Coconvex Approximation of Periodic Functions

Leviatan, D.; Shevchuk, I. A.

doi:10.1007/s00365-022-09597-y

Cited by 4 publications

(2 citation statements)

References 15 publications

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“…The final approximation method we implement is Lagrange polynomial approximation. Following mathematical theory [33], we apply the lagrange function from the SciPy package to obtain the Lagrange interpolating polynomial [31].…”

Section: Polynomial Approximationmentioning

confidence: 99%

“…As an alternative to interpolation, we may also apply the least squares method to obtain a Lagrange approximating polynomial, which leads to an exact approximation if the function to be approximated lies in the space spanned by the basis functions [33]. We implement a generic least squares protocol to perform a least squares fit to a given function over a given interval.…”

Section: Polynomial Approximationmentioning

confidence: 99%

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Efficient optimisation framework for convolutional neural networks with secure multiparty computation

Berry

Komninos

2022

Computers & Security

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Section: Polynomial Approximationmentioning

confidence: 99%

Section: Polynomial Approximationmentioning

confidence: 99%

Efficient optimisation framework for convolutional neural networks with secure multiparty computation

Berry

Komninos

2022

Computers & Security

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Fast Decreasing Trigonometric Polynomials and Applications

Leviatan,

Motorna,

Shevchuk

2024

J Fourier Anal Appl

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We construct trigonometric polynomials that fast decrease towards $$\pm \pi $$ ± π . We apply them to construct a trigonometric polynomial the derivative of which interpolates the derivative of a given $$2\pi $$ 2 π -periodic function, at some prescribed distinct points in $$[-\pi ,\pi )$$ [ - π , π ) , and vanishes at some other prescribed points in that interval. The construction requires that the function possesses derivatives where the interpolation is supposed to take place. Still, we are able to apply the result to trigonometric approximation of a $$2\pi $$ 2 π -periodic piecewise algebraic polynomial which is merely continuous, while interpolating its derivative at some points (that, obviously, are not knots).

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Shape Preserving Approximation of Periodic Functions: Conclusion

Leviatan,

Shevchuk

2024

Constr Approx

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We give here the final results about the validity of Jackson-type estimates in comonotone approximation of $$2\pi $$ 2 π -periodic functions by trigonometric polynomials ($$q=1$$ q = 1 ). For coconvex ($$q=2$$ q = 2 ) and the so called co-q-monotone, $$q>2$$ q > 2 , approximations, everything is known by now. Thus, this paper concludes the research on Jackson type estimates of Shape Preserving Approximation of periodic functions by trigonometric polynomials. It is interesting to point out that the results for comonotone approximation of a periodic function are substantially different than the analogous results for comonotone approximation of a continuous function on a finite interval, by algebraic polynomials.

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Coconvex Approximation of Periodic Functions

Cited by 4 publications

References 15 publications

Efficient optimisation framework for convolutional neural networks with secure multiparty computation

Efficient optimisation framework for convolutional neural networks with secure multiparty computation

Fast Decreasing Trigonometric Polynomials and Applications

Shape Preserving Approximation of Periodic Functions: Conclusion

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