identity. For a wide class of kernels we show that obtained convergence regions are optimal in some sense. It is also established a weak boundedness of the corresponding maximal operator in $L^p\,(1\le p<\infty)$.
In this paper we prove sharp $L^\infty$-$L^\infty$-$L^\infty$ decay for certain trilinear oscillatory integral forms of convolution type on $\mathbb R^2$. These estimates imply earlier $L^2$-$L^2$-$L^2$ results obtained by the second author as well as corresponding sharp, stable sublevel set estimates of the form studied by Christ and Christ, Li, Tao, and Thiele. New connections to the multilinear results of Phong, Stein, and Sturm are also considered.
We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained more quickly. We also give a characterization of the integrable functions and their primitives.
In this paper, we prove an $L^2-L^2-L^2$ decay estimate for a trilinear oscillatory integral of convolution type in $\mathbb{R}^d,$ which recovers the earlier result of Li (2013) when $d=1.$ We discuss the sharpness of our result in the $d=2$ case. Our main hypothesis has close connections to the property of simple nondegeneracy studied by Christ, Li, Tao and Thiele (2005).
We obtain estimates exact in order for deviations of Zygmund sums in metrics of spaces $L_{q}$, $1<q<\infty$, on classes of $2π$-periodic functions, that admit the representation in the form of convolution of functions that belong to unit ball of the space $L_{1}$ with fixed kernel $Ψ_β$. We show that at certain values of the parameters that define the class $L^ψ_{β,1}$ and method of approximation, Zygmund sums provide the order of best approximation of given classes by trigonometric polynomials in metric $L_{q}$
Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted $\alext$ and is a Banach space under the Alexiewicz norm, $\|f\|_\T =\sup_{|I|\leq 2π}|\int_I f|$, the supremum being taken over intervals of length not exceeding $2π$. It contains the periodic functions integrable in the sense of Lebesgue and Henstock-Kurzweil. Many of the properties of $L^1$ Fourier series continue to hold for this larger space, with the $L^1$ norm replaced by the Alexiewicz …
Spectral properties of many finite convolution integral operators have been understood by finding differential operators that commute with them. In this paper we compile a complete list of such commuting pairs, extending previous work to complex-valued and non self-adjoint operators. In addition, we introduce a new kind of commutation relation, which we call sesquicommutation, that also has implications for the spectral properties of the integral operator. In this case we also compute a complete list of sequicommuting pairs of integral and differential operators.
We approach the Riemann integral via generalized primitives to give a new proof for a general result on change of variable originally proven by Kestelman and Davies. Our proof is similar to Kestelman's, but we hope readers will find it clearer thanks to the use of a new test for the Riemann integrability (which we introduce in this paper) along with some ingredients from some other more recent proofs available in the literature. We also include a bibliographical review of related results and proofs. Although this paper emphasizes in the change of variable theorem, our contributions to the Riem…
This paper deals with the delta continuous Stieltjes variational integral generalized in the plane. In particular, this work presents about some fundamental properties of it. The delta continuous Stieltjes variational integral in the plane is considered as a form of generalization of the same type of integral defined in the real line [4]. The extremely difference between the delta continuous Stieltjes variational integrals defined in both the real line and the plane is especially related to their primitive functions. Where, if $F$ is a primitive function of the delta continuous Stieltjes varia…
The generalized Parseval equality for the Mellin transform is employed to prove the inversion theorem in L_2 with the respective inverse operator related to the Hartley transform on the nonnegative half-axis (the half-Hartley transform). Moreover, involving the convolution method, which is based on the double Mellin-Barnes integrals, the corresponding convolution and Titchmarsh's theorems for the half-Hartley transform are established. As an application, we consider solvability conditions for a homogeneous integral equation of the second kind involving the Hartley kernel.
In metric of spaces $L_{s}, \ 1< s\leq\infty$, we obtain exact order estimates of best approximations and approximations by Fourier sums of classes of convolutions the periodic functions that belong to unit ball of space $L_{1}$, with generating kernel $Ψ_β(t)=\sum\limits_{k=1}^{\infty}ψ(k)\cos(kt-\frac{βπ}{2})$, $β\in\mathbb{R}$, whose coefficients $ψ(k)$ are such that product $ψ(n)n^{1-\frac{1}{s}}$, $1<s\leq\infty$, can't tend to nought faster than every power function and besides, if $1<s<\infty$, then $\sum\limits_{k=1}^{\infty}ψ^{s}(k)k^{s-2}<\infty$ and if $s=\infty$, then $\sum\limits_…
A different application of the familiar integral representation for the modifed Bessel function drives to a new Kontorovich-Lebedev-like integral transformation of a general complex index. Mapping and operational properties, a convolution operator and inversion formula are established. Solvability conditions and explicit solutions of the corresponding class of convolution integral equations are exhibited.
We prove that the weakly singular, non-linear convolution integral equation $\int_{\mathbb{R}^n}|x-y|^{-λ}f(y)dy=f(x)^{p-1}$, where $0<λ<n$, and $p=2n/(2n-λ)$ has at least two non-equivalent solutions. This answers a problem of Elliott Lieb. We also prove certain orthogonality relations among linear differential forms with constant coefficients related to the corresponding type of convolution operators. Finally, we discuss the regularity of the solutions of such non-linear integral equations over not necessarily bounded open subsets of $\mathbb{R}^n$.
While exploiting the generalized Parseval equality for the Mellin transform, we derive the reciprocal inverse operator in the weighted L_2-space related to the Hilbert transform on the nonnegative half-axis. Moreover, employing the convolution method, which is based on the Mellin-Barnes integrals, we prove the corresponding convolution and Titchmarsh's theorems for the half-Hilbert transform. Some applications to the solvability of a new class of singular integral equations are demonstrated. Our technique does not require the use of methods of the Riemann-Hilbert boundary value problems for an…
A $q$-deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear $q$-difference equation associated with the $q$-Painlevé VI equation. Then we obtain integral transformations. We investigate the $q$-middle convolution in terms of the affine Weyl group symmetry of the $q$-Painlevé VI equation. We deduce an integral transformation on the $q$-Heun equation.
