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Erik Talvila

Publications and source records attributed to Erik Talvila.

At least 19 recordsLinked to original sources
arXiv2026

Fourier transforms of bounded functions

The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a H\"older continuous function. The Fourier transform is written as the difference of $\int_{-1}^1 e^{-ist}f(t)\,dt$ and the second distributional derivative of the integral $\int_{\lvert{t}\rvert>1}e^{-ist}f(t)\,dt/t^2$. The space of such Fourier transforms is isometrically isomorphic to $L^\infty(\mathbb{R})$. There is an exchange theorem, inversion and convolution results. The Fourier transform of the functions $x\mapsto\cos^m(a/x)$ for each natural number $m$ are computed. Also for $x\mapsto x\sin(a/x)$ and $x\mapsto\arctan(x/a)$.

math.CA↗
arXiv2026

Summing series using recurrence relations

Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation yields a closed form for the series. This method is used to sum several series and to solve two {\it American Mathematical Monthly} problems.

math.GM↗
arXiv2025

The Fourier transform with Henstock--Kurzweil and continuous primitive integrals

For each $f\!:\!\mathbb{R}\to\mathbb{C}$ that is Henstock--Kurzweil integrable on the real line, or is a distribution in the completion of the space of Henstock--Kurzweil integrable functions in the Alexiewicz norm, it is shown that the Fourier transform is the second distributional derivative of a H\"older continuous function. The space of such Fourier transforms is isometrically isomorphic to the completion of the Henstock--Kurzweil integrable functions. There is an exchange theorem, inversion in norm and convolution results. Sufficient conditions are given for an $L^1$ function to have a Fourier transform that is of bounded variation. Pointwise inversion of the Fourier transform is proved for functions in $L^p$ spaces for $1<p<\infty$. The exchange theorem is used to evaluate an integral that does not appear in published tables.

math.CA↗
arXiv2023

The heat equation with the $L^p$ primitive integral

For each $1\leq p<\infty$ a Banach space of integrable Schwartz distributions is defined by taking the distributional derivative of all functions in $L^p({\mathbb R})$. Such distributions can be integrated when multiplied by a function that is the integral of a function in $L^q({\mathbb R})$, where $q$ is the conjugate exponent of $p$. The heat equation on the real line is solved in this space of distributions. The initial data is taken to be the distributional derivative of an $L^p({\mathbb R})$ function. The solutions are shown to be smooth functions. Initial conditions are taken on in norm. Sharp estimates of solutions are obtained and a uniqueness theorem is proved.

math.AP↗
arXiv2023

The Fourier transform in Lebesgue spaces

For each $f\in L^p({\mathbb R)}$ ($1\leq p<\infty$) it is shown that the Fourier transform is the distributional derivative of a H\"older continuous function. For each $p$ a norm is defined so that the space Fourier transforms is isometrically isomorphic to $L^p({\mathbb R)}$. There is an exchange theorem and inversion in norm.

math.CA↗
arXiv2022

Fourier transform inversion: Bounded variation, polynomial growth, Henstock--Stieltjes integration

In this paper we prove pointwise and distributional Fourier transform inversion theorems for functions on the real line that are locally of bounded variation, while in a neighbourhood of infinity are Lebesgue integrable or have polynomial growth. We also allow the Fourier transform to exist in the principal value sense. A function is called regulated if it has a left limit and a right limit at each point. The main inversion theorem is obtained by solving the differential equation $df(t)-iωf(t)=g(t)$ for a regulated function $f$, where $ω$ is a complex number with positive imaginary part. This is done using the Henstock--Stieltjes integral. This is an integral defined with Riemann sums and a gauge. Some variants of the integration by parts formula are also proved for this integral. When the function is of polynomial growth its Fourier transform exists in a distributional sense, although the inversion formula only involves integration of functions and returns pointwise values.

math.CA↗
arXiv2022

Fourier transform inversion in the Alexiewicz norm

If $f\in L^1({\mathbb R})$ it is proved that $\lim_{S\to\infty}\lVert f-f\ast D_S\rVert=0$, where $D_S(x)=\sin(Sx)/(πx)$ is the Dirichlet kernel and $\lVert f\rVert = \sup_{α<β}|\int_α^βf(x)\,dx|$ is the Alexiewicz norm. This gives a symmetric inversion of the Fourier transform on the real line. An asymmetric inversion is also proved. The results also hold for a measure given by $dF$ where $F$ is a continuous function of bounded variation. Such measures need not be absolutely continuous with respect to Lebesgue measure. An example shows there is $f\in L^1({\mathbb R})$ such that $\lim_{S\to\infty} \rVert f-f\ast D_S\lVert_1\neq 0$.

math.CA↗
arXiv2020

Variations on least squares

Three methods of least squares are examined for fitting a line to points in the plane. Two well known methods are to minimize sums of squares of vertical or horizontal distances to the line. Less known is to minimize sums of squares of distances to the line. Concise proofs are given for each method using a combination of the first derivative test for functions of two variables and completing the square. The three methods are compared and the distances to the line method appears to be favourable in most circumstances. They generally draw different regression lines. The method of vertical displacements typically gives a slope of too small magnitude while the method of horizontal displacements typically gives a slope of too large magnitude. An inequality involving the three slopes is proved. Rotating all the data points in the same way with these two methods does not result in the regression line being rotated the same way. However, the distance to the line method is invariant under rotations.

math.CA↗
arXiv2020

The continuous primitive integral in the plane

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 $f=\partial^2/(\partial x\partial y) F$ then the integral over interval $[a,b]\times [c,d] \subseteq[-\infty,\infty]^2$ is $\int_a^b\int_c^d f=F(a,c)+F(b,d)-F(a,d)-F(b,c)$ and $\int_{-\infty}^\infty \int_{-\infty}^\infty f=F(\infty,\infty)$. The definition then builds in the fundamental theorem of calculus. The Alexiewicz norm is ${\lVert f\rVert}={\lVert F\rVert}_\infty$ where $F$ is the unique primitive of $f$. The space of integrable distributions is then a separable Banach space isometrically isomorphic to the space of primitives. The space of integrable distributions is the completion of both $L^1$ and the space of Henstock--Kurzweil integrable functions. The Banach lattice and Banach algebra structures of the continuous functions in ${\lVert \cdot\rVert}_\infty$ are also inherited by the integrable distributions. It is shown that the dual space are the functions of bounded Hardy--Krause variation. Various tools that make these integrals useful in applications are proved: integration by parts, Hölder inequality, second mean value theorem, Fubini theorem, a convergence theorem, change of variables, convolution. The changes necessary to define the integral in ${\mathbb R}^n$ are sketched out.

math.CA↗
arXiv2019

Elementary numerical methods for double integrals

Approximations to the integral $\int_a^b\int_c^d f(x,y)\,dy\,dx$ are obtained under the assumption that the partial derivatives of the integrand are in an $L^p$ space, for some $1\leq p\leq\infty$. We assume ${\lVert f_{xy}\rVert}_p$ is bounded (integration over $[a,b]\times[c,d]$), assume ${\lVert f_x(\cdot,c)\rVert}_p$ and ${\lVert f_x(\cdot,d)\rVert}_p$ are bounded (integration over $[a,b]$), and assume ${\lVert f_y(a,\cdot)\rVert}_p$ and ${\lVert f_y(b,\cdot)\rVert}_p$ are bounded (integration over $[c,d]$). The methods are elementary, using only integration by parts and H\"older's inequality. Versions of the trapezoidal rule, composite trapezoidal rule, midpoint rule and composite midpoint rule are given, with error estimates in terms of the above norms.

math.NA↗
arXiv2018

Fourier transform inversion using an elementary differential equation and a contour integral

Let $f$ be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that $f$ is absolutely continuous such that $f$ and $f'$ are Lebesgue integrable. A function $g$ is defined by $f'(t)-iwf(t)=g(t)$. This differential equation has a well known integral solution using the Heaviside step function. An elementary calculation with residues is used to write the Heaviside step function as a simple contour integral. The rest of the proof requires elementary manipulation of integrals. Hence, the Fourier transform inversion theorem is proved with very little machinery. With only minor changes the method is also used to prove the inversion theorem for functions of several variables and to prove Riemann's localization theorem.

math.CA↗
arXiv2016

Higher order corrected trapezoidal rules in Lebesgue and Alexiewicz spaces

If $f\!:\![a,b]\to\R$ such that $f^{(n)}$ is integrable then integration by parts gives the formula \begin{align*} &\intab f(x)\,dx = &\frac{(-1)^n}{n!}\sum_{k=0}^{n-1}(-1)^{n-k-1}\left[ \phi_n^{(n-k-1)}(a)f^{(k)}(a)- \phi_n^{(n-k-1)}(b)f^{(k)}(b)\right] +E_n(f), \end{align*} where $\phi_n$ is a monic polynomial of degree $n$ and the error is given by $E_n(f)=\frac{(-1)^n}{n!}\int_a^b f^{(n)}(x)\phi_n(x)\,dx$. This then gives a quadrature formula for $\int_a^bf(x)\,dx$. The polynomial $\phi_n$ is chosen to optimize the error estimate under the assumption that $f^{(n)}\in L^p([a,b])$ for some $1\leq p\leq\infty$ or if $f^{(n)}$ is integrable in the distributional or Henstock--Kurzweil sense. Sharp error estimates are obtained. It is shown that this formula is exact for all such $\phi_n$ if $f$ is a polynomial of degree at most $n-1$. If $\phi_n$ is a Legendre polynomial then the formula is exact for $f$ a polynomial of degree at most $2n-1$.

math.CA↗
arXiv2015

The one-dimensional heat equation in the Alexiewicz norm

A distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock--Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let $Θ_t(x)=\exp(-x^2/(4t))/\sqrt{4πt}$ be the heat kernel. With initial data $f$ that is the distributional derivative of a continuous function, it is shown that $u_t(x):=u(x,t):=f\astΘ_t(x)$ is a classical solution of the heat equation $u_{11}=u_2$. The estimate $\|f\astΘ_t\|_\infty\leq\|f\|/\sqrt{πt}$ holds. The Alexiewicz norm is $\|f\|=\sup_I|\int_If|$, the supremum taken over all intervals. The initial data is taken on in the Alexiewicz norm, $\|u_t-f\|\to 0$ as $t\to 0^+$. The solution of the heat equation is unique under the assumptions that $\|u_t\|$ is bounded and $u_t\to f$ in the Alexiewicz norm for some integrable $f$. The heat equation is also considered with initial data that is the $n$th derivative of a continuous function and in weighted spaces such that $\int_{-\infty}^\infty f(x)\exp(-ax^2)\,dx$ exists for some $a>0$. Similar results are obtained.

math.AP↗
arXiv2013

Distributions, their primitives and integrals with applications to differential equations

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.

math.CA↗
arXiv2012

The $L^p$ primitive integral

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 $L^'^{\,p}$ share many properties with functions in $L^p$. Hence, $L^'^{\,p}$ is reflexive, its dual space is identified with $L^q$, there is a type of Hölder inequality, continuity in norm, convergence theorems, Gateaux derivative. It is a Banach lattice and abstract $L$-space. Convolutions and Fourier transforms are defined. Convolution with the Poisson kernel is well-defined and provides a solution to the half plane Dirichlet problem, boundary values being taken on in the new norm. A product is defined that makes $L^'^{\,1}$ into a Banach algebra isometrically isomorphic to the convolution algebra on $L^1$. Spaces of higher order derivatives of $L^p$ functions are defined. These are also Banach spaces isometrically isomorphic to $L^p$.

math.CA↗
arXiv2012

Trigonometry of The Gold-Bug

The classic Edgar Allan Poe story The Gold-Bug involves digging for pirate treasure. Locating the digging sites requires some simple trigonometry.

math.HO↗
arXiv2012

Optimal error estimates for corrected trapezoidal rules

Corrected trapezoidal rules are proved for $\int_a^b f(x)\,dx$ under the assumption that $f"\in L^p([a,b])$ for some $1\leq p\leq\infty$. Such quadrature rules involve the trapezoidal rule modified by the addition of a term $k[f'(a)-f'(b)]$. The coefficient $k$ in the quadrature formula is found that minimizes the error estimates. It is shown that when $f'$ is merely assumed to be continuous then the optimal rule is the trapezoidal rule itself. In this case error estimates are in terms of the Alexiewicz norm. This includes the case when $f"$ is integrable in the Henstock--Kurzweil sense or as a distribution. All error estimates are shown to be sharp for the given assumptions on $f"$. It is shown how to make these formulas exact for all cubic polynomials $f$. Composite formulas are computed for uniform partitions.

math.CA↗