Abstract
The present paper deals mainly with seven fundamental theorems of mathematical analysis, numerical analysis, and number theory, namely the generalized Parseval decomposition formula (GPDF), introduced 15 years ago, the well-known approximate sampling theorem (ASF), the new approximate reproducing kernel theorem, the basic Poisson summation formula, already known to Gauß, a newer version of the GPDF having a structure similar to that of the Poisson summation formula, namely, the Parseval decomposition–Poisson summation formula, the functional equation of Riemann’s zeta function, as well as the Euler–Maclaurin summation formula. It will in fact be shown that these seven theorems are all equivalent to one another, in the sense that each is a corollary of the others. Since these theorems can all be deduced from each other, one of them has to be proven independently in order to verify all. It is convenient to choose the ASF, introduced in 1963. The epilogue treats possible extensions to the more general contexts of reproducing kernel theory and of abstract harmonic analysis, using locally compact abelian groups. This paper is expository in the sense that it treats a number of mathematical theorems, their interconnections, their equivalence to one another. On the other hand, the proofs of the many intricate interconnections among these theorems are new in their essential steps and conclusions.
Highlights
A provoking result of Fourier analysis has turned up in the past 15 years, namely1.1 Generalized Parseval decomposition formula (GPDF) For f ∈ F2 ∩ Sw1, w > 0, and g ∈ F2, there holds Rw f ∈ L2(R) and R f (u)g(u) du =1 w k∈Z f k w g k w − 1 w f k∈Z√1 2π g(v)eikv/w dv +|v|≥π w (Rw f )(u)g(u) du, (1.1)where ( Rw f )(t)
In the present paper, dealing with the grouping (A): GPDF, approximate sampling formula (ASF), Approximate reproducing kernel formula (ARKF), Poisson summation formula (PSF), PDPS, FERZ, as well as the Euler–Maclaurin summation formula EMSF—all for non-bandlimited functions—the basic result is that all seven theorems are equivalent amongst themselves
We shall deduce the basic new ARKF theorem for L2(R)-function by means of the ASF
Summary
A provoking result of Fourier analysis has turned up in the past 15 years, namely. 1.1 Generalized Parseval decomposition formula (GPDF) For f ∈ F2 ∩ Sw1 , w > 0, and g ∈ F2, there holds Rw f ∈ L2(R) and. A formula of number theory, (externally) unrelated to periodicity and Fourier series, can yield a basic, general formula of analysis, one connecting Fourier series and integrals It is well-known that the FERZ is equivalent to the second famous functional equation, namely FEJT θ −1. Each of the seven formulae GPDF, ASF, ARKF, PSF, PDPS, FERZ and EMSF is equivalent to each other, in the sense that each is a corollary of each of the others. In the present paper, dealing with the grouping (A): GPDF, ASF, ARKF, PSF, PDPS, FERZ, as well as the Euler–Maclaurin summation formula EMSF (handled in [16])—all for non-bandlimited functions—the basic result is that all seven theorems are equivalent amongst themselves. Our paper can be regarded as a further justification of what electrical engineers do in their real life work
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
