Fredholmness of Convolution Type Operators

AbstractWe present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

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The Calderón–Zygmund Theorem with an $$L^1$$ Mean Hörmander Condition

Journal of Fourier Analysis and Applications ◽

10.1007/s00041-021-09810-9 ◽

2021 ◽

Vol 27 (2) ◽

Author(s):

Soichiro Suzuki

Keyword(s):

Singular Integral ◽

Integral Operators ◽

Acta Math ◽

Limited Range ◽

Singular Integral Operators ◽

Convolution Type ◽

Worst Case ◽

Hörmander Condition

AbstractIn 2019, Grafakos and Stockdale introduced an $$L^q$$ L q mean Hörmander condition and proved a “limited-range” Calderón–Zygmund theorem. Comparing their theorem with the classical one, it requires weaker assumptions and implies the $$L^p$$ L p boundedness for the “limited-range” instead of $$1< p < \infty $$ 1 < p < ∞ . However, in this paper, we show that the $$L^q$$ L q mean Hörmander condition is actually enough to obtain the $$L^p$$ L p boundedness for all $$1< p < \infty $$ 1 < p < ∞ even in the worst case $$q=1$$ q = 1 . We use a similar method to that used by Fefferman (Acta Math 124:9–36, 1970): form the Calderón–Zygmund decomposition with the bounded overlap property and approximate the bad part. Also we give a criterion of the $$L^2$$ L 2 boundedness for convolution type singular integral operators under the $$L^1$$ L 1 mean Hörmander condition.

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Classes of uniqueness for integrodifferential equations with volterra operators of convolution type

Functional Analysis and Its Applications ◽

10.1007/bf01077250 ◽

1979 ◽

Vol 13 (2) ◽

pp. 143-144 ◽

Cited By ~ 1

Author(s):

A. S. Kalashnikov

Keyword(s):

Integrodifferential Equations ◽

Convolution Type ◽

Volterra Operators

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Factorization of convolution type vector integro-differential equations

Journal of Contemporary Mathematical Analysis ◽

10.3103/s1068362308030035 ◽

2008 ◽

Vol 43 (3) ◽

pp. 145-156

Author(s):

Kh. A. Khachatryan ◽

M. G. Kostanyan

Keyword(s):

Differential Equations ◽

Convolution Type

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Convolution-type derivatives and transforms of Colombeau generalized stochastic processes

Integral Transforms and Special Functions ◽

10.1080/10652469.2010.541047 ◽

2011 ◽

Vol 22 (4-5) ◽

pp. 319-326 ◽

Cited By ~ 5

Author(s):

Danijela Rajter-Ćirić ◽

Mirjana Stojanović

Keyword(s):

Stochastic Processes ◽

Convolution Type

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Pointwise estimates for higher order convexity preserving polynomial approximation

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000010365 ◽

1994 ◽

Vol 36 (2) ◽

pp. 213-233 ◽

Cited By ~ 6

Author(s):

Jia-Ding Cao ◽

Heinz H. Gonska

Keyword(s):

Polynomial Approximation ◽

Convex Functions ◽

Higher Order ◽

Second Order ◽

Convolution Type ◽

Shape Preservation ◽

Moduli Of Continuity ◽

Pointwise Estimates ◽

Higher Order Convexity ◽

Boolean Sum

AbstractDeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

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