Searcharxiv⌕ Search

arXiv subjects

Erik Talvila

Publications and source records attributed to Erik Talvila.

36 records · Page 2Linked to original sources
arXiv2012

Simple derivation of basic quadrature formulas

Simple proofs of the midpoint, trapezoidal and Simpson's rules are proved for numerical integration on a compact interval. The integrand is assumed to be twice continuously differentiable for the midpoint and trapezoidal rules, and to be four times continuously differentiable for Simpson's rule. Errors are estimated in terms of the uniform norm of second or fourth derivatives of the integrand. The proof uses only integration by parts, applied to the second or fourth derivative of the integrand, multiplied by an appropriate polynomial or piecewise polynomial function. A corrected trapezoidal rule that includes the first derivative of the integrand at the endpoints of the integration interval is also proved in this manner, the coefficient in the error estimate being smaller than for the midpoint and trapezoidal rules. The proofs are suitable for presentation in a calculus or elementary numerical analysis class. Several student projects are suggested.

math.CA↗
arXiv2011

Integrals and Banach spaces for finite order distributions

Let $\Bc$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty$. Let $\Br$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty$. Define $\acn$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\Bc$. Similarly with $\arn$ from $\Br$. A type of integral is defined on distributions in $\acn$ and $\arn$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in\N$, the spaces $\acn$ and $\arn$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\Bc$ and $\Br$, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space $\ac^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space $\ar^1$ contains all finite signed Borel measures. Many of the usual properties of integrals hold: Hölder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.

math.CA↗
arXiv2011

Fourier series with the continuous primitive integral

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 norm. The Riemann-Lebesgue lemma takes the form $\fhat(n)=o(n)$ as $|n|\to\infty$. The convolution is defined for $f\in\alext$ and $g$ a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate $\|f\ast g\|_\infty\leq \|f\|_\T \|g\|_\bv$. For $g\in L^1(\T)$, $\|f\ast g\|_\T\leq \|f\|_\T \|g\|_1$. As well, $\widehat{f\ast g}(n)=\fhatn \hat{g}(n)$. There are versions of the Salem-Zygmund-Rudin-Cohen factorization theorem, Fejér's lemma and the Parseval equality. The trigonometric polynomials are dense in $\alext$. The convolution of $f$ with a sequence of summability kernels converges to $f$ in the Alexiewicz norm. Let $D_n$ be the Dirichlet kernel and let $f\in L^1(\T)$. Then $\|D_n\ast f-f\|_\T\to 0$ as $n\to\infty$. Fourier coefficients of functions of bounded variation are characterized. An appendix contains a type of Fubini theorem.

math.CA↗
arXiv2009

The regulated primitive integral

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 the space of signed Radon measures in the Alexiewicz norm. Functions of bounded variation form the dual space and the space of multipliers. The integrable distributions are a module over the functions of bounded variation. Properties such as integration by parts, change of variables, Hölder inequality, Taylor's theorem and convergence theorems are proved.

math.CA↗
arXiv2009

Convolutions with the continuous primitive integral

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.

math.CA↗
arXiv2007

The distributional Denjoy integral

Let $f$ be a distribution (generalised function) on the real line. If there is a continuous function $F$ with real limits at infinity such that $F'=f$ (distributional derivative) then the distributional integral of $f$ is defined as $\int_{-\infty}^\infty f = F(\infty) - F(-\infty)$. It is shown that this simple definition gives an integral that includes the Lebesgue and Henstock--Kurzweil integrals. The Alexiewicz norm leads to a Banach space of integrable distributions that is isometrically isomorphic to the space of continuous functions on the extended real line with uniform norm. The dual space is identified with the functions of bounded variation. Basic properties of integrals are established using elementary properties of distributions: integration by parts, Hölder inequality, change of variables, convergence theorems, Banach lattice structure, Hake theorem, Taylor theorem, second mean value theorem. Applications are made to the half plane Poisson integral and Laplace transform. The paper includes a short history of Denjoy's descriptive integral definitions. Distributional integrals in Euclidean spaces are discussed and a more general distributional integral that also integrates Radon measures is proposed.

math.CA↗
arXiv2006

Continuity in the Alexiewicz norm

If $f$ is a Henstock--Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $\|f\|=\sup_I|\int_I f|$ where the supremum is taken over all intervals $I\subset\R$. Define the translation $τ_x$ by $τ_xf(y)=f(y-x)$. Then $\|τ_xf-f\|$ tends to 0 as $x$ tends to 0, i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $\|τ_xf-f\|$ can tend to 0 arbitrarily slowly. In general, $\|τ_xf-f\|\geq {\rm osc}f |x|$ as $x\to 0$, where ${\rm osc}f$ is the oscillation of $f$. It is shown that if $F$ is a primitive of $f$ then $\|τ_xF-F\|\leq \|f\||x|$. An example shows that the function $y\mapsto τ_xF(y)-F(y)$ need not be in $L^1$. However, if $f\in L^1$ then $\|τ_xF-F\|_1\leq \|f\|_1|x|$. For a positive weight function $w$ on the real line, necessary and sufficient conditions on $w$ are given so that $\|(τ_xf-f)w\|\to 0$ as $x\to 0$ whenever $fw$ is Henstock--Kurzweil integrable. Applications are made to the Poisson integral on the disc and half-plane. All of the results also hold with the distributional Denjoy integral, which arises from the completion of the space of Henstock--Kurzweil integrable functions as a subspace of Schwartz distributions.

math.CA↗
arXiv2004

Lusin's Theorem and Bochner Integration

It is shown that the approximating functions used to define the Bochner integral can be formed using geometrically nice sets, such as balls, from a differentiation basis. Moreover, every appropriate sum of this form will be within a preassigned $ε$ of the integral, with the sum for the local errors also less than $ε$. All of this follows from the ubiquity of Lebesgue points, which is a consequence of Lusin's theorem, for which a simple proof is included in the discussion.

math.CA↗
arXiv2004

Estimates of Henstock--Kurzweil Poisson integrals

If $f$ is a real-valued function on $[-π,π]$ that is Henstock--Kurzweil integrable, let $u_r(θ)$ be its Poisson integral. It is shown that $\|u_r\|_p=o(1/(1-r))$ as $r\to 1$ and this estimate is sharp for $1\leq p\leq\infty$. If $μ$ is a finite Borel measure and $u_r(θ)$ is its Poisson integral then for each $1\leq p\leq \infty$ the estimate $\|u_r\|_p=O((1-r)^{1/p-1})$ as $r\to 1$ is sharp. The Alexiewicz norm estimates $\|u_r\|\leq\|f\|$ ($0\leq r<1$) and $\|u_r-f\|\to 0$ ($r\to 1$) hold. These estimates lead to two uniqueness theorems for the Dirichlet problem in the unit disc with Henstock--Kurzweil integrable boundary data. There are similar growth estimates when $u$ is in the harmonic Hardy space associated with the Alexiewicz norm and when $f$ is of bounded variation.

math.CA↗
arXiv2004

Estimates of the remainder in Taylor's theorem using the Henstock--Kurzweil integral

When a real-valued function of one variable is approximated by its $n^{th}$ degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue $p$-norms in cases where $f^{(n)}$ or $f^{(n+1)}$ are Henstock--Kurzweil integrable. When the only assumption is that $f^{(n)}$ is Henstock--Kurzweil integrable then a modified form of the $n^{th}$ degree Taylor polynomial is used. When the only assumption is that $f^{(n)}\in C^0$ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.

math.CA↗
arXiv2003

A product convergence theorem for Henstock--Kurzweil integrals

Necessary and sufficient for $\int_a^bfg_n\to \int_a^bfg$ for all Henstock--Kurzweil integrable functions $f$ is that $g$ be of bounded variation, $g_n$ be uniformly bounded and of uniform bounded variation and, on each compact interval in $(a,b)$, $g_n\to g$ in measure or in the $L^1$ norm. The same conditions are necessary and sufficient for $\|f(g_n-g)\|\to 0$ for all Henstock--Kurzweil integrable functions $f$. If $g_n\to g$ a.e. then convergence $\|fg_n\|\to\|fg\|$ for all Henstock--Kurzweil integrable functions $f$ is equivalent to $\|f(g_n-g)\|\to 0$. This extends a theorem due to Lee Peng-Yee.

math.CA↗
arXiv2002

Henstock--Kurzweil Fourier transforms

The Fourier transform is considered as a Henstock--Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The Riemann--Lebesgue lemma fails: Henstock--Kurzweil Fourier transforms can have arbitrarily large point-wise growth. Convolution and inversion theorems are established. An appendix gives sufficient conditions for interchanging repeated Henstock--Kurzweil integrals and gives an estimate on the integral of a product.

math.CA↗
arXiv2001

Some divergent trigonometric integrals

Some divergent trigonometric integrals have appeared in standard tables for many years, listed as converging. We give a simple proof that these integrals diverge and trace their history. The original error was made when a (startlingly) famous mathematician incorrectly differentiated under the integral sign with some convergent integrals that depend on a paramater.

math.CA↗
arXiv2001

Rapidly growing Fourier integrals

The Riemann-Lebesgue Lemma says that the Fourier transform of an absolutely integrable function on the real line tends to zero as the transform parameter tends to infinity. When the integral is allowed to converge conditionally, the transform can have arbitrarily rapid pointwise growth as the transform parameter tends to infinity. Smoothness of the function to be transformed need not decrease growth of the transform.

math.CA↗
arXiv2001

Covering theorems and Lebesgue integration

This paper shows how the Lebesgue integral can be obtained as a Riemann sum and provides an extension of the Morse Covering Theorem to open sets. Let $X$ be a finite dimensional normed space; let $μ$ be a Radon measure on $X$ and let $Ω\subseteq X$ be a $μ$-measurable set. For $λ\geq1$, a $μ$-measurable set $S_λ(a)\subseteq X$ is a $λ$-Morse set with tag $a\in S_λ(a)$ if there is $r>0$ such that $B(a,r)\subseteq S_{λ}(a)\subseteq B(a,λr)$ and $S_λ(a)$ is starlike with respect to all points in the closed ball $B(a,r)$. Given a gauge $δ:Ω\to(0,1]$ we say $S_λ(a)$ is $δ$-fine if $B(a,λr)\subseteq B(a,δ(a))$. If $f\geq0$ is a $μ$-measurable function on $Ω$ then $\int_Ωf dμ=F\in\mathbb{R}$ if and only if for some $λ\geq1$ and all $ε>0$ there is a gauge function $δ$ so that $|\sum_{n}f(x_{n}) μ(S(x_{n}))-F|<ε$ for all sequences of disjoint $λ$-Morse sets that are $δ$-fine and cover all but a $μ$-null subset of $Ω$. This procedure can be applied separately to the positive and negative parts of a real-valued function on $Ω$. The covering condition $μ(Ω\setminus\cup_{n}S(x_{n}))=0$ can be satisfied due to the Morse Covering Theorem. The improved version given here says that for a fixed $λ\geq1$, if $A$ is the set of centers of a family of $λ$-Morse sets then $A$ can be covered with the interiors of sets from at most $κ$ pairwise disjoint subfamilies of the original family; an estimate for $κ$ is given in terms of $λ$, $X$ and its norm.

math.CA↗
arXiv2001

Sharp Growth Estimates for Modified Poisson Integrals in a Half Space

For continuous boundary data, including data of polynomial growth, modified Poisson integrals are used to write solutions to the half space Dirichlet and Neumann problems in $\mathbb{R}^{n}$. Pointwise growth estimates for these integrals are given and the estimates are proved sharp in a strong sense. For decaying data, a new type of modified Poisson integral is introduced and used to develop asymptotic expansions for solutions of these half space problems.

math.CA↗