Abstract
We present some new, best possible, integral estimates involving the \(L^2\)-norm of the second derivative of smooth (periodic and non-periodic) functions.
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Notes
Let us point out that b is not (the trace) of a polynomial since for \(0\le x\le 1/2\), we have \(B_4(x+1/2)=x^4-(1/2)x^2+7/240\) and for \(1/2\le x\le 1\), we have \(B_4(x-1/2)=x^4-4x^3+(11/2)x^2-3x+127/240\). Posted also on [4]. We don’t know whether a polynomial can ever be extremal for (2).
References
Bary, N.K.: A Treatise on Trigonometric Series, vol. 1. Pergamon Press Book, MacMillan Company, New York (1964)
Garcia, A.: Problem 4615. Crux 47, 97 (2021)
Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications (East European Series), vol. 53. Kluwer Academic Publishers, Dordrecht (1991)
Mortini, R.: https://math.stackexchange.com/questions/4105664/the-alternating-fourier-series-associated-with-the-fourth-bernoulli-polynomial
Mortini, R., Rupp, R.: Extension Problems and Stable Ranks: A Space Odyssey, p. 2150. Birkhäuser, Cham (2021)
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Mortini , R., Rupp, R. Some best possible integral estimates involving Bernoulli polynomials. Arch. Math. 117, 411–422 (2021). https://doi.org/10.1007/s00013-021-01640-x
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DOI: https://doi.org/10.1007/s00013-021-01640-x